{"upwards":["Set (mathematics)"],"downwards":["History","How sets are defined and set notation","Membership","The empty set","Singleton sets","Subsets","Euler and Venn diagrams","Special sets of numbers in mathematics","Functions","Cardinality","Power sets","Partitions","Basic operations","Applications","Principle of inclusion and exclusion"]}
{"upwards":["Set (mathematics)","History"],"downwards":["Naive set theory","Axiomatic set theory"]}
{"upwards":["Set (mathematics)","History","Naive set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","History","Axiomatic set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation"],"downwards":["Roster notation","Semantic definition","Set-builder notation","Classifying methods of definition"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation"],"downwards":["Infinite sets in roster notation"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation","Infinite sets in roster notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Semantic definition"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Set-builder notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Classifying methods of definition"],"downwards":[]}
{"upwards":["Set (mathematics)","Membership"],"downwards":[]}
{"upwards":["Set (mathematics)","The empty set"],"downwards":[]}
{"upwards":["Set (mathematics)","Singleton sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Subsets"],"downwards":[]}
{"upwards":["Set (mathematics)","Euler and Venn diagrams"],"downwards":[]}
{"upwards":["Set (mathematics)","Special sets of numbers in mathematics"],"downwards":[]}
{"upwards":["Set (mathematics)","Functions"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality"],"downwards":["Infinite sets and infinite cardinality","The continuum hypothesis"]}
{"upwards":["Set (mathematics)","Cardinality","Infinite sets and infinite cardinality"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality","The continuum hypothesis"],"downwards":[]}
{"upwards":["Set (mathematics)","Power sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Partitions"],"downwards":[]}
{"upwards":["Set (mathematics)","Basic operations"],"downwards":[]}
{"upwards":["Set (mathematics)","Applications"],"downwards":[]}
{"upwards":["Set (mathematics)","Principle of inclusion and exclusion"],"downwards":[]}
{"upwards":["Paradoxes of the Infinite"],"downwards":["References"]}
{"upwards":["Paradoxes of the Infinite","References"],"downwards":[]}
{"upwards":["Wikipedia, the free encyclopedia"],"downwards":[]}
{"upwards":["Mathematical object"],"downwards":["List of mathematical objects by branch"]}
{"upwards":["Mathematical object","List of mathematical objects by branch"],"downwards":[]}
{"upwards":["Ordered pair"],"downwards":["Generalities","Informal and formal definitions","Defining the ordered pair using set theory","Category theory"]}
{"upwards":["Ordered pair","Generalities"],"downwards":[]}
{"upwards":["Ordered pair","Informal and formal definitions"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory"],"downwards":["Wiener's definition","Hausdorff's definition","Kuratowski's definition","Quine\u2013Rosser definition","Cantor\u2013Frege definition","Morse definition","Axiomatic definition"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Wiener's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Hausdorff's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition"],"downwards":["Variants","Proving that definitions satisfy the characteristic property"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Variants"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Proving that definitions satisfy the characteristic property"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Quine\u2013Rosser definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Cantor\u2013Frege definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Morse definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Axiomatic definition"],"downwards":[]}
{"upwards":["Ordered pair","Category theory"],"downwards":[]}
{"upwards":["Naive set theory"],"downwards":["Method","Sets, membership and equality","Specifying sets","Subsets","Universal sets and absolute complements","Unions, intersections, and relative complements","Ordered pairs and Cartesian products","Some important sets","Paradoxes in early set theory"]}
{"upwards":["Naive set theory","Method"],"downwards":["Paradoxes","Cantor's theory","Axiomatic theories","Consistency","Utility"]}
{"upwards":["Naive set theory","Method","Paradoxes"],"downwards":[]}
{"upwards":["Naive set theory","Method","Cantor's theory"],"downwards":[]}
{"upwards":["Naive set theory","Method","Axiomatic theories"],"downwards":[]}
{"upwards":["Naive set theory","Method","Consistency"],"downwards":[]}
{"upwards":["Naive set theory","Method","Utility"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality"],"downwards":["Note on consistency","Membership","Equality","Empty set"]}
{"upwards":["Naive set theory","Sets, membership and equality","Note on consistency"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Membership"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Equality"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Empty set"],"downwards":[]}
{"upwards":["Naive set theory","Specifying sets"],"downwards":[]}
{"upwards":["Naive set theory","Subsets"],"downwards":[]}
{"upwards":["Naive set theory","Universal sets and absolute complements"],"downwards":[]}
{"upwards":["Naive set theory","Unions, intersections, and relative complements"],"downwards":[]}
{"upwards":["Naive set theory","Ordered pairs and Cartesian products"],"downwards":[]}
{"upwards":["Naive set theory","Some important sets"],"downwards":[]}
{"upwards":["Naive set theory","Paradoxes in early set theory"],"downwards":[]}
{"upwards":["Rational number"],"downwards":["Terminology","Arithmetic","Continued fraction representation","Other representations","Formal construction","Properties","Real numbers and topological properties","p-adic numbers"]}
{"upwards":["Rational number","Terminology"],"downwards":["Etymology"]}
{"upwards":["Rational number","Terminology","Etymology"],"downwards":[]}
{"upwards":["Rational number","Arithmetic"],"downwards":["Irreducible fraction","Embedding of integers","Equality","Ordering","Addition","Subtraction","Multiplication","Inverse","Division","Exponentiation to integer power"]}
{"upwards":["Rational number","Arithmetic","Irreducible fraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Embedding of integers"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Equality"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Ordering"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Addition"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Multiplication"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Inverse"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Division"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Exponentiation to integer power"],"downwards":[]}
{"upwards":["Rational number","Continued fraction representation"],"downwards":[]}
{"upwards":["Rational number","Other representations"],"downwards":[]}
{"upwards":["Rational number","Formal construction"],"downwards":[]}
{"upwards":["Rational number","Properties"],"downwards":["Countability"]}
{"upwards":["Rational number","Properties","Countability"],"downwards":[]}
{"upwards":["Rational number","Real numbers and topological properties"],"downwards":[]}
{"upwards":["Rational number","p-adic numbers"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Singleton (mathematics)"],"downwards":["Properties","In category theory","Definition by indicator functions","Definition in Principia Mathematica"]}
{"upwards":["Singleton (mathematics)","Properties"],"downwards":[]}
{"upwards":["Singleton (mathematics)","In category theory"],"downwards":[]}
{"upwards":["Singleton (mathematics)","Definition by indicator functions"],"downwards":[]}
{"upwards":["Singleton (mathematics)","Definition in Principia Mathematica"],"downwards":[]}
{"upwards":["Primitive notion"],"downwards":["Details","Examples","Russell's primitives"]}
{"upwards":["Primitive notion","Details"],"downwards":[]}
{"upwards":["Primitive notion","Examples"],"downwards":[]}
{"upwards":["Primitive notion","Russell's primitives"],"downwards":[]}
{"upwards":["Cantor's paradox"],"downwards":["Statements and proofs","Discussion and consequences","Historical notes","References","External links"]}
{"upwards":["Cantor's paradox","Statements and proofs"],"downwards":[]}
{"upwards":["Cantor's paradox","Discussion and consequences"],"downwards":[]}
{"upwards":["Cantor's paradox","Historical notes"],"downwards":[]}
{"upwards":["Cantor's paradox","References"],"downwards":[]}
{"upwards":["Cantor's paradox","External links"],"downwards":[]}
{"upwards":["Extensionality"],"downwards":["Example","In mathematics"]}
{"upwards":["Extensionality","Example"],"downwards":[]}
{"upwards":["Extensionality","In mathematics"],"downwards":[]}
{"upwards":["Euler diagram"],"downwards":["History","Relation between Euler and Venn diagrams","Gallery"]}
{"upwards":["Euler diagram","History"],"downwards":[]}
{"upwards":["Euler diagram","Relation between Euler and Venn diagrams"],"downwards":["Example: Euler- to Venn-diagram and Karnaugh map"]}
{"upwards":["Euler diagram","Relation between Euler and Venn diagrams","Example: Euler- to Venn-diagram and Karnaugh map"],"downwards":[]}
{"upwards":["Euler diagram","Gallery"],"downwards":[]}
{"upwards":["Blackboard bold"],"downwards":["History","Encoding","Usage"]}
{"upwards":["Blackboard bold","History"],"downwards":["Origin","Use in textbooks"]}
{"upwards":["Blackboard bold","History","Origin"],"downwards":[]}
{"upwards":["Blackboard bold","History","Use in textbooks"],"downwards":[]}
{"upwards":["Blackboard bold","Encoding"],"downwards":[]}
{"upwards":["Blackboard bold","Usage"],"downwards":[]}
{"upwards":["Well-defined expression"],"downwards":["Example","\"Definition\" as anticipation of definition","Independence of representative","Well-defined notation","Other uses of the term"]}
{"upwards":["Well-defined expression","Example"],"downwards":[]}
{"upwards":["Well-defined expression","\"Definition\" as anticipation of definition"],"downwards":[]}
{"upwards":["Well-defined expression","Independence of representative"],"downwards":["Functions with one argument","Operations"]}
{"upwards":["Well-defined expression","Independence of representative","Functions with one argument"],"downwards":[]}
{"upwards":["Well-defined expression","Independence of representative","Operations"],"downwards":[]}
{"upwards":["Well-defined expression","Well-defined notation"],"downwards":[]}
{"upwards":["Well-defined expression","Other uses of the term"],"downwards":[]}
{"upwards":["De Morgan's laws"],"downwards":["Formal notation","History","Informal proof","Formal proof","Generalising De Morgan duality","Extension to predicate and modal logic","In intuitionistic logic","In computer engineering"]}
{"upwards":["De Morgan's laws","Formal notation"],"downwards":["Substitution form","Set theory and Boolean algebra","Engineering","Text searching"]}
{"upwards":["De Morgan's laws","Formal notation","Substitution form"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Set theory and Boolean algebra"],"downwards":["Unions and intersections of any number of sets"]}
{"upwards":["De Morgan's laws","Formal notation","Set theory and Boolean algebra","Unions and intersections of any number of sets"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Engineering"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Text searching"],"downwards":[]}
{"upwards":["De Morgan's laws","History"],"downwards":[]}
{"upwards":["De Morgan's laws","Informal proof"],"downwards":["Negation of a disjunction","Negation of a conjunction"]}
{"upwards":["De Morgan's laws","Informal proof","Negation of a disjunction"],"downwards":[]}
{"upwards":["De Morgan's laws","Informal proof","Negation of a conjunction"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof"],"downwards":["Part 1","Part 2","Conclusion"]}
{"upwards":["De Morgan's laws","Formal proof","Part 1"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof","Part 2"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof","Conclusion"],"downwards":[]}
{"upwards":["De Morgan's laws","Generalising De Morgan duality"],"downwards":[]}
{"upwards":["De Morgan's laws","Extension to predicate and modal logic"],"downwards":[]}
{"upwards":["De Morgan's laws","In intuitionistic logic"],"downwards":[]}
{"upwards":["De Morgan's laws","In computer engineering"],"downwards":[]}
{"upwards":["Algebraic number"],"downwards":["Examples","Properties","Field","Related fields","Algebraic integers","Special classes","Notes","References"]}
{"upwards":["Algebraic number","Examples"],"downwards":[]}
{"upwards":["Algebraic number","Properties"],"downwards":[]}
{"upwards":["Algebraic number","Field"],"downwards":[]}
{"upwards":["Algebraic number","Related fields"],"downwards":["Numbers defined by radicals","Closed-form number"]}
{"upwards":["Algebraic number","Related fields","Numbers defined by radicals"],"downwards":[]}
{"upwards":["Algebraic number","Related fields","Closed-form number"],"downwards":[]}
{"upwards":["Algebraic number","Algebraic integers"],"downwards":[]}
{"upwards":["Algebraic number","Special classes"],"downwards":[]}
{"upwards":["Algebraic number","Notes"],"downwards":[]}
{"upwards":["Algebraic number","References"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Mathematical model"],"downwards":["Elements of a mathematical model","Classifications","Construction","Significance in the natural sciences","Some applications","Examples"]}
{"upwards":["Mathematical model","Elements of a mathematical model"],"downwards":[]}
{"upwards":["Mathematical model","Classifications"],"downwards":[]}
{"upwards":["Mathematical model","Construction"],"downwards":["A priori information","Complexity","Training, tuning, and fitting","Evaluation and assessment"]}
{"upwards":["Mathematical model","Construction","A priori information"],"downwards":["Subjective information"]}
{"upwards":["Mathematical model","Construction","A priori information","Subjective information"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Complexity"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Training, tuning, and fitting"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment"],"downwards":["Prediction of empirical data","Scope of the model","Philosophical considerations"]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Prediction of empirical data"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Scope of the model"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Philosophical considerations"],"downwards":[]}
{"upwards":["Mathematical model","Significance in the natural sciences"],"downwards":[]}
{"upwards":["Mathematical model","Some applications"],"downwards":[]}
{"upwards":["Mathematical model","Examples"],"downwards":[]}
{"upwards":["Equality (mathematics)"],"downwards":["Etymology","Basic properties","Equality as predicate","Identities","Equations","Approximate equality","Relation with equivalence, congruence, and isomorphism","Logical definitions","Equality in set theory"]}
{"upwards":["Equality (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Equality (mathematics)","Basic properties"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality as predicate"],"downwards":[]}
{"upwards":["Equality (mathematics)","Identities"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equations"],"downwards":[]}
{"upwards":["Equality (mathematics)","Approximate equality"],"downwards":[]}
{"upwards":["Equality (mathematics)","Relation with equivalence, congruence, and isomorphism"],"downwards":[]}
{"upwards":["Equality (mathematics)","Logical definitions"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality in set theory"],"downwards":["Set equality based on first-order logic with equality","Set equality based on first-order logic without equality"]}
{"upwards":["Equality (mathematics)","Equality in set theory","Set equality based on first-order logic with equality"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality in set theory","Set equality based on first-order logic without equality"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Theorem"],"downwards":["Theoremhood and truth","Epistemological considerations","Informal account of theorems","Relation with scientific theories","Terminology","Layout","Lore","Theorems in logic"]}
{"upwards":["Theorem","Theoremhood and truth"],"downwards":[]}
{"upwards":["Theorem","Epistemological considerations"],"downwards":[]}
{"upwards":["Theorem","Informal account of theorems"],"downwards":[]}
{"upwards":["Theorem","Relation with scientific theories"],"downwards":[]}
{"upwards":["Theorem","Terminology"],"downwards":[]}
{"upwards":["Theorem","Layout"],"downwards":[]}
{"upwards":["Theorem","Lore"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic"],"downwards":["Syntax and semantics","Interpretation of a formal theorem","Theorems and theories"]}
{"upwards":["Theorem","Theorems in logic","Syntax and semantics"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Interpretation of a formal theorem"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Theorems and theories"],"downwards":[]}
{"upwards":["Bernard Bolzano"],"downwards":["Family","Career","Mathematical work","Philosophical work","Philosophical legacy","Writings"]}
{"upwards":["Bernard Bolzano","Family"],"downwards":[]}
{"upwards":["Bernard Bolzano","Career"],"downwards":[]}
{"upwards":["Bernard Bolzano","Mathematical work"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work"],"downwards":["Wissenschaftslehre (Theory of Science)"]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)"],"downwards":["Introduction to Wissenschaftslehre","Metaphysics","Satz an Sich (proposition in itself)","Ideas and objects","Sensation and simple ideas","Logic","Truth","Judgments and cognitions"]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Introduction to Wissenschaftslehre"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Metaphysics"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Satz an Sich (proposition in itself)"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Ideas and objects"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Sensation and simple ideas"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Logic"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Truth"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Judgments and cognitions"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical legacy"],"downwards":[]}
{"upwards":["Bernard Bolzano","Writings"],"downwards":["Translations and compilations"]}
{"upwards":["Bernard Bolzano","Writings","Translations and compilations"],"downwards":[]}
{"upwards":["Foundations of mathematics"],"downwards":["Historical context","Foundational crisis","Toward resolution of the crisis"]}
{"upwards":["Foundations of mathematics","Historical context"],"downwards":["Ancient Greek mathematics","Platonism as a philosophy of mathematics","Aristotelian realism","Middle Ages and Renaissance","19th century"]}
{"upwards":["Foundations of mathematics","Historical context","Ancient Greek mathematics"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Platonism as a philosophy of mathematics"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Aristotelian realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Middle Ages and Renaissance"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century"],"downwards":["Real analysis","Group theory","Non-Euclidean geometries","Projective geometry","Boolean algebra and logic","Peano arithmetic"]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Real analysis"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Group theory"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Non-Euclidean geometries"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Projective geometry"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Boolean algebra and logic"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Peano arithmetic"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis"],"downwards":["Philosophical views","More paradoxes"]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views"],"downwards":["Formalism","Intuitionism","Logicism","Set-theoretic Platonism","Indispensability argument for realism","Rough-and-ready realism","Philosophical consequences of G\u00f6del's completeness theorem"]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Formalism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Intuitionism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Logicism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Set-theoretic Platonism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Indispensability argument for realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Rough-and-ready realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Philosophical consequences of G\u00f6del's completeness theorem"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","More paradoxes"],"downwards":[]}
{"upwards":["Foundations of mathematics","Toward resolution of the crisis"],"downwards":[]}
{"upwards":["Cardinality"],"downwards":["History","Comparing sets","Cardinal numbers","Finite, countable and uncountable sets","Infinite sets","Examples and properties","Union and intersection"]}
{"upwards":["Cardinality","History"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets"],"downwards":["Definition 1: |A| = |B|","Definition 2: |A| \u2264 |B|","Definition 3: |A| < |B|"]}
{"upwards":["Cardinality","Comparing sets","Definition 1: |A| = |B|"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets","Definition 2: |A| \u2264 |B|"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets","Definition 3: |A| < |B|"],"downwards":[]}
{"upwards":["Cardinality","Cardinal numbers"],"downwards":[]}
{"upwards":["Cardinality","Finite, countable and uncountable sets"],"downwards":[]}
{"upwards":["Cardinality","Infinite sets"],"downwards":["Cardinality of the continuum"]}
{"upwards":["Cardinality","Infinite sets","Cardinality of the continuum"],"downwards":[]}
{"upwards":["Cardinality","Examples and properties"],"downwards":[]}
{"upwards":["Cardinality","Union and intersection"],"downwards":[]}
{"upwards":["Integer"],"downwards":["History","Algebraic properties","Order-theoretic properties","Construction","Computer science","Cardinality"]}
{"upwards":["Integer","History"],"downwards":[]}
{"upwards":["Integer","Algebraic properties"],"downwards":[]}
{"upwards":["Integer","Order-theoretic properties"],"downwards":[]}
{"upwards":["Integer","Construction"],"downwards":["Traditional development","Equivalence classes of ordered pairs","Other approaches"]}
{"upwards":["Integer","Construction","Traditional development"],"downwards":[]}
{"upwards":["Integer","Construction","Equivalence classes of ordered pairs"],"downwards":[]}
{"upwards":["Integer","Construction","Other approaches"],"downwards":[]}
{"upwards":["Integer","Computer science"],"downwards":[]}
{"upwards":["Integer","Cardinality"],"downwards":[]}
{"upwards":["Set-builder notation"],"downwards":["Sets defined by enumeration","Sets defined by a predicate","More complex expressions on the left side of the notation","Equivalent predicates yield equal sets","Set existence axiom","In programming languages"]}
{"upwards":["Set-builder notation","Sets defined by enumeration"],"downwards":[]}
{"upwards":["Set-builder notation","Sets defined by a predicate"],"downwards":["Specifying the domain","Examples"]}
{"upwards":["Set-builder notation","Sets defined by a predicate","Specifying the domain"],"downwards":[]}
{"upwards":["Set-builder notation","Sets defined by a predicate","Examples"],"downwards":[]}
{"upwards":["Set-builder notation","More complex expressions on the left side of the notation"],"downwards":[]}
{"upwards":["Set-builder notation","Equivalent predicates yield equal sets"],"downwards":[]}
{"upwards":["Set-builder notation","Set existence axiom"],"downwards":[]}
{"upwards":["Set-builder notation","In programming languages"],"downwards":[]}
{"upwards":["Injective function"],"downwards":["Definition","Examples","Injections can be undone","Injections may be made invertible","Other properties","Proving that functions are injective","Gallery"]}
{"upwards":["Injective function","Definition"],"downwards":[]}
{"upwards":["Injective function","Examples"],"downwards":[]}
{"upwards":["Injective function","Injections can be undone"],"downwards":[]}
{"upwards":["Injective function","Injections may be made invertible"],"downwards":[]}
{"upwards":["Injective function","Other properties"],"downwards":[]}
{"upwards":["Injective function","Proving that functions are injective"],"downwards":[]}
{"upwards":["Injective function","Gallery"],"downwards":[]}
{"upwards":["Transcendental number"],"downwards":["History","Properties","Numbers proven to be transcendental","Possible transcendental numbers","Sketch of a proof that e is transcendental"]}
{"upwards":["Transcendental number","History"],"downwards":[]}
{"upwards":["Transcendental number","Properties"],"downwards":[]}
{"upwards":["Transcendental number","Numbers proven to be transcendental"],"downwards":[]}
{"upwards":["Transcendental number","Possible transcendental numbers"],"downwards":[]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental"],"downwards":["The transcendence of \u03c0"]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental","The transcendence of \u03c0"],"downwards":[]}
{"upwards":["Extensional and intensional definitions"],"downwards":["Intensional definition","Extensional definition","History"]}
{"upwards":["Extensional and intensional definitions","Intensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","Extensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","History"],"downwards":[]}
{"upwards":["Euclidean space"],"downwards":["Definition","Prototypical examples","Affine structure","Metric structure","Isometries","Topology","Axiomatic definitions","Usage","Other geometric spaces"]}
{"upwards":["Euclidean space","Definition"],"downwards":["History of the definition","Motivation of the modern definition","Technical definition"]}
{"upwards":["Euclidean space","Definition","History of the definition"],"downwards":[]}
{"upwards":["Euclidean space","Definition","Motivation of the modern definition"],"downwards":[]}
{"upwards":["Euclidean space","Definition","Technical definition"],"downwards":[]}
{"upwards":["Euclidean space","Prototypical examples"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure"],"downwards":["Subspaces","Lines and segments","Parallelism"]}
{"upwards":["Euclidean space","Affine structure","Subspaces"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure","Lines and segments"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure","Parallelism"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure"],"downwards":["Distance and length","Orthogonality","Angle","Cartesian coordinates","Other coordinates"]}
{"upwards":["Euclidean space","Metric structure","Distance and length"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Orthogonality"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Angle"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Cartesian coordinates"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Other coordinates"],"downwards":[]}
{"upwards":["Euclidean space","Isometries"],"downwards":["Isometry with prototypical examples","Euclidean group"]}
{"upwards":["Euclidean space","Isometries","Isometry with prototypical examples"],"downwards":[]}
{"upwards":["Euclidean space","Isometries","Euclidean group"],"downwards":[]}
{"upwards":["Euclidean space","Topology"],"downwards":[]}
{"upwards":["Euclidean space","Axiomatic definitions"],"downwards":[]}
{"upwards":["Euclidean space","Usage"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces"],"downwards":["Affine space","Projective space","Non-Euclidean geometries","Curved spaces","Pseudo-Euclidean space"]}
{"upwards":["Euclidean space","Other geometric spaces","Affine space"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Projective space"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Non-Euclidean geometries"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Curved spaces"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Pseudo-Euclidean space"],"downwards":[]}
{"upwards":["e (mathematical constant)"],"downwards":["History","Applications","In calculus","Properties","Representations","Computing the digits","In computer culture","References","Further reading","External links"]}
{"upwards":["e (mathematical constant)","History"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications"],"downwards":["Compound interest","Bernoulli trials","Standard normal distribution","Derangements","Optimal planning problems","Asymptotics"]}
{"upwards":["e (mathematical constant)","Applications","Compound interest"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Bernoulli trials"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Standard normal distribution"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Derangements"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Optimal planning problems"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Asymptotics"],"downwards":[]}
{"upwards":["e (mathematical constant)","In calculus"],"downwards":["Alternative characterizations"]}
{"upwards":["e (mathematical constant)","In calculus","Alternative characterizations"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties"],"downwards":["Calculus","Inequalities","Exponential-like functions","Number theory","Complex numbers"]}
{"upwards":["e (mathematical constant)","Properties","Calculus"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Inequalities"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Exponential-like functions"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Number theory"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Complex numbers"],"downwards":[]}
{"upwards":["e (mathematical constant)","Representations"],"downwards":["Stochastic representations","Known digits"]}
{"upwards":["e (mathematical constant)","Representations","Stochastic representations"],"downwards":[]}
{"upwards":["e (mathematical constant)","Representations","Known digits"],"downwards":[]}
{"upwards":["e (mathematical constant)","Computing the digits"],"downwards":[]}
{"upwards":["e (mathematical constant)","In computer culture"],"downwards":[]}
{"upwards":["e (mathematical constant)","References"],"downwards":[]}
{"upwards":["e (mathematical constant)","Further reading"],"downwards":[]}
{"upwards":["e (mathematical constant)","External links"],"downwards":[]}
{"upwards":["Enumerative definition"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Relation (mathematics)"],"downwards":["Definition","Representation of relations","Properties of relations","Operations on relations","Examples","Generalizations"]}
{"upwards":["Relation (mathematics)","Definition"],"downwards":[]}
{"upwards":["Relation (mathematics)","Representation of relations"],"downwards":[]}
{"upwards":["Relation (mathematics)","Properties of relations"],"downwards":["Combinations of properties"]}
{"upwards":["Relation (mathematics)","Properties of relations","Combinations of properties"],"downwards":[]}
{"upwards":["Relation (mathematics)","Operations on relations"],"downwards":[]}
{"upwards":["Relation (mathematics)","Examples"],"downwards":[]}
{"upwards":["Relation (mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Power set"],"downwards":["Example","Properties","Representing subsets as functions","Relation to binomial theorem","Recursive definition","Subsets of limited cardinality","Power object","Functors and quantifiers"]}
{"upwards":["Power set","Example"],"downwards":[]}
{"upwards":["Power set","Properties"],"downwards":[]}
{"upwards":["Power set","Representing subsets as functions"],"downwards":[]}
{"upwards":["Power set","Relation to binomial theorem"],"downwards":[]}
{"upwards":["Power set","Recursive definition"],"downwards":[]}
{"upwards":["Power set","Subsets of limited cardinality"],"downwards":[]}
{"upwards":["Power set","Power object"],"downwards":[]}
{"upwards":["Power set","Functors and quantifiers"],"downwards":[]}
{"upwards":["Map (mathematics)"],"downwards":["Maps as functions","As morphisms"]}
{"upwards":["Map (mathematics)","Maps as functions"],"downwards":[]}
{"upwards":["Map (mathematics)","As morphisms"],"downwards":[]}
{"upwards":["Aleph number"],"downwards":["Aleph-nought","Aleph-one","Continuum hypothesis","Aleph-omega","Aleph-\u03b1 for general \u03b1","Fixed points of omega","Role of axiom of choice"]}
{"upwards":["Aleph number","Aleph-nought"],"downwards":[]}
{"upwards":["Aleph number","Aleph-one"],"downwards":[]}
{"upwards":["Aleph number","Continuum hypothesis"],"downwards":[]}
{"upwards":["Aleph number","Aleph-omega"],"downwards":[]}
{"upwards":["Aleph number","Aleph-\u03b1 for general \u03b1"],"downwards":[]}
{"upwards":["Aleph number","Fixed points of omega"],"downwards":[]}
{"upwards":["Aleph number","Role of axiom of choice"],"downwards":[]}
{"upwards":["Rational number"],"downwards":["Terminology","Arithmetic","Continued fraction representation","Other representations","Formal construction","Properties","Real numbers and topological properties","p-adic numbers"]}
{"upwards":["Rational number","Terminology"],"downwards":["Etymology"]}
{"upwards":["Rational number","Terminology","Etymology"],"downwards":[]}
{"upwards":["Rational number","Arithmetic"],"downwards":["Irreducible fraction","Embedding of integers","Equality","Ordering","Addition","Subtraction","Multiplication","Inverse","Division","Exponentiation to integer power"]}
{"upwards":["Rational number","Arithmetic","Irreducible fraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Embedding of integers"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Equality"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Ordering"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Addition"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Multiplication"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Inverse"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Division"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Exponentiation to integer power"],"downwards":[]}
{"upwards":["Rational number","Continued fraction representation"],"downwards":[]}
{"upwards":["Rational number","Other representations"],"downwards":[]}
{"upwards":["Rational number","Formal construction"],"downwards":[]}
{"upwards":["Rational number","Properties"],"downwards":["Countability"]}
{"upwards":["Rational number","Properties","Countability"],"downwards":[]}
{"upwards":["Rational number","Real numbers and topological properties"],"downwards":[]}
{"upwards":["Rational number","p-adic numbers"],"downwards":[]}
{"upwards":["Dimension"],"downwards":["In mathematics","In physics","In computer graphics and spatial data","More dimensions","List of topics by dimension"]}
{"upwards":["Dimension","In mathematics"],"downwards":["Vector spaces","Manifolds","Varieties","Krull dimension","Topological spaces","Hausdorff dimension","Hilbert spaces"]}
{"upwards":["Dimension","In mathematics","Vector spaces"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Manifolds"],"downwards":["Complex dimension"]}
{"upwards":["Dimension","In mathematics","Manifolds","Complex dimension"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Varieties"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Krull dimension"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Topological spaces"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Hausdorff dimension"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Hilbert spaces"],"downwards":[]}
{"upwards":["Dimension","In physics"],"downwards":["Spatial dimensions","Time","Additional dimensions"]}
{"upwards":["Dimension","In physics","Spatial dimensions"],"downwards":[]}
{"upwards":["Dimension","In physics","Time"],"downwards":[]}
{"upwards":["Dimension","In physics","Additional dimensions"],"downwards":[]}
{"upwards":["Dimension","In computer graphics and spatial data"],"downwards":[]}
{"upwards":["Dimension","More dimensions"],"downwards":[]}
{"upwards":["Dimension","List of topics by dimension"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Uncountable set"],"downwards":["Characterizations","Properties","Examples","Without the axiom of choice"]}
{"upwards":["Uncountable set","Characterizations"],"downwards":[]}
{"upwards":["Uncountable set","Properties"],"downwards":[]}
{"upwards":["Uncountable set","Examples"],"downwards":[]}
{"upwards":["Uncountable set","Without the axiom of choice"],"downwards":[]}
{"upwards":["Surjective function"],"downwards":["Definition","Examples","Properties","Space of surjections","Gallery"]}
{"upwards":["Surjective function","Definition"],"downwards":[]}
{"upwards":["Surjective function","Examples"],"downwards":[]}
{"upwards":["Surjective function","Properties"],"downwards":["Surjections as right invertible functions","Surjections as epimorphisms","Surjections as binary relations","Cardinality of the domain of a surjection","Composition and decomposition","Induced surjection and induced bijection"]}
{"upwards":["Surjective function","Properties","Surjections as right invertible functions"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as epimorphisms"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as binary relations"],"downwards":[]}
{"upwards":["Surjective function","Properties","Cardinality of the domain of a surjection"],"downwards":[]}
{"upwards":["Surjective function","Properties","Composition and decomposition"],"downwards":[]}
{"upwards":["Surjective function","Properties","Induced surjection and induced bijection"],"downwards":[]}
{"upwards":["Surjective function","Space of surjections"],"downwards":[]}
{"upwards":["Surjective function","Gallery"],"downwards":[]}
{"upwards":["Continuum hypothesis"],"downwards":["History","Cardinality of infinite sets","Independence from ZFC","Arguments for and against the continuum hypothesis","The generalized continuum hypothesis"]}
{"upwards":["Continuum hypothesis","History"],"downwards":[]}
{"upwards":["Continuum hypothesis","Cardinality of infinite sets"],"downwards":[]}
{"upwards":["Continuum hypothesis","Independence from ZFC"],"downwards":[]}
{"upwards":["Continuum hypothesis","Arguments for and against the continuum hypothesis"],"downwards":[]}
{"upwards":["Continuum hypothesis","The generalized continuum hypothesis"],"downwards":["Implications of GCH for cardinal exponentiation"]}
{"upwards":["Continuum hypothesis","The generalized continuum hypothesis","Implications of GCH for cardinal exponentiation"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Cartesian product"],"downwards":["Set-theoretic definition","Examples","Most common implementation (set theory)","Cartesian products of several sets","Other forms","Definitions outside set theory"]}
{"upwards":["Cartesian product","Set-theoretic definition"],"downwards":[]}
{"upwards":["Cartesian product","Examples"],"downwards":["A deck of cards","A two-dimensional coordinate system"]}
{"upwards":["Cartesian product","Examples","A deck of cards"],"downwards":[]}
{"upwards":["Cartesian product","Examples","A two-dimensional coordinate system"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)"],"downwards":["Non-commutativity and non-associativity","Intersections, unions, and subsets","Cardinality"]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Non-commutativity and non-associativity"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Intersections, unions, and subsets"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Cardinality"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets"],"downwards":["n-ary Cartesian product","n-ary Cartesian power","Infinite Cartesian products"]}
{"upwards":["Cartesian product","Cartesian products of several sets","n-ary Cartesian product"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets","n-ary Cartesian power"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets","Infinite Cartesian products"],"downwards":[]}
{"upwards":["Cartesian product","Other forms"],"downwards":["Abbreviated form","Cartesian product of functions","Cylinder"]}
{"upwards":["Cartesian product","Other forms","Abbreviated form"],"downwards":[]}
{"upwards":["Cartesian product","Other forms","Cartesian product of functions"],"downwards":[]}
{"upwards":["Cartesian product","Other forms","Cylinder"],"downwards":[]}
{"upwards":["Cartesian product","Definitions outside set theory"],"downwards":["Category theory","Graph theory"]}
{"upwards":["Cartesian product","Definitions outside set theory","Category theory"],"downwards":[]}
{"upwards":["Cartesian product","Definitions outside set theory","Graph theory"],"downwards":[]}
{"upwards":["Union (set theory)"],"downwards":["Union of two sets","Algebraic properties","Finite unions","Arbitrary unions","Notation encoding"]}
{"upwards":["Union (set theory)","Union of two sets"],"downwards":[]}
{"upwards":["Union (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Union (set theory)","Finite unions"],"downwards":[]}
{"upwards":["Union (set theory)","Arbitrary unions"],"downwards":["Notations"]}
{"upwards":["Union (set theory)","Arbitrary unions","Notations"],"downwards":[]}
{"upwards":["Union (set theory)","Notation encoding"],"downwards":[]}
{"upwards":["Complement (set theory)"],"downwards":["Absolute complement","Relative complement","Complementary relation","LaTeX notation","In programming languages"]}
{"upwards":["Complement (set theory)","Absolute complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Absolute complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Relative complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Complementary relation"],"downwards":[]}
{"upwards":["Complement (set theory)","LaTeX notation"],"downwards":[]}
{"upwards":["Complement (set theory)","In programming languages"],"downwards":[]}
{"upwards":["Bijection"],"downwards":["Definition","Examples","More mathematical examples","Inverses","Composition","Cardinality","Properties","Category theory","Generalization to partial functions","Gallery"]}
{"upwards":["Bijection","Definition"],"downwards":[]}
{"upwards":["Bijection","Examples"],"downwards":["Batting line-up of a baseball or cricket team","Seats and students of a classroom"]}
{"upwards":["Bijection","Examples","Batting line-up of a baseball or cricket team"],"downwards":[]}
{"upwards":["Bijection","Examples","Seats and students of a classroom"],"downwards":[]}
{"upwards":["Bijection","More mathematical examples"],"downwards":[]}
{"upwards":["Bijection","Inverses"],"downwards":[]}
{"upwards":["Bijection","Composition"],"downwards":[]}
{"upwards":["Bijection","Cardinality"],"downwards":[]}
{"upwards":["Bijection","Properties"],"downwards":[]}
{"upwards":["Bijection","Category theory"],"downwards":[]}
{"upwards":["Bijection","Generalization to partial functions"],"downwards":[]}
{"upwards":["Bijection","Gallery"],"downwards":[]}
{"upwards":["Plane (mathematics)"],"downwards":["Euclidean plane","Elliptic plane","Projective plane","Further generalizations","Topological and differential geometric notions"]}
{"upwards":["Plane (mathematics)","Euclidean plane"],"downwards":["Embedding in three-dimensional space"]}
{"upwards":["Plane (mathematics)","Euclidean plane","Embedding in three-dimensional space"],"downwards":[]}
{"upwards":["Plane (mathematics)","Elliptic plane"],"downwards":[]}
{"upwards":["Plane (mathematics)","Projective plane"],"downwards":[]}
{"upwards":["Plane (mathematics)","Further generalizations"],"downwards":[]}
{"upwards":["Plane (mathematics)","Topological and differential geometric notions"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Irrational number"],"downwards":["History","Examples","Types","Decimal expansions","Irrational powers","Open questions","In constructive mathematics","Set of all irrationals"]}
{"upwards":["Irrational number","History"],"downwards":["Ancient Greece","India","Middle Ages","Modern period"]}
{"upwards":["Irrational number","History","Ancient Greece"],"downwards":[]}
{"upwards":["Irrational number","History","India"],"downwards":[]}
{"upwards":["Irrational number","History","Middle Ages"],"downwards":[]}
{"upwards":["Irrational number","History","Modern period"],"downwards":[]}
{"upwards":["Irrational number","Examples"],"downwards":["Square roots","General roots","Logarithms"]}
{"upwards":["Irrational number","Examples","Square roots"],"downwards":[]}
{"upwards":["Irrational number","Examples","General roots"],"downwards":[]}
{"upwards":["Irrational number","Examples","Logarithms"],"downwards":[]}
{"upwards":["Irrational number","Types"],"downwards":["Transcendental\/algebraic"]}
{"upwards":["Irrational number","Types","Transcendental\/algebraic"],"downwards":[]}
{"upwards":["Irrational number","Decimal expansions"],"downwards":[]}
{"upwards":["Irrational number","Irrational powers"],"downwards":[]}
{"upwards":["Irrational number","Open questions"],"downwards":[]}
{"upwards":["Irrational number","In constructive mathematics"],"downwards":[]}
{"upwards":["Irrational number","Set of all irrationals"],"downwards":[]}
{"upwards":["Axiom"],"downwards":["Etymology","Historical development","Mathematical logic"]}
{"upwards":["Axiom","Etymology"],"downwards":[]}
{"upwards":["Axiom","Historical development"],"downwards":["Early Greeks","Modern development","Other sciences"]}
{"upwards":["Axiom","Historical development","Early Greeks"],"downwards":[]}
{"upwards":["Axiom","Historical development","Modern development"],"downwards":[]}
{"upwards":["Axiom","Historical development","Other sciences"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic"],"downwards":["Logical axioms","Non-logical axioms","Role in mathematical logic","Further discussion"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic"],"downwards":["Deductive systems and completeness"]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic","Deductive systems and completeness"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Further discussion"],"downwards":[]}
{"upwards":["Infinite set"],"downwards":["Properties","Examples"]}
{"upwards":["Infinite set","Properties"],"downwards":[]}
{"upwards":["Infinite set","Examples"],"downwards":["Countably infinite sets","Uncountably infinite sets"]}
{"upwards":["Infinite set","Examples","Countably infinite sets"],"downwards":[]}
{"upwards":["Infinite set","Examples","Uncountably infinite sets"],"downwards":[]}
{"upwards":["Real number"],"downwards":["Characterizing properties","Arithmetic","Dedekind completeness","Decimal representation","Topological completeness","Cardinality","Other properties","History","Formal definitions","Applications and connections","Vocabulary and notation","Generalizations and extensions"]}
{"upwards":["Real number","Characterizing properties"],"downwards":[]}
{"upwards":["Real number","Arithmetic"],"downwards":["Auxiliary operations","Auxiliary order relations","Integers and fractions as real numbers"]}
{"upwards":["Real number","Arithmetic","Auxiliary operations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Auxiliary order relations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Integers and fractions as real numbers"],"downwards":[]}
{"upwards":["Real number","Dedekind completeness"],"downwards":[]}
{"upwards":["Real number","Decimal representation"],"downwards":[]}
{"upwards":["Real number","Topological completeness"],"downwards":["\"The complete ordered field\""]}
{"upwards":["Real number","Topological completeness","\"The complete ordered field\""],"downwards":[]}
{"upwards":["Real number","Cardinality"],"downwards":[]}
{"upwards":["Real number","Other properties"],"downwards":[]}
{"upwards":["Real number","History"],"downwards":[]}
{"upwards":["Real number","Formal definitions"],"downwards":["Axiomatic approach","Construction from the rational numbers"]}
{"upwards":["Real number","Formal definitions","Axiomatic approach"],"downwards":[]}
{"upwards":["Real number","Formal definitions","Construction from the rational numbers"],"downwards":[]}
{"upwards":["Real number","Applications and connections"],"downwards":["Physics","Logic","Computation","Set theory"]}
{"upwards":["Real number","Applications and connections","Physics"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Logic"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Computation"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Set theory"],"downwards":[]}
{"upwards":["Real number","Vocabulary and notation"],"downwards":[]}
{"upwards":["Real number","Generalizations and extensions"],"downwards":[]}
{"upwards":["Russell's paradox"],"downwards":["Informal presentation","Formal presentation","Philosophical implications","Set-theoretic responses","History","Applied versions","Applications and related topics","Related paradoxes"]}
{"upwards":["Russell's paradox","Informal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Formal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Philosophical implications"],"downwards":[]}
{"upwards":["Russell's paradox","Set-theoretic responses"],"downwards":[]}
{"upwards":["Russell's paradox","History"],"downwards":[]}
{"upwards":["Russell's paradox","Applied versions"],"downwards":[]}
{"upwards":["Russell's paradox","Applications and related topics"],"downwards":["Russell-like paradoxes"]}
{"upwards":["Russell's paradox","Applications and related topics","Russell-like paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox","Related paradoxes"],"downwards":[]}
{"upwards":["Empty set"],"downwards":["Notation","Properties","In other areas of mathematics","Questioned existence"]}
{"upwards":["Empty set","Notation"],"downwards":[]}
{"upwards":["Empty set","Properties"],"downwards":["Operations on the empty set"]}
{"upwards":["Empty set","Properties","Operations on the empty set"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics"],"downwards":["Extended real numbers","Topology","Category theory","Set theory"]}
{"upwards":["Empty set","In other areas of mathematics","Extended real numbers"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Topology"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Category theory"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Set theory"],"downwards":[]}
{"upwards":["Empty set","Questioned existence"],"downwards":["Axiomatic set theory","Philosophical issues"]}
{"upwards":["Empty set","Questioned existence","Axiomatic set theory"],"downwards":[]}
{"upwards":["Empty set","Questioned existence","Philosophical issues"],"downwards":[]}
{"upwards":["Tuple"],"downwards":["Etymology","Properties","Definitions","n-tuples of m-sets","Type theory"]}
{"upwards":["Tuple","Etymology"],"downwards":["Names for tuples of specific lengths"]}
{"upwards":["Tuple","Etymology","Names for tuples of specific lengths"],"downwards":[]}
{"upwards":["Tuple","Properties"],"downwards":[]}
{"upwards":["Tuple","Definitions"],"downwards":["Tuples as functions","Tuples as nested ordered pairs","Tuples as nested sets"]}
{"upwards":["Tuple","Definitions","Tuples as functions"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested ordered pairs"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested sets"],"downwards":[]}
{"upwards":["Tuple","n-tuples of m-sets"],"downwards":[]}
{"upwards":["Tuple","Type theory"],"downwards":[]}
{"upwards":["Boolean ring"],"downwards":["Notations","Examples","Relation to Boolean algebras","Properties of Boolean rings","Unification"]}
{"upwards":["Boolean ring","Notations"],"downwards":[]}
{"upwards":["Boolean ring","Examples"],"downwards":[]}
{"upwards":["Boolean ring","Relation to Boolean algebras"],"downwards":[]}
{"upwards":["Boolean ring","Properties of Boolean rings"],"downwards":[]}
{"upwards":["Boolean ring","Unification"],"downwards":[]}
{"upwards":["Function (mathematics)"],"downwards":["Definition","Notation","Other terms","Specifying a function","Representing a function","General properties","Multivariate function","In calculus","Function space","Multi-valued functions","In the foundations of mathematics and set theory","In computer science"]}
{"upwards":["Function (mathematics)","Definition"],"downwards":["Total, univalent relation","Set exponentiation"]}
{"upwards":["Function (mathematics)","Definition","Total, univalent relation"],"downwards":[]}
{"upwards":["Function (mathematics)","Definition","Set exponentiation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation"],"downwards":["Functional notation","Arrow notation","Index notation","Dot notation","Specialized notations"]}
{"upwards":["Function (mathematics)","Notation","Functional notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Arrow notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Index notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Dot notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Specialized notations"],"downwards":[]}
{"upwards":["Function (mathematics)","Other terms"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function"],"downwards":["By listing function values","By a formula","Inverse and implicit functions","Using differential calculus","By recurrence"]}
{"upwards":["Function (mathematics)","Specifying a function","By listing function values"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","By a formula"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","Inverse and implicit functions"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","Using differential calculus"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","By recurrence"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function"],"downwards":["Graphs and plots","Tables","Bar chart"]}
{"upwards":["Function (mathematics)","Representing a function","Graphs and plots"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function","Tables"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function","Bar chart"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties"],"downwards":["Standard functions","Function composition","Image and preimage","Injective, surjective and bijective functions","Restriction and extension"]}
{"upwards":["Function (mathematics)","General properties","Standard functions"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Function composition"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Image and preimage"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Injective, surjective and bijective functions"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Restriction and extension"],"downwards":[]}
{"upwards":["Function (mathematics)","Multivariate function"],"downwards":[]}
{"upwards":["Function (mathematics)","In calculus"],"downwards":["Real function","Vector-valued function"]}
{"upwards":["Function (mathematics)","In calculus","Real function"],"downwards":[]}
{"upwards":["Function (mathematics)","In calculus","Vector-valued function"],"downwards":[]}
{"upwards":["Function (mathematics)","Function space"],"downwards":[]}
{"upwards":["Function (mathematics)","Multi-valued functions"],"downwards":[]}
{"upwards":["Function (mathematics)","In the foundations of mathematics and set theory"],"downwards":[]}
{"upwards":["Function (mathematics)","In computer science"],"downwards":[]}
{"upwards":["Extensional and intensional definitions"],"downwards":["Intensional definition","Extensional definition","History"]}
{"upwards":["Extensional and intensional definitions","Intensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","Extensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","History"],"downwards":[]}
{"upwards":["Colon (punctuation)"],"downwards":["History","Usage in English","Usage in other languages","Mathematics and logic","Computing"]}
{"upwards":["Colon (punctuation)","History"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English"],"downwards":["Syntactical-deductive","Syntactical-descriptive","Appositive","Segmental","Use of capitals","Spacing and parentheses"]}
{"upwards":["Colon (punctuation)","Usage in English","Syntactical-deductive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Syntactical-descriptive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Appositive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Segmental"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Use of capitals"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Spacing and parentheses"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages"],"downwards":["Suffix separator","Abbreviation mark","Word separator","End of sentence or verse","Score divider"]}
{"upwards":["Colon (punctuation)","Usage in other languages","Suffix separator"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Abbreviation mark"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Word separator"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","End of sentence or verse"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Score divider"],"downwards":[]}
{"upwards":["Colon (punctuation)","Mathematics and logic"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing"],"downwards":["Programming languages","Addresses","Text markup"]}
{"upwards":["Colon (punctuation)","Computing","Programming languages"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing","Addresses"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing","Text markup"],"downwards":[]}
{"upwards":["Disjoint sets"],"downwards":["Generalizations","Intersections","Disjoint unions and partitions"]}
{"upwards":["Disjoint sets","Generalizations"],"downwards":[]}
{"upwards":["Disjoint sets","Intersections"],"downwards":[]}
{"upwards":["Disjoint sets","Disjoint unions and partitions"],"downwards":[]}
{"upwards":["Independence (mathematical logic)"],"downwards":["Usage note","Independence results in set theory","Applications to physical theory"]}
{"upwards":["Independence (mathematical logic)","Usage note"],"downwards":[]}
{"upwards":["Independence (mathematical logic)","Independence results in set theory"],"downwards":[]}
{"upwards":["Independence (mathematical logic)","Applications to physical theory"],"downwards":[]}
{"upwards":["Line (geometry)"],"downwards":["Properties","Definition","Other representations","Generalizations of the Euclidean line","Extensions"]}
{"upwards":["Line (geometry)","Properties"],"downwards":["In higher dimensions","Types","In axiomatic systems"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions"],"downwards":["Collinear points"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions","Collinear points"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","Types"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","In axiomatic systems"],"downwards":[]}
{"upwards":["Line (geometry)","Definition"],"downwards":["Linear equation","Parametric equation","Hesse normal form"]}
{"upwards":["Line (geometry)","Definition","Linear equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Parametric equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Hesse normal form"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations"],"downwards":["Vectors","Polar coordinates"]}
{"upwards":["Line (geometry)","Other representations","Vectors"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations","Polar coordinates"],"downwards":[]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line"],"downwards":["Projective geometry"]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line","Projective geometry"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions"],"downwards":["Ray","Line segment","Number line"]}
{"upwards":["Line (geometry)","Extensions","Ray"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Line segment"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Number line"],"downwards":[]}
{"upwards":["Axiom of choice"],"downwards":["Statement","Usage","Examples","Criticism and acceptance","In constructive mathematics","Independence","Stronger axioms","Equivalents","Weaker forms","Stronger forms of the negation of AC","Statements consistent with the negation of AC","Axiom of choice in type theory","Quotations","Notes","References","External links"]}
{"upwards":["Axiom of choice","Statement"],"downwards":["Nomenclature ZF, AC, and ZFC","Variants","Restriction to finite sets"]}
{"upwards":["Axiom of choice","Statement","Nomenclature ZF, AC, and ZFC"],"downwards":[]}
{"upwards":["Axiom of choice","Statement","Variants"],"downwards":[]}
{"upwards":["Axiom of choice","Statement","Restriction to finite sets"],"downwards":[]}
{"upwards":["Axiom of choice","Usage"],"downwards":[]}
{"upwards":["Axiom of choice","Examples"],"downwards":[]}
{"upwards":["Axiom of choice","Criticism and acceptance"],"downwards":[]}
{"upwards":["Axiom of choice","In constructive mathematics"],"downwards":[]}
{"upwards":["Axiom of choice","Independence"],"downwards":[]}
{"upwards":["Axiom of choice","Stronger axioms"],"downwards":[]}
{"upwards":["Axiom of choice","Equivalents"],"downwards":["Category theory"]}
{"upwards":["Axiom of choice","Equivalents","Category theory"],"downwards":[]}
{"upwards":["Axiom of choice","Weaker forms"],"downwards":["Results requiring AC (or weaker forms) but weaker than it","Possibly equivalent implications of AC"]}
{"upwards":["Axiom of choice","Weaker forms","Results requiring AC (or weaker forms) but weaker than it"],"downwards":[]}
{"upwards":["Axiom of choice","Weaker forms","Possibly equivalent implications of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Stronger forms of the negation of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Statements consistent with the negation of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Axiom of choice in type theory"],"downwards":[]}
{"upwards":["Axiom of choice","Quotations"],"downwards":[]}
{"upwards":["Axiom of choice","Notes"],"downwards":[]}
{"upwards":["Axiom of choice","References"],"downwards":[]}
{"upwards":["Axiom of choice","External links"],"downwards":[]}
{"upwards":["Intersection (set theory)"],"downwards":["Notation and terminology","Definition","Algebraic properties","Arbitrary intersections","Nullary intersection"]}
{"upwards":["Intersection (set theory)","Notation and terminology"],"downwards":[]}
{"upwards":["Intersection (set theory)","Definition"],"downwards":["Intersecting and disjoint sets"]}
{"upwards":["Intersection (set theory)","Definition","Intersecting and disjoint sets"],"downwards":[]}
{"upwards":["Intersection (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Intersection (set theory)","Arbitrary intersections"],"downwards":[]}
{"upwards":["Intersection (set theory)","Nullary intersection"],"downwards":[]}
{"upwards":["Complement (set theory)"],"downwards":["Absolute complement","Relative complement","Complementary relation","LaTeX notation","In programming languages"]}
{"upwards":["Complement (set theory)","Absolute complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Absolute complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Relative complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Complementary relation"],"downwards":[]}
{"upwards":["Complement (set theory)","LaTeX notation"],"downwards":[]}
{"upwards":["Complement (set theory)","In programming languages"],"downwards":[]}
{"upwards":["Algebra of sets"],"downwards":["Fundamentals","The fundamental properties of set algebra","The principle of duality","Some additional laws for unions and intersections","Some additional laws for complements","The algebra of inclusion","The algebra of relative complements"]}
{"upwards":["Algebra of sets","Fundamentals"],"downwards":[]}
{"upwards":["Algebra of sets","The fundamental properties of set algebra"],"downwards":[]}
{"upwards":["Algebra of sets","The principle of duality"],"downwards":[]}
{"upwards":["Algebra of sets","Some additional laws for unions and intersections"],"downwards":[]}
{"upwards":["Algebra of sets","Some additional laws for complements"],"downwards":[]}
{"upwards":["Algebra of sets","The algebra of inclusion"],"downwards":[]}
{"upwards":["Algebra of sets","The algebra of relative complements"],"downwards":[]}
{"upwards":["Paul Cohen"],"downwards":["Early life and education","Career","Death","Selected publications"]}
{"upwards":["Paul Cohen","Early life and education"],"downwards":[]}
{"upwards":["Paul Cohen","Career"],"downwards":["Continuum hypothesis"]}
{"upwards":["Paul Cohen","Career","Continuum hypothesis"],"downwards":[]}
{"upwards":["Paul Cohen","Death"],"downwards":[]}
{"upwards":["Paul Cohen","Selected publications"],"downwards":[]}
{"upwards":["Universe (mathematics)"],"downwards":["In a specific context","In ordinary mathematics","In set theory","In predicate calculus","In category theory","In type theory"]}
{"upwards":["Universe (mathematics)","In a specific context"],"downwards":[]}
{"upwards":["Universe (mathematics)","In ordinary mathematics"],"downwards":[]}
{"upwards":["Universe (mathematics)","In set theory"],"downwards":[]}
{"upwards":["Universe (mathematics)","In predicate calculus"],"downwards":[]}
{"upwards":["Universe (mathematics)","In category theory"],"downwards":[]}
{"upwards":["Universe (mathematics)","In type theory"],"downwards":[]}
{"upwards":["Line segment"],"downwards":["In real or complex vector spaces","Properties","In proofs","As a degenerate ellipse","In other geometric shapes","Directed line segment","Generalizations","Types of line segments"]}
{"upwards":["Line segment","In real or complex vector spaces"],"downwards":[]}
{"upwards":["Line segment","Properties"],"downwards":[]}
{"upwards":["Line segment","In proofs"],"downwards":[]}
{"upwards":["Line segment","As a degenerate ellipse"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes"],"downwards":["Triangles","Quadrilaterals","Circles and ellipses"]}
{"upwards":["Line segment","In other geometric shapes","Triangles"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes","Quadrilaterals"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes","Circles and ellipses"],"downwards":[]}
{"upwards":["Line segment","Directed line segment"],"downwards":[]}
{"upwards":["Line segment","Generalizations"],"downwards":[]}
{"upwards":["Line segment","Types of line segments"],"downwards":[]}
{"upwards":["Element (mathematics)"],"downwards":["Sets","Notation and terminology","Cardinality of sets","Examples","Formal relation"]}
{"upwards":["Element (mathematics)","Sets"],"downwards":[]}
{"upwards":["Element (mathematics)","Notation and terminology"],"downwards":[]}
{"upwards":["Element (mathematics)","Cardinality of sets"],"downwards":[]}
{"upwards":["Element (mathematics)","Examples"],"downwards":[]}
{"upwards":["Element (mathematics)","Formal relation"],"downwards":[]}
{"upwards":["Russell's paradox"],"downwards":["Informal presentation","Formal presentation","Philosophical implications","Set-theoretic responses","History","Applied versions","Applications and related topics","Related paradoxes"]}
{"upwards":["Russell's paradox","Informal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Formal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Philosophical implications"],"downwards":[]}
{"upwards":["Russell's paradox","Set-theoretic responses"],"downwards":[]}
{"upwards":["Russell's paradox","History"],"downwards":[]}
{"upwards":["Russell's paradox","Applied versions"],"downwards":[]}
{"upwards":["Russell's paradox","Applications and related topics"],"downwards":["Russell-like paradoxes"]}
{"upwards":["Russell's paradox","Applications and related topics","Russell-like paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox","Related paradoxes"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory"],"downwards":["History","Axioms","Motivation via the cumulative hierarchy","Metamathematics","Criticisms"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","History"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms"],"downwards":["1. Axiom of extensionality","2. Axiom of regularity (also called the axiom of foundation)","3. Axiom schema of specification (or of separation, or of restricted comprehension)","4. Axiom of pairing","5. Axiom of union","6. Axiom schema of replacement","7. Axiom of infinity","8. Axiom of power set","9. Well-ordering axiom","Zorn's lemma"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","1. Axiom of extensionality"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","2. Axiom of regularity (also called the axiom of foundation)"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","3. Axiom schema of specification (or of separation, or of restricted comprehension)"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","4. Axiom of pairing"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","5. Axiom of union"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","6. Axiom schema of replacement"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","7. Axiom of infinity"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","8. Axiom of power set"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","9. Well-ordering axiom"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","Zorn's lemma"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Motivation via the cumulative hierarchy"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics"],"downwards":["Virtual classes","Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Consistency","Independence","Proposed additions"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Virtual classes"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Von Neumann\u2013Bernays\u2013G\u00f6del set theory"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Consistency"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Independence"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Proposed additions"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Criticisms"],"downwards":[]}
{"upwards":["Ellipsis"],"downwards":["Background","In writing","In different languages","Usage in computer system menus","In mathematical notation","Computer science","On Internet chat rooms and in text messaging","Computer representations"]}
{"upwards":["Ellipsis","Background"],"downwards":[]}
{"upwards":["Ellipsis","In writing"],"downwards":[]}
{"upwards":["Ellipsis","In different languages"],"downwards":["In English","In Polish","In Russian","In Japanese","In Chinese","In Spanish","In French","In German","In Italian"]}
{"upwards":["Ellipsis","In different languages","In English"],"downwards":["American English","British English"]}
{"upwards":["Ellipsis","In different languages","In English","American English"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In English","British English"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Polish"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Russian"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Japanese"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Chinese"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Spanish"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In French"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In German"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Italian"],"downwards":[]}
{"upwards":["Ellipsis","Usage in computer system menus"],"downwards":[]}
{"upwards":["Ellipsis","In mathematical notation"],"downwards":[]}
{"upwards":["Ellipsis","Computer science"],"downwards":["Programming languages","HTML and CSS"]}
{"upwards":["Ellipsis","Computer science","Programming languages"],"downwards":[]}
{"upwards":["Ellipsis","Computer science","HTML and CSS"],"downwards":[]}
{"upwards":["Ellipsis","On Internet chat rooms and in text messaging"],"downwards":[]}
{"upwards":["Ellipsis","Computer representations"],"downwards":["Input"]}
{"upwards":["Ellipsis","Computer representations","Input"],"downwards":[]}
{"upwards":["Bracket"],"downwards":["History","Typography","Round brackets or parentheses","Square brackets or brackets","Curly brackets or braces","Angle brackets or chevrons","Other brackets","Unicode"]}
{"upwards":["Bracket","History"],"downwards":[]}
{"upwards":["Bracket","Typography"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses"],"downwards":["Uses of ( )"]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )"],"downwards":["Enumerations","Accounting","Parentheses in mathematics","Parentheses in programming languages","Taxonomy","Chemistry and physics","Pop songs"]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Enumerations"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Accounting"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Parentheses in mathematics"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Parentheses in programming languages"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Taxonomy"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Chemistry and physics"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Pop songs"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets"],"downwards":["Uses of [ ]"]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]"],"downwards":["Proofreading","Law","Square brackets in mathematics","Chemistry","Square brackets in programming languages"]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Proofreading"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Law"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Square brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Chemistry"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Square brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces"],"downwards":["Uses of { }"]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }"],"downwards":["Music","Curly brackets in programming languages","Curly brackets in mathematics"]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Music"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Curly brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Curly brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons"],"downwards":["Shape","Uses of \u27e8 \u27e9"]}
{"upwards":["Bracket","Angle brackets or chevrons","Shape"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9"],"downwards":["Angle brackets in mathematics","Physics and mechanics","Angle brackets in programming languages"]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Angle brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Physics and mechanics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Angle brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Other brackets"],"downwards":["Lenticular brackets\u3010\u3011","Floor \u230a \u230b and ceiling \u2308 \u2309 corner brackets","Quine corners \u231c\u231d and half brackets \u2e24 \u2e25 or \u2e22 \u2e23","Double brackets \u27e6 \u27e7","Brackets with quills \u2045 \u2046"]}
{"upwards":["Bracket","Other brackets","Lenticular brackets\u3010\u3011"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Floor \u230a \u230b and ceiling \u2308 \u2309 corner brackets"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Quine corners \u231c\u231d and half brackets \u2e24 \u2e25 or \u2e22 \u2e23"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Double brackets \u27e6 \u27e7"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Brackets with quills \u2045 \u2046"],"downwards":[]}
{"upwards":["Bracket","Unicode"],"downwards":[]}
{"upwards":["Integer"],"downwards":["History","Algebraic properties","Order-theoretic properties","Construction","Computer science","Cardinality"]}
{"upwards":["Integer","History"],"downwards":[]}
{"upwards":["Integer","Algebraic properties"],"downwards":[]}
{"upwards":["Integer","Order-theoretic properties"],"downwards":[]}
{"upwards":["Integer","Construction"],"downwards":["Traditional development","Equivalence classes of ordered pairs","Other approaches"]}
{"upwards":["Integer","Construction","Traditional development"],"downwards":[]}
{"upwards":["Integer","Construction","Equivalence classes of ordered pairs"],"downwards":[]}
{"upwards":["Integer","Construction","Other approaches"],"downwards":[]}
{"upwards":["Integer","Computer science"],"downwards":[]}
{"upwards":["Integer","Cardinality"],"downwards":[]}
{"upwards":["Permutation"],"downwards":["History","Permutations without repetitions","Definition","Notations","Other uses of the term permutation","Properties","Permutations of totally ordered sets","Permutations in computing"]}
{"upwards":["Permutation","History"],"downwards":[]}
{"upwards":["Permutation","Permutations without repetitions"],"downwards":[]}
{"upwards":["Permutation","Definition"],"downwards":[]}
{"upwards":["Permutation","Notations"],"downwards":["Two-line notation","One-line notation","Cycle notation","Canonical cycle notation","Composition of permutations"]}
{"upwards":["Permutation","Notations","Two-line notation"],"downwards":[]}
{"upwards":["Permutation","Notations","One-line notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Cycle notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Canonical cycle notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Composition of permutations"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation"],"downwards":["k-permutations of n","Permutations with repetition","Permutations of multisets","Circular permutations"]}
{"upwards":["Permutation","Other uses of the term permutation","k-permutations of n"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Permutations with repetition"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Permutations of multisets"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Circular permutations"],"downwards":[]}
{"upwards":["Permutation","Properties"],"downwards":["Cycle type","Conjugating permutations","Permutation order","Parity of a permutation","Matrix representation"]}
{"upwards":["Permutation","Properties","Cycle type"],"downwards":[]}
{"upwards":["Permutation","Properties","Conjugating permutations"],"downwards":[]}
{"upwards":["Permutation","Properties","Permutation order"],"downwards":[]}
{"upwards":["Permutation","Properties","Parity of a permutation"],"downwards":[]}
{"upwards":["Permutation","Properties","Matrix representation"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets"],"downwards":["Ascents, descents, runs and exceedances","Foata's transition lemma","Inversions"]}
{"upwards":["Permutation","Permutations of totally ordered sets","Ascents, descents, runs and exceedances"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets","Foata's transition lemma"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets","Inversions"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing"],"downwards":["Numbering permutations","Algorithms to generate permutations","Applications"]}
{"upwards":["Permutation","Permutations in computing","Numbering permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations"],"downwards":["Random generation of permutations","Generation in lexicographic order","Generation with minimal changes","Meandric permutations"]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Random generation of permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Generation in lexicographic order"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Generation with minimal changes"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Meandric permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Applications"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems"],"downwards":["Formal systems: completeness, consistency, and effective axiomatization","First incompleteness theorem","Second incompleteness theorem","Examples of undecidable statements","Relationship with computability","Proof sketch for the first theorem","Proof sketch for the second theorem","Discussion and implications","History"]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization"],"downwards":["Effective axiomatization","Completeness","Consistency","Systems which contain arithmetic","Conflicting goals"]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Effective axiomatization"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Completeness"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Consistency"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Systems which contain arithmetic"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Conflicting goals"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem"],"downwards":["Syntactic form of the G\u00f6del sentence","Truth of the G\u00f6del sentence","Relationship with the liar paradox","Extensions of G\u00f6del's original result"]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Syntactic form of the G\u00f6del sentence"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Truth of the G\u00f6del sentence"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Relationship with the liar paradox"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Extensions of G\u00f6del's original result"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem"],"downwards":["Expressing consistency","The Hilbert\u2013Bernays conditions","Implications for consistency proofs"]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","Expressing consistency"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","The Hilbert\u2013Bernays conditions"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","Implications for consistency proofs"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Examples of undecidable statements"],"downwards":["Undecidable statements provable in larger systems"]}
{"upwards":["G\u00f6del's incompleteness theorems","Examples of undecidable statements","Undecidable statements provable in larger systems"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Relationship with computability"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem"],"downwards":["Arithmetization of syntax","Construction of a statement about \"provability\"","Diagonalization","Proof via Berry's paradox","Computer verified proofs"]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Arithmetization of syntax"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Construction of a statement about \"provability\""],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Diagonalization"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Proof via Berry's paradox"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Computer verified proofs"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the second theorem"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications"],"downwards":["Consequences for logicism and Hilbert's second problem","Minds and machines","Paraconsistent logic","Appeals to the incompleteness theorems in other fields"]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Consequences for logicism and Hilbert's second problem"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Minds and machines"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Paraconsistent logic"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Appeals to the incompleteness theorems in other fields"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History"],"downwards":["Announcement","Generalization and acceptance","Criticisms"]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Announcement"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Generalization and acceptance"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms"],"downwards":["Finsler","Zermelo","Wittgenstein"]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Finsler"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Zermelo"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Wittgenstein"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Flag of France"],"downwards":["Design","Symbolism","History","Regimental flags","Naval flags","Colonial flags","Other","Gallery"]}
{"upwards":["Flag of France","Design"],"downwards":[]}
{"upwards":["Flag of France","Symbolism"],"downwards":[]}
{"upwards":["Flag of France","History"],"downwards":["Kingdom of France","The Tricolore"]}
{"upwards":["Flag of France","History","Kingdom of France"],"downwards":[]}
{"upwards":["Flag of France","History","The Tricolore"],"downwards":[]}
{"upwards":["Flag of France","Regimental flags"],"downwards":[]}
{"upwards":["Flag of France","Naval flags"],"downwards":[]}
{"upwards":["Flag of France","Colonial flags"],"downwards":[]}
{"upwards":["Flag of France","Other"],"downwards":[]}
{"upwards":["Flag of France","Gallery"],"downwards":[]}
{"upwards":["Sequence"],"downwards":["Examples and notation","Formal definition and basic properties","Limits and convergence","Series","Use in other fields of mathematics"]}
{"upwards":["Sequence","Examples and notation"],"downwards":["Examples","Indexing","Defining a sequence by recursion"]}
{"upwards":["Sequence","Examples and notation","Examples"],"downwards":[]}
{"upwards":["Sequence","Examples and notation","Indexing"],"downwards":[]}
{"upwards":["Sequence","Examples and notation","Defining a sequence by recursion"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties"],"downwards":["Definition","Finite and infinite","Increasing and decreasing","Bounded","Subsequences","Other types of sequences"]}
{"upwards":["Sequence","Formal definition and basic properties","Definition"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Finite and infinite"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Increasing and decreasing"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Bounded"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Subsequences"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Other types of sequences"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence"],"downwards":["Formal definition of convergence","Applications and important results","Cauchy sequences","Infinite limits"]}
{"upwards":["Sequence","Limits and convergence","Formal definition of convergence"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Applications and important results"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Cauchy sequences"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Infinite limits"],"downwards":[]}
{"upwards":["Sequence","Series"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics"],"downwards":["Topology","Analysis","Linear algebra","Abstract algebra","Set theory","Computing","Streams"]}
{"upwards":["Sequence","Use in other fields of mathematics","Topology"],"downwards":["Product topology"]}
{"upwards":["Sequence","Use in other fields of mathematics","Topology","Product topology"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Analysis"],"downwards":["Sequence spaces"]}
{"upwards":["Sequence","Use in other fields of mathematics","Analysis","Sequence spaces"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Linear algebra"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra"],"downwards":["Free monoid","Exact sequences","Spectral sequences"]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Free monoid"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Exact sequences"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Spectral sequences"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Set theory"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Computing"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Streams"],"downwards":[]}
{"upwards":["Italic type"],"downwards":["History","Examples","Usage","Oblique type compared to italics","More complex usage"]}
{"upwards":["Italic type","History"],"downwards":[]}
{"upwards":["Italic type","Examples"],"downwards":[]}
{"upwards":["Italic type","Usage"],"downwards":[]}
{"upwards":["Italic type","Oblique type compared to italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage"],"downwards":["Italics within italics","Left-leaning italics","Upright italics","Parentheses","Substitutes","OpenType","Web pages","Unicode"]}
{"upwards":["Italic type","More complex usage","Italics within italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Left-leaning italics"],"downwards":["Iranic font style"]}
{"upwards":["Italic type","More complex usage","Left-leaning italics","Iranic font style"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Upright italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Parentheses"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Substitutes"],"downwards":[]}
{"upwards":["Italic type","More complex usage","OpenType"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Web pages"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Unicode"],"downwards":[]}
{"upwards":["Bijection"],"downwards":["Definition","Examples","More mathematical examples","Inverses","Composition","Cardinality","Properties","Category theory","Generalization to partial functions","Gallery"]}
{"upwards":["Bijection","Definition"],"downwards":[]}
{"upwards":["Bijection","Examples"],"downwards":["Batting line-up of a baseball or cricket team","Seats and students of a classroom"]}
{"upwards":["Bijection","Examples","Batting line-up of a baseball or cricket team"],"downwards":[]}
{"upwards":["Bijection","Examples","Seats and students of a classroom"],"downwards":[]}
{"upwards":["Bijection","More mathematical examples"],"downwards":[]}
{"upwards":["Bijection","Inverses"],"downwards":[]}
{"upwards":["Bijection","Composition"],"downwards":[]}
{"upwards":["Bijection","Cardinality"],"downwards":[]}
{"upwards":["Bijection","Properties"],"downwards":[]}
{"upwards":["Bijection","Category theory"],"downwards":[]}
{"upwards":["Bijection","Generalization to partial functions"],"downwards":[]}
{"upwards":["Bijection","Gallery"],"downwards":[]}
{"upwards":["Complex number"],"downwards":["Definition","Notation","Visualization","History","Relations and operations","Properties","Formal construction","Complex analysis","Applications","Generalizations and related notions"]}
{"upwards":["Complex number","Definition"],"downwards":[]}
{"upwards":["Complex number","Notation"],"downwards":[]}
{"upwards":["Complex number","Visualization"],"downwards":["Cartesian complex plane","Polar complex plane","Complex graphs"]}
{"upwards":["Complex number","Visualization","Cartesian complex plane"],"downwards":[]}
{"upwards":["Complex number","Visualization","Polar complex plane"],"downwards":["Modulus and argument"]}
{"upwards":["Complex number","Visualization","Polar complex plane","Modulus and argument"],"downwards":[]}
{"upwards":["Complex number","Visualization","Complex graphs"],"downwards":[]}
{"upwards":["Complex number","History"],"downwards":[]}
{"upwards":["Complex number","Relations and operations"],"downwards":["Equality","Ordering","Conjugate","Addition and subtraction","Multiplication and square","Reciprocal and division","Multiplication and division in polar form","Square root","Exponential function","Complex logarithm","Exponentiation"]}
{"upwards":["Complex number","Relations and operations","Equality"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Ordering"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Conjugate"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and square"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Reciprocal and division"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and division in polar form"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Square root"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function"],"downwards":["Functional equation","Euler's formula"]}
{"upwards":["Complex number","Relations and operations","Exponential function","Functional equation"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function","Euler's formula"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Complex logarithm"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponentiation"],"downwards":["Integer and fractional exponents"]}
{"upwards":["Complex number","Relations and operations","Exponentiation","Integer and fractional exponents"],"downwards":[]}
{"upwards":["Complex number","Properties"],"downwards":["Field structure","Solutions of polynomial equations","Algebraic characterization","Characterization as a topological field"]}
{"upwards":["Complex number","Properties","Field structure"],"downwards":[]}
{"upwards":["Complex number","Properties","Solutions of polynomial equations"],"downwards":[]}
{"upwards":["Complex number","Properties","Algebraic characterization"],"downwards":[]}
{"upwards":["Complex number","Properties","Characterization as a topological field"],"downwards":[]}
{"upwards":["Complex number","Formal construction"],"downwards":["Construction as ordered pairs","Construction as a quotient field","Matrix representation of complex numbers"]}
{"upwards":["Complex number","Formal construction","Construction as ordered pairs"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Construction as a quotient field"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Matrix representation of complex numbers"],"downwards":[]}
{"upwards":["Complex number","Complex analysis"],"downwards":["Complex exponential and related functions","Holomorphic functions"]}
{"upwards":["Complex number","Complex analysis","Complex exponential and related functions"],"downwards":[]}
{"upwards":["Complex number","Complex analysis","Holomorphic functions"],"downwards":[]}
{"upwards":["Complex number","Applications"],"downwards":["Geometry","Algebraic number theory","Analytic number theory","Improper integrals","Dynamic equations","Linear algebra","In applied mathematics","In physics"]}
{"upwards":["Complex number","Applications","Geometry"],"downwards":["Shapes","Fractal geometry","Triangles"]}
{"upwards":["Complex number","Applications","Geometry","Shapes"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Fractal geometry"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Triangles"],"downwards":[]}
{"upwards":["Complex number","Applications","Algebraic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Analytic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Improper integrals"],"downwards":[]}
{"upwards":["Complex number","Applications","Dynamic equations"],"downwards":[]}
{"upwards":["Complex number","Applications","Linear algebra"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics"],"downwards":["Control theory","Signal analysis"]}
{"upwards":["Complex number","Applications","In applied mathematics","Control theory"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics","Signal analysis"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics"],"downwards":["Electromagnetism and electrical engineering","Fluid dynamics","Quantum mechanics","Relativity"]}
{"upwards":["Complex number","Applications","In physics","Electromagnetism and electrical engineering"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Fluid dynamics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Quantum mechanics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Relativity"],"downwards":[]}
{"upwards":["Complex number","Generalizations and related notions"],"downwards":[]}
{"upwards":["Uncountable set"],"downwards":["Characterizations","Properties","Examples","Without the axiom of choice"]}
{"upwards":["Uncountable set","Characterizations"],"downwards":[]}
{"upwards":["Uncountable set","Properties"],"downwards":[]}
{"upwards":["Uncountable set","Examples"],"downwards":[]}
{"upwards":["Uncountable set","Without the axiom of choice"],"downwards":[]}
{"upwards":["Symmetric difference"],"downwards":["Properties","n-ary symmetric difference","Symmetric difference on measure spaces","Hausdorff distance vs. symmetric difference"]}
{"upwards":["Symmetric difference","Properties"],"downwards":[]}
{"upwards":["Symmetric difference","n-ary symmetric difference"],"downwards":[]}
{"upwards":["Symmetric difference","Symmetric difference on measure spaces"],"downwards":[]}
{"upwards":["Symmetric difference","Hausdorff distance vs. symmetric difference"],"downwards":[]}
{"upwards":["Venn diagram"],"downwards":["Details","Example","History","Overview","Extensions to higher numbers of sets","Related concepts"]}
{"upwards":["Venn diagram","Details"],"downwards":[]}
{"upwards":["Venn diagram","Example"],"downwards":[]}
{"upwards":["Venn diagram","History"],"downwards":[]}
{"upwards":["Venn diagram","Overview"],"downwards":[]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets"],"downwards":["Edwards\u2013Venn diagrams","Other diagrams"]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets","Edwards\u2013Venn diagrams"],"downwards":[]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets","Other diagrams"],"downwards":[]}
{"upwards":["Venn diagram","Related concepts"],"downwards":[]}
{"upwards":["Partition of a set"],"downwards":["Definition and notation","Examples","Partitions and equivalence relations","Refinement of partitions","Noncrossing partitions","Counting partitions"]}
{"upwards":["Partition of a set","Definition and notation"],"downwards":[]}
{"upwards":["Partition of a set","Examples"],"downwards":[]}
{"upwards":["Partition of a set","Partitions and equivalence relations"],"downwards":[]}
{"upwards":["Partition of a set","Refinement of partitions"],"downwards":[]}
{"upwards":["Partition of a set","Noncrossing partitions"],"downwards":[]}
{"upwards":["Partition of a set","Counting partitions"],"downwards":[]}
{"upwards":["Philosophy"],"downwards":["Definitions","Historical overview","Branches of philosophy","Methods of philosophy","Applied and professional philosophy"]}
{"upwards":["Philosophy","Definitions"],"downwards":[]}
{"upwards":["Philosophy","Historical overview"],"downwards":["Western philosophy","Middle Eastern philosophy","Eastern philosophy","African philosophy","Indigenous American philosophy"]}
{"upwards":["Philosophy","Historical overview","Western philosophy"],"downwards":["Ancient era","Medieval era","Modern era"]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Ancient era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Medieval era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Modern era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy"],"downwards":["Pre-Islamic philosophy","Islamic philosophy"]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy","Pre-Islamic philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy","Islamic philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy"],"downwards":["Indian philosophy","Buddhist philosophy","East Asian philosophy"]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","Indian philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","Buddhist philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","East Asian philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","African philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Indigenous American philosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy"],"downwards":["Aesthetics","Ethics","Epistemology","Metaphysics","Logic","Mind and language","Philosophy of science","Political philosophy","Philosophy of religion","Metaphilosophy","Other subdivisions"]}
{"upwards":["Philosophy","Branches of philosophy","Aesthetics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Ethics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Epistemology"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Metaphysics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Logic"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Mind and language"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Philosophy of science"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Political philosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Philosophy of religion"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Metaphilosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Other subdivisions"],"downwards":[]}
{"upwards":["Philosophy","Methods of philosophy"],"downwards":["Philosophical progress"]}
{"upwards":["Philosophy","Methods of philosophy","Philosophical progress"],"downwards":[]}
{"upwards":["Philosophy","Applied and professional philosophy"],"downwards":["Women in philosophy"]}
{"upwards":["Philosophy","Applied and professional philosophy","Women in philosophy"],"downwards":[]}
{"upwards":["Real number"],"downwards":["Characterizing properties","Arithmetic","Dedekind completeness","Decimal representation","Topological completeness","Cardinality","Other properties","History","Formal definitions","Applications and connections","Vocabulary and notation","Generalizations and extensions"]}
{"upwards":["Real number","Characterizing properties"],"downwards":[]}
{"upwards":["Real number","Arithmetic"],"downwards":["Auxiliary operations","Auxiliary order relations","Integers and fractions as real numbers"]}
{"upwards":["Real number","Arithmetic","Auxiliary operations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Auxiliary order relations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Integers and fractions as real numbers"],"downwards":[]}
{"upwards":["Real number","Dedekind completeness"],"downwards":[]}
{"upwards":["Real number","Decimal representation"],"downwards":[]}
{"upwards":["Real number","Topological completeness"],"downwards":["\"The complete ordered field\""]}
{"upwards":["Real number","Topological completeness","\"The complete ordered field\""],"downwards":[]}
{"upwards":["Real number","Cardinality"],"downwards":[]}
{"upwards":["Real number","Other properties"],"downwards":[]}
{"upwards":["Real number","History"],"downwards":[]}
{"upwards":["Real number","Formal definitions"],"downwards":["Axiomatic approach","Construction from the rational numbers"]}
{"upwards":["Real number","Formal definitions","Axiomatic approach"],"downwards":[]}
{"upwards":["Real number","Formal definitions","Construction from the rational numbers"],"downwards":[]}
{"upwards":["Real number","Applications and connections"],"downwards":["Physics","Logic","Computation","Set theory"]}
{"upwards":["Real number","Applications and connections","Physics"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Logic"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Computation"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Set theory"],"downwards":[]}
{"upwards":["Real number","Vocabulary and notation"],"downwards":[]}
{"upwards":["Real number","Generalizations and extensions"],"downwards":[]}
{"upwards":["Letter case"],"downwards":["Terminology","Typographical considerations","Bicameral script","Stylistic or specialised usage","Case folding and case conversion","History"]}
{"upwards":["Letter case","Terminology"],"downwards":[]}
{"upwards":["Letter case","Typographical considerations"],"downwards":[]}
{"upwards":["Letter case","Bicameral script"],"downwards":["Capitalisation","Exceptional letters and digraphs","Related features"]}
{"upwards":["Letter case","Bicameral script","Capitalisation"],"downwards":[]}
{"upwards":["Letter case","Bicameral script","Exceptional letters and digraphs"],"downwards":[]}
{"upwards":["Letter case","Bicameral script","Related features"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage"],"downwards":["Headings and publication titles","Multi-word proper nouns","Unit symbols and prefixes in the metric system","Use within programming languages","Studly caps"]}
{"upwards":["Letter case","Stylistic or specialised usage","Headings and publication titles"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Multi-word proper nouns"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Unit symbols and prefixes in the metric system"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages"],"downwards":["Camel case","Snake case","Kebab case"]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages","Camel case"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages","Snake case"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages","Kebab case"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Studly caps"],"downwards":[]}
{"upwards":["Letter case","Case folding and case conversion"],"downwards":["Unicode case folding and script identification","Methods in word processing","Methods in programming"]}
{"upwards":["Letter case","Case folding and case conversion","Unicode case folding and script identification"],"downwards":[]}
{"upwards":["Letter case","Case folding and case conversion","Methods in word processing"],"downwards":[]}
{"upwards":["Letter case","Case folding and case conversion","Methods in programming"],"downwards":[]}
{"upwards":["Letter case","History"],"downwards":["Type cases"]}
{"upwards":["Letter case","History","Type cases"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
{"upwards":["Fraction","Vocabulary"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions"],"downwards":["Simple, common, or vulgar fractions","Proper and improper fractions","Reciprocals and the \"invisible denominator\"","Ratios","Decimal fractions and percentages","Mixed numbers","Historical notions"]}
{"upwards":["Fraction","Forms of fractions","Simple, common, or vulgar fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Proper and improper fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Reciprocals and the \"invisible denominator\""],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Ratios"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Decimal fractions and percentages"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions"],"downwards":["Egyptian fraction","Complex and compound fractions"]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Egyptian fraction"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Complex and compound fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions"],"downwards":["Equivalent fractions","Comparing fractions","Addition","Subtraction","Multiplication","Division","Converting between decimals and fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions"],"downwards":["Simplifying (reducing) fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions","Simplifying (reducing) fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Comparing fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Addition"],"downwards":["Adding unlike quantities"]}
{"upwards":["Fraction","Arithmetic with fractions","Addition","Adding unlike quantities"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Subtraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication"],"downwards":["Multiplying a fraction by another fraction","Multiplying a fraction by a whole number","Multiplying mixed numbers"]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by another fraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by a whole number"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Division"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions"],"downwards":["Converting repeating decimals to fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
{"upwards":["Fraction","Algebraic fractions"],"downwards":[]}
{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Georg Cantor"],"downwards":["Biography","Mathematical work","Philosophy, religion, literature and Cantor's mathematics","Cantor's ancestry","Biographies"]}
{"upwards":["Georg Cantor","Biography"],"downwards":["Youth and studies","Teacher and researcher","Later years and death"]}
{"upwards":["Georg Cantor","Biography","Youth and studies"],"downwards":[]}
{"upwards":["Georg Cantor","Biography","Teacher and researcher"],"downwards":[]}
{"upwards":["Georg Cantor","Biography","Later years and death"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work"],"downwards":["Number theory, trigonometric series and ordinals","Set theory"]}
{"upwards":["Georg Cantor","Mathematical work","Number theory, trigonometric series and ordinals"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory"],"downwards":["One-to-one correspondence","Continuum hypothesis","Absolute infinite, well-ordering theorem, and paradoxes"]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","One-to-one correspondence"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","Continuum hypothesis"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","Absolute infinite, well-ordering theorem, and paradoxes"],"downwards":[]}
{"upwards":["Georg Cantor","Philosophy, religion, literature and Cantor's mathematics"],"downwards":[]}
{"upwards":["Georg Cantor","Cantor's ancestry"],"downwards":[]}
{"upwards":["Georg Cantor","Biographies"],"downwards":[]}
{"upwards":["First-order logic"],"downwards":["Introduction","Syntax","Semantics","Deductive systems","Equality and its axioms","Metalogical properties","Limitations","Restrictions, extensions, and variations","Automated theorem proving and formal methods"]}
{"upwards":["First-order logic","Introduction"],"downwards":[]}
{"upwards":["First-order logic","Syntax"],"downwards":["Alphabet","Formation rules","Free and bound variables","Example: ordered abelian groups"]}
{"upwards":["First-order logic","Syntax","Alphabet"],"downwards":["Logical symbols","Non-logical symbols"]}
{"upwards":["First-order logic","Syntax","Alphabet","Logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Alphabet","Non-logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules"],"downwards":["Terms","Formulas","Notational conventions"]}
{"upwards":["First-order logic","Syntax","Formation rules","Terms"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Formulas"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Notational conventions"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Free and bound variables"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Example: ordered abelian groups"],"downwards":[]}
{"upwards":["First-order logic","Semantics"],"downwards":["First-order structures","Evaluation of truth values","Validity, satisfiability, and logical consequence","Algebraizations","First-order theories, models, and elementary classes","Empty domains"]}
{"upwards":["First-order logic","Semantics","First-order structures"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Evaluation of truth values"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Validity, satisfiability, and logical consequence"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Algebraizations"],"downwards":[]}
{"upwards":["First-order logic","Semantics","First-order theories, models, and elementary classes"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Empty domains"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems"],"downwards":["Rules of inference","Hilbert-style systems and natural deduction","Sequent calculus","Tableaux method","Resolution","Provable identities"]}
{"upwards":["First-order logic","Deductive systems","Rules of inference"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Hilbert-style systems and natural deduction"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Sequent calculus"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Tableaux method"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Resolution"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Provable identities"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms"],"downwards":["First-order logic without equality","Defining equality within a theory"]}
{"upwards":["First-order logic","Equality and its axioms","First-order logic without equality"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms","Defining equality within a theory"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties"],"downwards":["Completeness and undecidability","The L\u00f6wenheim\u2013Skolem theorem","The compactness theorem","Lindstr\u00f6m's theorem"]}
{"upwards":["First-order logic","Metalogical properties","Completeness and undecidability"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The compactness theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","Lindstr\u00f6m's theorem"],"downwards":[]}
{"upwards":["First-order logic","Limitations"],"downwards":["Expressiveness","Formalizing natural languages"]}
{"upwards":["First-order logic","Limitations","Expressiveness"],"downwards":[]}
{"upwards":["First-order logic","Limitations","Formalizing natural languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations"],"downwards":["Restricted languages","Many-sorted logic","Additional quantifiers","Infinitary logics","Non-classical and modal logics","Fixpoint logic","Higher-order logics"]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Restricted languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Many-sorted logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Additional quantifiers"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Infinitary logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Non-classical and modal logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Fixpoint logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Higher-order logics"],"downwards":[]}
{"upwards":["First-order logic","Automated theorem proving and formal methods"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
{"upwards":["Fraction","Vocabulary"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions"],"downwards":["Simple, common, or vulgar fractions","Proper and improper fractions","Reciprocals and the \"invisible denominator\"","Ratios","Decimal fractions and percentages","Mixed numbers","Historical notions"]}
{"upwards":["Fraction","Forms of fractions","Simple, common, or vulgar fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Proper and improper fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Reciprocals and the \"invisible denominator\""],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Ratios"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Decimal fractions and percentages"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions"],"downwards":["Egyptian fraction","Complex and compound fractions"]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Egyptian fraction"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Complex and compound fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions"],"downwards":["Equivalent fractions","Comparing fractions","Addition","Subtraction","Multiplication","Division","Converting between decimals and fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions"],"downwards":["Simplifying (reducing) fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions","Simplifying (reducing) fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Comparing fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Addition"],"downwards":["Adding unlike quantities"]}
{"upwards":["Fraction","Arithmetic with fractions","Addition","Adding unlike quantities"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Subtraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication"],"downwards":["Multiplying a fraction by another fraction","Multiplying a fraction by a whole number","Multiplying mixed numbers"]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by another fraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by a whole number"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Division"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions"],"downwards":["Converting repeating decimals to fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
{"upwards":["Fraction","Algebraic fractions"],"downwards":[]}
{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Ostensive definition"],"downwards":["Overview"]}
{"upwards":["Ostensive definition","Overview"],"downwards":[]}
{"upwards":["Complex number"],"downwards":["Definition","Notation","Visualization","History","Relations and operations","Properties","Formal construction","Complex analysis","Applications","Generalizations and related notions"]}
{"upwards":["Complex number","Definition"],"downwards":[]}
{"upwards":["Complex number","Notation"],"downwards":[]}
{"upwards":["Complex number","Visualization"],"downwards":["Cartesian complex plane","Polar complex plane","Complex graphs"]}
{"upwards":["Complex number","Visualization","Cartesian complex plane"],"downwards":[]}
{"upwards":["Complex number","Visualization","Polar complex plane"],"downwards":["Modulus and argument"]}
{"upwards":["Complex number","Visualization","Polar complex plane","Modulus and argument"],"downwards":[]}
{"upwards":["Complex number","Visualization","Complex graphs"],"downwards":[]}
{"upwards":["Complex number","History"],"downwards":[]}
{"upwards":["Complex number","Relations and operations"],"downwards":["Equality","Ordering","Conjugate","Addition and subtraction","Multiplication and square","Reciprocal and division","Multiplication and division in polar form","Square root","Exponential function","Complex logarithm","Exponentiation"]}
{"upwards":["Complex number","Relations and operations","Equality"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Ordering"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Conjugate"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and square"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Reciprocal and division"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and division in polar form"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Square root"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function"],"downwards":["Functional equation","Euler's formula"]}
{"upwards":["Complex number","Relations and operations","Exponential function","Functional equation"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function","Euler's formula"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Complex logarithm"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponentiation"],"downwards":["Integer and fractional exponents"]}
{"upwards":["Complex number","Relations and operations","Exponentiation","Integer and fractional exponents"],"downwards":[]}
{"upwards":["Complex number","Properties"],"downwards":["Field structure","Solutions of polynomial equations","Algebraic characterization","Characterization as a topological field"]}
{"upwards":["Complex number","Properties","Field structure"],"downwards":[]}
{"upwards":["Complex number","Properties","Solutions of polynomial equations"],"downwards":[]}
{"upwards":["Complex number","Properties","Algebraic characterization"],"downwards":[]}
{"upwards":["Complex number","Properties","Characterization as a topological field"],"downwards":[]}
{"upwards":["Complex number","Formal construction"],"downwards":["Construction as ordered pairs","Construction as a quotient field","Matrix representation of complex numbers"]}
{"upwards":["Complex number","Formal construction","Construction as ordered pairs"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Construction as a quotient field"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Matrix representation of complex numbers"],"downwards":[]}
{"upwards":["Complex number","Complex analysis"],"downwards":["Complex exponential and related functions","Holomorphic functions"]}
{"upwards":["Complex number","Complex analysis","Complex exponential and related functions"],"downwards":[]}
{"upwards":["Complex number","Complex analysis","Holomorphic functions"],"downwards":[]}
{"upwards":["Complex number","Applications"],"downwards":["Geometry","Algebraic number theory","Analytic number theory","Improper integrals","Dynamic equations","Linear algebra","In applied mathematics","In physics"]}
{"upwards":["Complex number","Applications","Geometry"],"downwards":["Shapes","Fractal geometry","Triangles"]}
{"upwards":["Complex number","Applications","Geometry","Shapes"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Fractal geometry"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Triangles"],"downwards":[]}
{"upwards":["Complex number","Applications","Algebraic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Analytic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Improper integrals"],"downwards":[]}
{"upwards":["Complex number","Applications","Dynamic equations"],"downwards":[]}
{"upwards":["Complex number","Applications","Linear algebra"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics"],"downwards":["Control theory","Signal analysis"]}
{"upwards":["Complex number","Applications","In applied mathematics","Control theory"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics","Signal analysis"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics"],"downwards":["Electromagnetism and electrical engineering","Fluid dynamics","Quantum mechanics","Relativity"]}
{"upwards":["Complex number","Applications","In physics","Electromagnetism and electrical engineering"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Fluid dynamics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Quantum mechanics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Relativity"],"downwards":[]}
{"upwards":["Complex number","Generalizations and related notions"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
{"upwards":["Subset","Definition"],"downwards":[]}
{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["Vertical bar"],"downwards":["Usage","Encoding"]}
{"upwards":["Vertical bar","Usage"],"downwards":["Mathematics","Physics","Computing","Phonetics and orthography","Literature"]}
{"upwards":["Vertical bar","Usage","Mathematics"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Physics"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing"],"downwards":["Pipe","Disjunction","Concatenation","Delimiter","Backus\u2013Naur form","Concurrency operator","APL","List comprehensions","Text markup"]}
{"upwards":["Vertical bar","Usage","Computing","Pipe"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Disjunction"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Concatenation"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Delimiter"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Backus\u2013Naur form"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Concurrency operator"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","APL"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","List comprehensions"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Text markup"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Phonetics and orthography"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature"],"downwards":["Punctuation","Poetry","Notation","Music scoring"]}
{"upwards":["Vertical bar","Usage","Literature","Punctuation"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature","Poetry"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature","Notation"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature","Music scoring"],"downwards":[]}
{"upwards":["Vertical bar","Encoding"],"downwards":["Solid vertical bar vs broken bar","Unicode code points","Code pages and other historical encodings"]}
{"upwards":["Vertical bar","Encoding","Solid vertical bar vs broken bar"],"downwards":[]}
{"upwards":["Vertical bar","Encoding","Unicode code points"],"downwards":[]}
{"upwards":["Vertical bar","Encoding","Code pages and other historical encodings"],"downwards":[]}
{"upwards":["Algebraic structure"],"downwards":["Introduction","Common axioms","Common algebraic structures","Hybrid structures","Universal algebra","Category theory","Different meanings of \"structure\""]}
{"upwards":["Algebraic structure","Introduction"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms"],"downwards":["Equational axioms","Existential axioms","Non-equational axioms"]}
{"upwards":["Algebraic structure","Common axioms","Equational axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms","Existential axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms","Non-equational axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common algebraic structures"],"downwards":["One set with operations","Two sets with operations"]}
{"upwards":["Algebraic structure","Common algebraic structures","One set with operations"],"downwards":[]}
{"upwards":["Algebraic structure","Common algebraic structures","Two sets with operations"],"downwards":[]}
{"upwards":["Algebraic structure","Hybrid structures"],"downwards":[]}
{"upwards":["Algebraic structure","Universal algebra"],"downwards":[]}
{"upwards":["Algebraic structure","Category theory"],"downwards":[]}
{"upwards":["Algebraic structure","Different meanings of \"structure\""],"downwards":[]}
{"upwards":["Abstract algebra"],"downwards":["History","Basic concepts","Branches of abstract algebra","Applications"]}
{"upwards":["Abstract algebra","History"],"downwards":["Elementary algebra","Early group theory","Early ring theory","Early field theory","Other major areas","Modern algebra"]}
{"upwards":["Abstract algebra","History","Elementary algebra"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early group theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early ring theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early field theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Other major areas"],"downwards":[]}
{"upwards":["Abstract algebra","History","Modern algebra"],"downwards":[]}
{"upwards":["Abstract algebra","Basic concepts"],"downwards":[]}
{"upwards":["Abstract algebra","Branches of abstract algebra"],"downwards":["Group theory","Ring theory"]}
{"upwards":["Abstract algebra","Branches of abstract algebra","Group theory"],"downwards":[]}
{"upwards":["Abstract algebra","Branches of abstract algebra","Ring theory"],"downwards":[]}
{"upwards":["Abstract algebra","Applications"],"downwards":[]}
{"upwards":["Group (mathematics)"],"downwards":["Definition and illustration","History","Elementary consequences of the group axioms","Basic concepts","Examples and applications","Finite groups","Groups with additional structure","Generalizations"]}
{"upwards":["Group (mathematics)","Definition and illustration"],"downwards":["First example: the integers","Definition","Notation and terminology","Second example: a symmetry group"]}
{"upwards":["Group (mathematics)","Definition and illustration","First example: the integers"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Definition"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Notation and terminology"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Second example: a symmetry group"],"downwards":[]}
{"upwards":["Group (mathematics)","History"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms"],"downwards":["Uniqueness of identity element","Uniqueness of inverses","Division","Equivalent definition with relaxed axioms"]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Uniqueness of identity element"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Uniqueness of inverses"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Division"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Equivalent definition with relaxed axioms"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts"],"downwards":["Group homomorphisms","Subgroups","Cosets","Quotient groups","Presentations"]}
{"upwards":["Group (mathematics)","Basic concepts","Group homomorphisms"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Subgroups"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Cosets"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Quotient groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Presentations"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications"],"downwards":["Numbers","Modular arithmetic","Cyclic groups","Symmetry groups","General linear group and representation theory","Galois groups"]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers"],"downwards":["Integers","Rationals"]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers","Integers"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers","Rationals"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Modular arithmetic"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Cyclic groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Symmetry groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","General linear group and representation theory"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Galois groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups"],"downwards":["Finite abelian groups","Simple groups","Classification of finite simple groups"]}
{"upwards":["Group (mathematics)","Finite groups","Finite abelian groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups","Simple groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups","Classification of finite simple groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Groups with additional structure"],"downwards":["Topological groups","Lie groups"]}
{"upwards":["Group (mathematics)","Groups with additional structure","Topological groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Groups with additional structure","Lie groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Field (mathematics)"],"downwards":["Definition","Examples","Elementary notions","Finite fields","History","Constructing fields","Fields with additional structure","Galois theory","Invariants of fields","Applications","Related notions","Notes","References","External links"]}
{"upwards":["Field (mathematics)","Definition"],"downwards":["Classic definition","Alternative definition"]}
{"upwards":["Field (mathematics)","Definition","Classic definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Definition","Alternative definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples"],"downwards":["Rational numbers","Real and complex numbers","Constructible numbers","A field with four elements"]}
{"upwards":["Field (mathematics)","Examples","Rational numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","Real and complex numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","Constructible numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","A field with four elements"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions"],"downwards":["Consequences of the definition","The additive and the multiplicative group of a field","Characteristic","Subfields and prime fields"]}
{"upwards":["Field (mathematics)","Elementary notions","Consequences of the definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","The additive and the multiplicative group of a field"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","Characteristic"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","Subfields and prime fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Finite fields"],"downwards":[]}
{"upwards":["Field (mathematics)","History"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields"],"downwards":["Constructing fields from rings","Constructing fields within a bigger field","Field extensions","Closure operations"]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings"],"downwards":["Field of fractions","Residue fields"]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings","Field of fractions"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings","Residue fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields within a bigger field"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions"],"downwards":["Algebraic extensions","Transcendence bases"]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions","Algebraic extensions"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions","Transcendence bases"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Closure operations"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure"],"downwards":["Ordered fields","Topological fields","Differential fields"]}
{"upwards":["Field (mathematics)","Fields with additional structure","Ordered fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure","Topological fields"],"downwards":["Local fields"]}
{"upwards":["Field (mathematics)","Fields with additional structure","Topological fields","Local fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure","Differential fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Galois theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields"],"downwards":["Model theory of fields","The absolute Galois group","K-theory"]}
{"upwards":["Field (mathematics)","Invariants of fields","Model theory of fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields","The absolute Galois group"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields","K-theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications"],"downwards":["Linear algebra and commutative algebra","Finite fields: cryptography and coding theory","Geometry: field of functions","Number theory: global fields"]}
{"upwards":["Field (mathematics)","Applications","Linear algebra and commutative algebra"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Finite fields: cryptography and coding theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Geometry: field of functions"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Number theory: global fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Related notions"],"downwards":["Division rings"]}
{"upwards":["Field (mathematics)","Related notions","Division rings"],"downwards":[]}
{"upwards":["Field (mathematics)","Notes"],"downwards":[]}
{"upwards":["Field (mathematics)","References"],"downwards":[]}
{"upwards":["Field (mathematics)","External links"],"downwards":[]}
{"upwards":["Ring (mathematics)"],"downwards":["Definition","Illustration","History","Basic examples","Basic concepts","Module","Constructions","Special kinds of rings","Rings with extra structure","Some examples of the ubiquity of rings","Category-theoretic description","Generalization","Other ring-like objects"]}
{"upwards":["Ring (mathematics)","Definition"],"downwards":["Notes on the definition"]}
{"upwards":["Ring (mathematics)","Definition","Notes on the definition"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration"],"downwards":["Some properties","Example: Integers modulo 4","Example: 2-by-2 matrices"]}
{"upwards":["Ring (mathematics)","Illustration","Some properties"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration","Example: Integers modulo 4"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration","Example: 2-by-2 matrices"],"downwards":[]}
{"upwards":["Ring (mathematics)","History"],"downwards":["Dedekind","Hilbert","Fraenkel and Noether","Multiplicative identity and the term \"ring\""]}
{"upwards":["Ring (mathematics)","History","Dedekind"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Hilbert"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Fraenkel and Noether"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Multiplicative identity and the term \"ring\""],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples"],"downwards":["Commutative rings","Noncommutative rings","Non-rings"]}
{"upwards":["Ring (mathematics)","Basic examples","Commutative rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples","Noncommutative rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples","Non-rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts"],"downwards":["Products and powers","Elements in a ring","Subring","Ideal","Homomorphism","Quotient ring"]}
{"upwards":["Ring (mathematics)","Basic concepts","Products and powers"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Elements in a ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Subring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Ideal"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Homomorphism"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Quotient ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Module"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions"],"downwards":["Direct product","Polynomial ring","Matrix ring and endomorphism ring","Limits and colimits of rings","Localization","Completion","Rings with generators and relations"]}
{"upwards":["Ring (mathematics)","Constructions","Direct product"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Polynomial ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Matrix ring and endomorphism ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Limits and colimits of rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Localization"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Completion"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Rings with generators and relations"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings"],"downwards":["Domains","Division ring","Semisimple rings","Central simple algebra and Brauer group","Valuation ring"]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Domains"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Division ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings"],"downwards":["Examples","Properties"]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings","Examples"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings","Properties"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Central simple algebra and Brauer group"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Valuation ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Rings with extra structure"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings"],"downwards":["Cohomology ring of a topological space","Burnside ring of a group","Representation ring of a group ring","Function field of an irreducible algebraic variety","Face ring of a simplicial complex"]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Cohomology ring of a topological space"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Burnside ring of a group"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Representation ring of a group ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Function field of an irreducible algebraic variety"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Face ring of a simplicial complex"],"downwards":[]}
{"upwards":["Ring (mathematics)","Category-theoretic description"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization"],"downwards":["Rng","Nonassociative ring","Semiring"]}
{"upwards":["Ring (mathematics)","Generalization","Rng"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization","Nonassociative ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization","Semiring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects"],"downwards":["Ring object in a category","Ring scheme","Ring spectrum"]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring object in a category"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring scheme"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring spectrum"],"downwards":[]}
{"upwards":["Closure (mathematics)"],"downwards":["Definitions","In topology","Binary relations","Other examples","Closure operator","Notes","References"]}
{"upwards":["Closure (mathematics)","Definitions"],"downwards":["In algebraic structures"]}
{"upwards":["Closure (mathematics)","Definitions","In algebraic structures"],"downwards":[]}
{"upwards":["Closure (mathematics)","In topology"],"downwards":[]}
{"upwards":["Closure (mathematics)","Binary relations"],"downwards":[]}
{"upwards":["Closure (mathematics)","Other examples"],"downwards":[]}
{"upwards":["Closure (mathematics)","Closure operator"],"downwards":["Closure operator vs. closed sets"]}
{"upwards":["Closure (mathematics)","Closure operator","Closure operator vs. closed sets"],"downwards":[]}
{"upwards":["Closure (mathematics)","Notes"],"downwards":[]}
{"upwards":["Closure (mathematics)","References"],"downwards":[]}
{"upwards":["Domain of a function"],"downwards":["Natural domain","Other uses","Set theoretical notions"]}
{"upwards":["Domain of a function","Natural domain"],"downwards":["Examples"]}
{"upwards":["Domain of a function","Natural domain","Examples"],"downwards":[]}
{"upwards":["Domain of a function","Other uses"],"downwards":[]}
{"upwards":["Domain of a function","Set theoretical notions"],"downwards":[]}
{"upwards":["Bertrand Russell"],"downwards":["Biography","Views","Selected works"]}
{"upwards":["Bertrand Russell","Biography"],"downwards":["Early life and background","Childhood and adolescence","University and first marriage","Early career","First World War","G. H. Hardy on the Trinity controversy","Between the wars","Second World War","Later life","Political causes","Final years, death and legacy","Titles and honours from birth"]}
{"upwards":["Bertrand Russell","Biography","Early life and background"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Childhood and adolescence"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","University and first marriage"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Early career"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","First World War"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","G. H. Hardy on the Trinity controversy"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Between the wars"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Second World War"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Later life"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Political causes"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Final years, death and legacy"],"downwards":["Marriages and issue"]}
{"upwards":["Bertrand Russell","Biography","Final years, death and legacy","Marriages and issue"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Titles and honours from birth"],"downwards":[]}
{"upwards":["Bertrand Russell","Views"],"downwards":["Philosophy","Religion","Society","Freedom of opinion and expression","Education"]}
{"upwards":["Bertrand Russell","Views","Philosophy"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Religion"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Society"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Freedom of opinion and expression"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Education"],"downwards":[]}
{"upwards":["Bertrand Russell","Selected works"],"downwards":[]}
{"upwards":["Codomain"],"downwards":["Examples"]}
{"upwards":["Codomain","Examples"],"downwards":[]}
{"upwards":["Rock paper scissors"],"downwards":["Etymology","Gameplay","History","Strategies","Variations","Analogues in real life","Instances of usage","Tournaments"]}
{"upwards":["Rock paper scissors","Etymology"],"downwards":[]}
{"upwards":["Rock paper scissors","Gameplay"],"downwards":[]}
{"upwards":["Rock paper scissors","History"],"downwards":["Origins","Spread beyond East Asia"]}
{"upwards":["Rock paper scissors","History","Origins"],"downwards":[]}
{"upwards":["Rock paper scissors","History","Spread beyond East Asia"],"downwards":[]}
{"upwards":["Rock paper scissors","Strategies"],"downwards":["Algorithms"]}
{"upwards":["Rock paper scissors","Strategies","Algorithms"],"downwards":[]}
{"upwards":["Rock paper scissors","Variations"],"downwards":["Adapted rules","Different weapons","Additional weapons"]}
{"upwards":["Rock paper scissors","Variations","Adapted rules"],"downwards":[]}
{"upwards":["Rock paper scissors","Variations","Different weapons"],"downwards":[]}
{"upwards":["Rock paper scissors","Variations","Additional weapons"],"downwards":[]}
{"upwards":["Rock paper scissors","Analogues in real life"],"downwards":["Lizard mating strategies","Bacteria","Mechanical devices and geometrical constructions"]}
{"upwards":["Rock paper scissors","Analogues in real life","Lizard mating strategies"],"downwards":[]}
{"upwards":["Rock paper scissors","Analogues in real life","Bacteria"],"downwards":[]}
{"upwards":["Rock paper scissors","Analogues in real life","Mechanical devices and geometrical constructions"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage"],"downwards":["American court case","Auction house selection","FA Women's Super League match","Play by chimpanzees","Game design"]}
{"upwards":["Rock paper scissors","Instances of usage","American court case"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","Auction house selection"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","FA Women's Super League match"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","Play by chimpanzees"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","Game design"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments"],"downwards":["World Rock Paper Scissors Association","World Rock Paper Scissors Society","UK championships","USARPS tournaments","National XtremeRPS Competition 2007\u20132008","Guinness Book of World Records","World Series","Jackpot En Poy of Eat Bulaga!"]}
{"upwards":["Rock paper scissors","Tournaments","World Rock Paper Scissors Association"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","World Rock Paper Scissors Society"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","UK championships"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","USARPS tournaments"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","National XtremeRPS Competition 2007\u20132008"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","Guinness Book of World Records"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","World Series"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","Jackpot En Poy of Eat Bulaga!"],"downwards":[]}
{"upwards":["Alternative set theory"],"downwards":["Vop\u011bnka's Alternative Set Theory","Other alternative set theories"]}
{"upwards":["Alternative set theory","Vop\u011bnka's Alternative Set Theory"],"downwards":[]}
{"upwards":["Alternative set theory","Other alternative set theories"],"downwards":[]}
{"upwards":["Category of sets"],"downwards":["Properties of the category of sets","Foundations for the category of sets"]}
{"upwards":["Category of sets","Properties of the category of sets"],"downwards":[]}
{"upwards":["Category of sets","Foundations for the category of sets"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle"],"downwards":["Formula","Examples","A special case","Formula generalization","In probability","Other formulas","Applications","Diluted inclusion\u2013exclusion principle","Proof of main statement"]}
{"upwards":["Inclusion\u2013exclusion principle","Formula"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Examples"],"downwards":["Counting integers","Counting derangements"]}
{"upwards":["Inclusion\u2013exclusion principle","Examples","Counting integers"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Examples","Counting derangements"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","A special case"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Formula generalization"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","In probability"],"downwards":["Special case"]}
{"upwards":["Inclusion\u2013exclusion principle","In probability","Special case"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Other formulas"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications"],"downwards":["Counting derangements","Counting intersections","Graph coloring","Bipartite graph perfect matchings","Number of onto functions","Permutations with forbidden positions","Stirling numbers of the second kind","Rook polynomials","Euler's phi function"]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Counting derangements"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Counting intersections"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Graph coloring"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Bipartite graph perfect matchings"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Number of onto functions"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Permutations with forbidden positions"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Stirling numbers of the second kind"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Rook polynomials"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Euler's phi function"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Diluted inclusion\u2013exclusion principle"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Proof of main statement"],"downwards":["Algebraic proof"]}
{"upwards":["Inclusion\u2013exclusion principle","Proof of main statement","Algebraic proof"],"downwards":[]}
{"upwards":["Dense set"],"downwards":["Definition","Examples","Properties","Related notions"]}
{"upwards":["Dense set","Definition"],"downwards":["Density in metric spaces"]}
{"upwards":["Dense set","Definition","Density in metric spaces"],"downwards":[]}
{"upwards":["Dense set","Examples"],"downwards":[]}
{"upwards":["Dense set","Properties"],"downwards":[]}
{"upwards":["Dense set","Related notions"],"downwards":[]}
{"upwards":["Family of sets"],"downwards":["Examples","Properties","Related concepts","Special types of set families"]}
{"upwards":["Family of sets","Examples"],"downwards":[]}
{"upwards":["Family of sets","Properties"],"downwards":[]}
{"upwards":["Family of sets","Related concepts"],"downwards":["Covers and topologies"]}
{"upwards":["Family of sets","Related concepts","Covers and topologies"],"downwards":[]}
{"upwards":["Family of sets","Special types of set families"],"downwards":[]}
{"upwards":["Fuzzy set"],"downwards":["Definition","Fuzzy logic","Fuzzy number and only number","Fuzzy categories","Fuzzy relation equation","Entropy","Extensions"]}
{"upwards":["Fuzzy set","Definition"],"downwards":["Crisp sets related to a fuzzy set","Other definitions","Fuzzy set operations","Disjoint fuzzy sets","Scalar cardinality","Distance and similarity","L-fuzzy sets","Neutrosophic fuzzy sets","Pythagorean fuzzy sets"]}
{"upwards":["Fuzzy set","Definition","Crisp sets related to a fuzzy set"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Other definitions"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Fuzzy set operations"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Disjoint fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Scalar cardinality"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Distance and similarity"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","L-fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Neutrosophic fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Pythagorean fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy logic"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy number and only number"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy categories"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy relation equation"],"downwards":[]}
{"upwards":["Fuzzy set","Entropy"],"downwards":[]}
{"upwards":["Fuzzy set","Extensions"],"downwards":[]}
{"upwards":["Internal set"],"downwards":["Internal sets in the ultrapower construction","Internal subsets of the reals"]}
{"upwards":["Internal set","Internal sets in the ultrapower construction"],"downwards":[]}
{"upwards":["Internal set","Internal subsets of the reals"],"downwards":[]}
{"upwards":["Mereology"],"downwards":["History","Axioms and primitive notions","Various systems","Set theory","Mathematics","Natural language","Metaphysics","Important surveys"]}
{"upwards":["Mereology","History"],"downwards":[]}
{"upwards":["Mereology","Axioms and primitive notions"],"downwards":[]}
{"upwards":["Mereology","Various systems"],"downwards":[]}
{"upwards":["Mereology","Set theory"],"downwards":[]}
{"upwards":["Mereology","Mathematics"],"downwards":[]}
{"upwards":["Mereology","Natural language"],"downwards":[]}
{"upwards":["Mereology","Metaphysics"],"downwards":["Mereological constitution","Mereological composition"]}
{"upwards":["Mereology","Metaphysics","Mereological constitution"],"downwards":[]}
{"upwards":["Mereology","Metaphysics","Mereological composition"],"downwards":["Fundamentality","Special composition question (SCQ)"]}
{"upwards":["Mereology","Metaphysics","Mereological composition","Fundamentality"],"downwards":[]}
{"upwards":["Mereology","Metaphysics","Mereological composition","Special composition question (SCQ)"],"downwards":[]}
{"upwards":["Mereology","Important surveys"],"downwards":[]}
{"upwards":["Multiset"],"downwards":["History","Examples","Definition","Basic properties and operations","Counting multisets","Applications","Generalizations"]}
{"upwards":["Multiset","History"],"downwards":[]}
{"upwards":["Multiset","Examples"],"downwards":[]}
{"upwards":["Multiset","Definition"],"downwards":[]}
{"upwards":["Multiset","Basic properties and operations"],"downwards":[]}
{"upwards":["Multiset","Counting multisets"],"downwards":["Recurrence relation","Generating series","Generalization and connection to the negative binomial series"]}
{"upwards":["Multiset","Counting multisets","Recurrence relation"],"downwards":[]}
{"upwards":["Multiset","Counting multisets","Generating series"],"downwards":[]}
{"upwards":["Multiset","Counting multisets","Generalization and connection to the negative binomial series"],"downwards":[]}
{"upwards":["Multiset","Applications"],"downwards":[]}
{"upwards":["Multiset","Generalizations"],"downwards":[]}
{"upwards":["Principia Mathematica"],"downwards":["Scope of foundations laid","Theoretical basis","Ramified types and the axiom of reducibility","Notation","Consistency and criticisms","Contents","Comparison with set theory","Differences between editions","Editions"]}
{"upwards":["Principia Mathematica","Scope of foundations laid"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis"],"downwards":["Contemporary construction of a formal theory","Construction","Primitive ideas","Primitive propositions"]}
{"upwards":["Principia Mathematica","Theoretical basis","Contemporary construction of a formal theory"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Construction"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Primitive ideas"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Primitive propositions"],"downwards":[]}
{"upwards":["Principia Mathematica","Ramified types and the axiom of reducibility"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation"],"downwards":["An introduction to the notation of \"Section A Mathematical Logic\" (formulas \u27381\u2013\u27385.71)","An introduction to the notation of \"Section B Theory of Apparent Variables\" (formulas \u27388\u2013\u273814.34)","Introduction to the notation of the theory of classes and relations"]}
{"upwards":["Principia Mathematica","Notation","An introduction to the notation of \"Section A Mathematical Logic\" (formulas \u27381\u2013\u27385.71)"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation","An introduction to the notation of \"Section B Theory of Apparent Variables\" (formulas \u27388\u2013\u273814.34)"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation","Introduction to the notation of the theory of classes and relations"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms"],"downwards":["G\u00f6del 1930, 1931","Wittgenstein 1919, 1939","G\u00f6del 1944"]}
{"upwards":["Principia Mathematica","Consistency and criticisms","G\u00f6del 1930, 1931"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms","Wittgenstein 1919, 1939"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms","G\u00f6del 1944"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents"],"downwards":["Part I Mathematical logic. Volume I \u27381 to \u273843","Part II Prolegomena to cardinal arithmetic. Volume I \u273850 to \u273897","Part III Cardinal arithmetic. Volume II \u2738100 to \u2738126","Part IV Relation-arithmetic. Volume II \u2738150 to \u2738186","Part V Series. Volume II \u2738200 to \u2738234 and volume III \u2738250 to \u2738276","Part VI Quantity. Volume III \u2738300 to \u2738375"]}
{"upwards":["Principia Mathematica","Contents","Part I Mathematical logic. Volume I \u27381 to \u273843"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part II Prolegomena to cardinal arithmetic. Volume I \u273850 to \u273897"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part III Cardinal arithmetic. Volume II \u2738100 to \u2738126"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part IV Relation-arithmetic. Volume II \u2738150 to \u2738186"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part V Series. Volume II \u2738200 to \u2738234 and volume III \u2738250 to \u2738276"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part VI Quantity. Volume III \u2738300 to \u2738375"],"downwards":[]}
{"upwards":["Principia Mathematica","Comparison with set theory"],"downwards":[]}
{"upwards":["Principia Mathematica","Differences between editions"],"downwards":[]}
{"upwards":["Principia Mathematica","Editions"],"downwards":[]}
{"upwards":["Pi"],"downwards":["Fundamentals","History","Modern quest for more digits","Role and characterizations in mathematics","Outside mathematics"]}
{"upwards":["Pi","Fundamentals"],"downwards":["Name","Definition","Irrationality and normality","Transcendence","Continued fractions","Approximate value and digits","Complex numbers and Euler's identity"]}
{"upwards":["Pi","Fundamentals","Name"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Definition"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Irrationality and normality"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Transcendence"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Continued fractions"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Approximate value and digits"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Complex numbers and Euler's identity"],"downwards":[]}
{"upwards":["Pi","History"],"downwards":["Antiquity","Polygon approximation era","Infinite series","Irrationality and transcendence","Adoption of the symbol \u03c0"]}
{"upwards":["Pi","History","Antiquity"],"downwards":[]}
{"upwards":["Pi","History","Polygon approximation era"],"downwards":[]}
{"upwards":["Pi","History","Infinite series"],"downwards":["Rate of convergence"]}
{"upwards":["Pi","History","Infinite series","Rate of convergence"],"downwards":[]}
{"upwards":["Pi","History","Irrationality and transcendence"],"downwards":[]}
{"upwards":["Pi","History","Adoption of the symbol \u03c0"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits"],"downwards":["Computer era and iterative algorithms","Motives for computing \u03c0","Rapidly convergent series","Monte Carlo methods","Spigot algorithms"]}
{"upwards":["Pi","Modern quest for more digits","Computer era and iterative algorithms"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Motives for computing \u03c0"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Rapidly convergent series"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Monte Carlo methods"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Spigot algorithms"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics"],"downwards":["Geometry and trigonometry","Units of angle","Eigenvalues","Inequalities","Fourier transform and Heisenberg uncertainty principle","Gaussian integrals","Topology","Cauchy's integral formula","Total curvature","The gamma function and Stirling's approximation","Number theory and Riemann zeta function","Fourier series","Modular forms and theta functions","Cauchy distribution and potential theory","In the Mandelbrot set"]}
{"upwards":["Pi","Role and characterizations in mathematics","Geometry and trigonometry"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Units of angle"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Eigenvalues"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Inequalities"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Fourier transform and Heisenberg uncertainty principle"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Gaussian integrals"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Topology"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Cauchy's integral formula"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Total curvature"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","The gamma function and Stirling's approximation"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Number theory and Riemann zeta function"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Fourier series"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Modular forms and theta functions"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Cauchy distribution and potential theory"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","In the Mandelbrot set"],"downwards":[]}
{"upwards":["Pi","Outside mathematics"],"downwards":["Describing physical phenomena","Memorizing digits","In popular culture","In computer culture"]}
{"upwards":["Pi","Outside mathematics","Describing physical phenomena"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","Memorizing digits"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","In popular culture"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","In computer culture"],"downwards":[]}
{"upwards":["Rough set"],"downwards":["Definitions","Rule extraction","Incomplete data","Applications","History","Extensions and generalizations"]}
{"upwards":["Rough set","Definitions"],"downwards":["Information system framework","Example: equivalence-class structure","Definition of a rough set","Definability","Reduct and core","Attribute dependency"]}
{"upwards":["Rough set","Definitions","Information system framework"],"downwards":[]}
{"upwards":["Rough set","Definitions","Example: equivalence-class structure"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set"],"downwards":["Lower approximation and positive region","Upper approximation and negative region","Boundary region","The rough set","Objective analysis"]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Lower approximation and positive region"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Upper approximation and negative region"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Boundary region"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","The rough set"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Objective analysis"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definability"],"downwards":[]}
{"upwards":["Rough set","Definitions","Reduct and core"],"downwards":[]}
{"upwards":["Rough set","Definitions","Attribute dependency"],"downwards":[]}
{"upwards":["Rough set","Rule extraction"],"downwards":["Decision matrices","LERS rule induction system"]}
{"upwards":["Rough set","Rule extraction","Decision matrices"],"downwards":[]}
{"upwards":["Rough set","Rule extraction","LERS rule induction system"],"downwards":[]}
{"upwards":["Rough set","Incomplete data"],"downwards":[]}
{"upwards":["Rough set","Applications"],"downwards":[]}
{"upwards":["Rough set","History"],"downwards":[]}
{"upwards":["Rough set","Extensions and generalizations"],"downwards":["Rough membership","Other generalizations"]}
{"upwards":["Rough set","Extensions and generalizations","Rough membership"],"downwards":[]}
{"upwards":["Rough set","Extensions and generalizations","Other generalizations"],"downwards":[]}
{"upwards":["ISBN"],"downwards":["History","Overview","Check digits","EAN format used in barcodes, and upgrading"]}
{"upwards":["ISBN","History"],"downwards":[]}
{"upwards":["ISBN","Overview"],"downwards":["Issuing process","Registration group element","Registrant element"]}
{"upwards":["ISBN","Overview","Issuing process"],"downwards":[]}
{"upwards":["ISBN","Overview","Registration group element"],"downwards":[]}
{"upwards":["ISBN","Overview","Registrant element"],"downwards":["English language pattern"]}
{"upwards":["ISBN","Overview","Registrant element","English language pattern"],"downwards":[]}
{"upwards":["ISBN","Check digits"],"downwards":["ISBN-10 check digits","ISBN-10 check digit calculation","ISBN-13 check digit calculation","ISBN-10 to ISBN-13 conversion","Errors in usage","eISBN"]}
{"upwards":["ISBN","Check digits","ISBN-10 check digits"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-10 check digit calculation"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-13 check digit calculation"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-10 to ISBN-13 conversion"],"downwards":[]}
{"upwards":["ISBN","Check digits","Errors in usage"],"downwards":[]}
{"upwards":["ISBN","Check digits","eISBN"],"downwards":[]}
{"upwards":["ISBN","EAN format used in barcodes, and upgrading"],"downwards":[]}
{"upwards":["The Principles of Mathematics"],"downwards":["Contents","Early reviews","Second edition","Later reviews"]}
{"upwards":["The Principles of Mathematics","Contents"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Early reviews"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Second edition"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Later reviews"],"downwards":[]}
{"upwards":["Crelle's Journal"],"downwards":["History","Past editors","References","External links"]}
{"upwards":["Crelle's Journal","History"],"downwards":[]}
{"upwards":["Crelle's Journal","Past editors"],"downwards":[]}
{"upwards":["Crelle's Journal","References"],"downwards":[]}
{"upwards":["Crelle's Journal","External links"],"downwards":[]}
{"upwards":["Digital object identifier"],"downwards":["Nomenclature and syntax","Content","Features and benefits","Comparison with other identifier schemes","Resolution","IDF organizational structure","Standardization"]}
{"upwards":["Digital object identifier","Nomenclature and syntax"],"downwards":["Display"]}
{"upwards":["Digital object identifier","Nomenclature and syntax","Display"],"downwards":[]}
{"upwards":["Digital object identifier","Content"],"downwards":[]}
{"upwards":["Digital object identifier","Features and benefits"],"downwards":[]}
{"upwards":["Digital object identifier","Comparison with other identifier schemes"],"downwards":[]}
{"upwards":["Digital object identifier","Resolution"],"downwards":[]}
{"upwards":["Digital object identifier","IDF organizational structure"],"downwards":[]}
{"upwards":["Digital object identifier","Standardization"],"downwards":[]}
{"upwards":["Bibcode"],"downwards":["Adoption","Format","Examples"]}
{"upwards":["Bibcode","Adoption"],"downwards":[]}
{"upwards":["Bibcode","Format"],"downwards":[]}
{"upwards":["Bibcode","Examples"],"downwards":[]}
{"upwards":["JSTOR"],"downwards":["History","Content","Access","Usage"]}
{"upwards":["JSTOR","History"],"downwards":[]}
{"upwards":["JSTOR","Content"],"downwards":[]}
{"upwards":["JSTOR","Access"],"downwards":["Aaron Swartz incident","Limitations","Increasing public access"]}
{"upwards":["JSTOR","Access","Aaron Swartz incident"],"downwards":[]}
{"upwards":["JSTOR","Access","Limitations"],"downwards":[]}
{"upwards":["JSTOR","Access","Increasing public access"],"downwards":[]}
{"upwards":["JSTOR","Usage"],"downwards":[]}
{"upwards":["PubMed Central"],"downwards":["Adoption","Technology","Reception","PMCID"]}
{"upwards":["PubMed Central","Adoption"],"downwards":[]}
{"upwards":["PubMed Central","Technology"],"downwards":[]}
{"upwards":["PubMed Central","Reception"],"downwards":[]}
{"upwards":["PubMed Central","PMCID"],"downwards":[]}
{"upwards":["PubMed"],"downwards":["Content","Characteristics"]}
{"upwards":["PubMed","Content"],"downwards":[]}
{"upwards":["PubMed","Characteristics"],"downwards":["Website design","Search","Journal article parameters"]}
{"upwards":["PubMed","Characteristics","Website design"],"downwards":["PubMed for handhelds\/mobiles"]}
{"upwards":["PubMed","Characteristics","Website design","PubMed for handhelds\/mobiles"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Search"],"downwards":["Standard search","Comprehensive search"]}
{"upwards":["PubMed","Characteristics","Search","Standard search"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Search","Comprehensive search"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Journal article parameters"],"downwards":["Publication Type: Clinical queries\/systematic reviews","Secondary ID"]}
{"upwards":["PubMed","Characteristics","Journal article parameters","Publication Type: Clinical queries\/systematic reviews"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Journal article parameters","Secondary ID"],"downwards":[]}
{"upwards":["Joseph Dauben"],"downwards":["Positions","Publications","Articles, reviews, and essays"]}
{"upwards":["Joseph Dauben","Positions"],"downwards":[]}
{"upwards":["Joseph Dauben","Publications"],"downwards":[]}
{"upwards":["Joseph Dauben","Articles, reviews, and essays"],"downwards":[]}
{"upwards":["Dover Publications"],"downwards":["Classic reprints","History","Publications","References","External links"]}
{"upwards":["Dover Publications","Classic reprints"],"downwards":[]}
{"upwards":["Dover Publications","History"],"downwards":[]}
{"upwards":["Dover Publications","Publications"],"downwards":[]}
{"upwards":["Dover Publications","References"],"downwards":[]}
{"upwards":["Dover Publications","External links"],"downwards":[]}
{"upwards":["Structure (mathematical logic)"],"downwards":["Definition","Induced substructures and closed subsets","Homomorphisms and embeddings","Structures and first-order logic","Many-sorted structures","Other generalizations"]}
{"upwards":["Structure (mathematical logic)","Definition"],"downwards":["Domain","Signature","Interpretation function","Examples"]}
{"upwards":["Structure (mathematical logic)","Definition","Domain"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Signature"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Interpretation function"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets"],"downwards":["Examples"]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings"],"downwards":["Homomorphisms","Embeddings","Example","Homomorphism problem"]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphisms"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Embeddings"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Example"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphism problem"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic"],"downwards":["Satisfaction relation","Definable relations"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Satisfaction relation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations"],"downwards":["Definability with parameters","Implicit definability"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Definability with parameters"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Implicit definability"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Many-sorted structures"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations"],"downwards":["Partial algebras","Structures for typed languages","Higher-order languages","Structures that are proper classes"]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Partial algebras"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures for typed languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Higher-order languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures that are proper classes"],"downwards":[]}
{"upwards":["Logical consequence"],"downwards":["Formal accounts","A priori property of logical consequence","Proofs and models","Modal accounts"]}
{"upwards":["Logical consequence","Formal accounts"],"downwards":[]}
{"upwards":["Logical consequence","A priori property of logical consequence"],"downwards":[]}
{"upwards":["Logical consequence","Proofs and models"],"downwards":["Syntactic consequence","Semantic consequence"]}
{"upwards":["Logical consequence","Proofs and models","Syntactic consequence"],"downwards":[]}
{"upwards":["Logical consequence","Proofs and models","Semantic consequence"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts"],"downwards":["Modal-formal accounts","Warrant-based accounts","Non-monotonic logical consequence"]}
{"upwards":["Logical consequence","Modal accounts","Modal-formal accounts"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts","Warrant-based accounts"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts","Non-monotonic logical consequence"],"downwards":[]}
{"upwards":["Cambridge University Press"],"downwards":["History","Print and typographic heritage","Current publications","Open access","Nobel prize winners published by Cambridge University Press","Organisational governance and operational structure","Cambridge University Press partnerships and acquisitions","Digital developments","Controversies","Corporate social responsibility","References","External links"]}
{"upwards":["Cambridge University Press","History"],"downwards":[]}
{"upwards":["Cambridge University Press","Print and typographic heritage"],"downwards":["People","Publications"]}
{"upwards":["Cambridge University Press","Print and typographic heritage","People"],"downwards":[]}
{"upwards":["Cambridge University Press","Print and typographic heritage","Publications"],"downwards":[]}
{"upwards":["Cambridge University Press","Current publications"],"downwards":[]}
{"upwards":["Cambridge University Press","Open access"],"downwards":[]}
{"upwards":["Cambridge University Press","Nobel prize winners published by Cambridge University Press"],"downwards":[]}
{"upwards":["Cambridge University Press","Organisational governance and operational structure"],"downwards":["Relationship with the University of Cambridge","Operational structure"]}
{"upwards":["Cambridge University Press","Organisational governance and operational structure","Relationship with the University of Cambridge"],"downwards":[]}
{"upwards":["Cambridge University Press","Organisational governance and operational structure","Operational structure"],"downwards":[]}
{"upwards":["Cambridge University Press","Cambridge University Press partnerships and acquisitions"],"downwards":[]}
{"upwards":["Cambridge University Press","Digital developments"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies"],"downwards":["Tax exemption controversy","Alms for Jihad","Cambridge University Press v. Patton","The China Quarterly","The Cambridge Handbook of Privatization"]}
{"upwards":["Cambridge University Press","Controversies","Tax exemption controversy"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","Alms for Jihad"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","Cambridge University Press v. Patton"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","The China Quarterly"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","The Cambridge Handbook of Privatization"],"downwards":[]}
{"upwards":["Cambridge University Press","Corporate social responsibility"],"downwards":["Community","Environment"]}
{"upwards":["Cambridge University Press","Corporate social responsibility","Community"],"downwards":[]}
{"upwards":["Cambridge University Press","Corporate social responsibility","Environment"],"downwards":[]}
{"upwards":["Cambridge University Press","References"],"downwards":["Citations","Sources"]}
{"upwards":["Cambridge University Press","References","Citations"],"downwards":[]}
{"upwards":["Cambridge University Press","References","Sources"],"downwards":[]}
{"upwards":["Cambridge University Press","External links"],"downwards":[]}
{"upwards":["Formal proof"],"downwards":["Background"]}
{"upwards":["Formal proof","Background"],"downwards":["Formal language","Formal grammar","Formal systems","Interpretations"]}
{"upwards":["Formal proof","Background","Formal language"],"downwards":[]}
{"upwards":["Formal proof","Background","Formal grammar"],"downwards":[]}
{"upwards":["Formal proof","Background","Formal systems"],"downwards":[]}
{"upwards":["Formal proof","Background","Interpretations"],"downwards":[]}
{"upwards":["Axiom"],"downwards":["Etymology","Historical development","Mathematical logic"]}
{"upwards":["Axiom","Etymology"],"downwards":[]}
{"upwards":["Axiom","Historical development"],"downwards":["Early Greeks","Modern development","Other sciences"]}
{"upwards":["Axiom","Historical development","Early Greeks"],"downwards":[]}
{"upwards":["Axiom","Historical development","Modern development"],"downwards":[]}
{"upwards":["Axiom","Historical development","Other sciences"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic"],"downwards":["Logical axioms","Non-logical axioms","Role in mathematical logic","Further discussion"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic"],"downwards":["Deductive systems and completeness"]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic","Deductive systems and completeness"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Further discussion"],"downwards":[]}
{"upwards":["Harvard University Press"],"downwards":["Related publishers, imprints, and series","Awards","Publications"]}
{"upwards":["Harvard University Press","Related publishers, imprints, and series"],"downwards":[]}
{"upwards":["Harvard University Press","Awards"],"downwards":[]}
{"upwards":["Harvard University Press","Publications"],"downwards":[]}
{"upwards":["Paul Halmos"],"downwards":["Early life and education","Career","Accomplishments","Books by Halmos"]}
{"upwards":["Paul Halmos","Early life and education"],"downwards":[]}
{"upwards":["Paul Halmos","Career"],"downwards":[]}
{"upwards":["Paul Halmos","Accomplishments"],"downwards":[]}
{"upwards":["Paul Halmos","Books by Halmos"],"downwards":[]}
{"upwards":["Mathematical logic"],"downwards":["Subfields and scope","History","Formal logical systems","Set theory","Model theory","Recursion theory","Proof theory and constructive mathematics","Applications","Connections with computer science","Foundations of mathematics"]}
{"upwards":["Mathematical logic","Subfields and scope"],"downwards":[]}
{"upwards":["Mathematical logic","History"],"downwards":["Early history","19th century","20th century"]}
{"upwards":["Mathematical logic","History","Early history"],"downwards":[]}
{"upwards":["Mathematical logic","History","19th century"],"downwards":["Foundational theories"]}
{"upwards":["Mathematical logic","History","19th century","Foundational theories"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century"],"downwards":["Set theory and paradoxes","Symbolic logic","Beginnings of the other branches"]}
{"upwards":["Mathematical logic","History","20th century","Set theory and paradoxes"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Symbolic logic"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Beginnings of the other branches"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems"],"downwards":["First-order logic","Other classical logics","Nonclassical and modal logic","Algebraic logic"]}
{"upwards":["Mathematical logic","Formal logical systems","First-order logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Other classical logics"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Nonclassical and modal logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Algebraic logic"],"downwards":[]}
{"upwards":["Mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Mathematical logic","Model theory"],"downwards":[]}
{"upwards":["Mathematical logic","Recursion theory"],"downwards":["Algorithmically unsolvable problems"]}
{"upwards":["Mathematical logic","Recursion theory","Algorithmically unsolvable problems"],"downwards":[]}
{"upwards":["Mathematical logic","Proof theory and constructive mathematics"],"downwards":[]}
{"upwards":["Mathematical logic","Applications"],"downwards":[]}
{"upwards":["Mathematical logic","Connections with computer science"],"downwards":[]}
{"upwards":["Mathematical logic","Foundations of mathematics"],"downwards":[]}
{"upwards":["Type theory"],"downwards":["History","Introduction","Differences from set theory","Technical details","Interpretations","List of type theories","Applications"]}
{"upwards":["Type theory","History"],"downwards":[]}
{"upwards":["Type theory","Introduction"],"downwards":["Basics","Functions","Common \"by rule\" types and terms"]}
{"upwards":["Type theory","Introduction","Basics"],"downwards":["Terms and types","Computation"]}
{"upwards":["Type theory","Introduction","Basics","Terms and types"],"downwards":[]}
{"upwards":["Type theory","Introduction","Basics","Computation"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions"],"downwards":["Lambda terms","Dependent typing","Universes"]}
{"upwards":["Type theory","Introduction","Functions","Lambda terms"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions","Dependent typing"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions","Universes"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms"],"downwards":["Empty type","Unit type","Boolean type","Product type","Sum type","Natural numbers","Identity type","Inductive types"]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Empty type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Unit type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Boolean type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Product type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Sum type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Natural numbers"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Identity type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Inductive types"],"downwards":[]}
{"upwards":["Type theory","Differences from set theory"],"downwards":[]}
{"upwards":["Type theory","Technical details"],"downwards":["Terms","Judgements","Rules","Properties of type theories","Axioms","Decision problems","Research areas"]}
{"upwards":["Type theory","Technical details","Terms"],"downwards":[]}
{"upwards":["Type theory","Technical details","Judgements"],"downwards":[]}
{"upwards":["Type theory","Technical details","Rules"],"downwards":[]}
{"upwards":["Type theory","Technical details","Properties of type theories"],"downwards":[]}
{"upwards":["Type theory","Technical details","Axioms"],"downwards":[]}
{"upwards":["Type theory","Technical details","Decision problems"],"downwards":["Type inhabitation","Type inference"]}
{"upwards":["Type theory","Technical details","Decision problems","Type inhabitation"],"downwards":[]}
{"upwards":["Type theory","Technical details","Decision problems","Type inference"],"downwards":[]}
{"upwards":["Type theory","Technical details","Research areas"],"downwards":[]}
{"upwards":["Type theory","Interpretations"],"downwards":["Types are propositions; terms are proofs","Curry-Howard correspondence","Intuitionistic logic","Constructive mathematics","Category theory"]}
{"upwards":["Type theory","Interpretations","Types are propositions; terms are proofs"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Curry-Howard correspondence"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Intuitionistic logic"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Constructive mathematics"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Category theory"],"downwards":[]}
{"upwards":["Type theory","List of type theories"],"downwards":["Major","Minor","Active research"]}
{"upwards":["Type theory","List of type theories","Major"],"downwards":[]}
{"upwards":["Type theory","List of type theories","Minor"],"downwards":[]}
{"upwards":["Type theory","List of type theories","Active research"],"downwards":[]}
{"upwards":["Type theory","Applications"],"downwards":["Mathematical foundations","Proof assistants","Programming languages","Linguistics","Social sciences"]}
{"upwards":["Type theory","Applications","Mathematical foundations"],"downwards":[]}
{"upwards":["Type theory","Applications","Proof assistants"],"downwards":[]}
{"upwards":["Type theory","Applications","Programming languages"],"downwards":[]}
{"upwards":["Type theory","Applications","Linguistics"],"downwards":[]}
{"upwards":["Type theory","Applications","Social sciences"],"downwards":[]}
{"upwards":["Lindstr\u00f6m's theorem"],"downwards":["Notes","References"]}
{"upwards":["Lindstr\u00f6m's theorem","Notes"],"downwards":[]}
{"upwards":["Lindstr\u00f6m's theorem","References"],"downwards":[]}
{"upwards":["Semantics of logic"],"downwards":["Overview"]}
{"upwards":["Semantics of logic","Overview"],"downwards":[]}
{"upwards":["Theorem"],"downwards":["Theoremhood and truth","Epistemological considerations","Informal account of theorems","Relation with scientific theories","Terminology","Layout","Lore","Theorems in logic"]}
{"upwards":["Theorem","Theoremhood and truth"],"downwards":[]}
{"upwards":["Theorem","Epistemological considerations"],"downwards":[]}
{"upwards":["Theorem","Informal account of theorems"],"downwards":[]}
{"upwards":["Theorem","Relation with scientific theories"],"downwards":[]}
{"upwards":["Theorem","Terminology"],"downwards":[]}
{"upwards":["Theorem","Layout"],"downwards":[]}
{"upwards":["Theorem","Lore"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic"],"downwards":["Syntax and semantics","Interpretation of a formal theorem","Theorems and theories"]}
{"upwards":["Theorem","Theorems in logic","Syntax and semantics"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Interpretation of a formal theorem"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Theorems and theories"],"downwards":[]}
{"upwards":["Paradoxes of set theory"],"downwards":["Basics","Paradoxes of the infinite sets","Paradoxes of the Supertask","Paradoxes of proof and definability"]}
{"upwards":["Paradoxes of set theory","Basics"],"downwards":["Cardinal numbers","Ordinal numbers","Power sets"]}
{"upwards":["Paradoxes of set theory","Basics","Cardinal numbers"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Basics","Ordinal numbers"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Basics","Power sets"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets"],"downwards":["Paradoxes of enumeration","Je le vois, mais je ne crois pas","Paradoxes of well-ordering"]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Paradoxes of enumeration"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Je le vois, mais je ne crois pas"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Paradoxes of well-ordering"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask"],"downwards":["The diary of Tristram Shandy","The Ross-Littlewood paradox"]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask","The diary of Tristram Shandy"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask","The Ross-Littlewood paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability"],"downwards":["Early paradoxes: the set of all sets","Paradoxes by change of language","Paradox of L\u00f6wenheim and Skolem"]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Early paradoxes: the set of all sets"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language"],"downwards":["K\u00f6nig's paradox","Richard's paradox"]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language","K\u00f6nig's paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language","Richard's paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradox of L\u00f6wenheim and Skolem"],"downwards":[]}
{"upwards":["Lemma (mathematics)"],"downwards":["Etymology","Comparison with theorem","Well-known lemmas"]}
{"upwards":["Lemma (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Lemma (mathematics)","Comparison with theorem"],"downwards":[]}
{"upwards":["Lemma (mathematics)","Well-known lemmas"],"downwards":[]}
{"upwards":["List of axioms"],"downwards":["ZF (the Zermelo\u2013Fraenkel axioms without the axiom of choice)","Axiom of choice","Other axioms of mathematical logic","Geometry","Other axioms"]}
{"upwards":["List of axioms","ZF (the Zermelo\u2013Fraenkel axioms without the axiom of choice)"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice"],"downwards":["Equivalents of AC","Stronger than AC","Weaker than AC","Alternates incompatible with AC"]}
{"upwards":["List of axioms","Axiom of choice","Equivalents of AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Stronger than AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Weaker than AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Alternates incompatible with AC"],"downwards":[]}
{"upwards":["List of axioms","Other axioms of mathematical logic"],"downwards":[]}
{"upwards":["List of axioms","Geometry"],"downwards":[]}
{"upwards":["List of axioms","Other axioms"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem"],"downwards":["Theorem","Discussion","Proof sketch","In other logics","Historical notes","References","Sources","External links"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Theorem"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion"],"downwards":["Concepts","Consequences"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts"],"downwards":["Signatures","Structures \/ Models"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts","Signatures"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts","Structures \/ Models"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Consequences"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch"],"downwards":["Downward part","Upward part"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch","Downward part"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch","Upward part"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","In other logics"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Historical notes"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","References"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources"],"downwards":["Historical publications","Secondary sources"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources","Historical publications"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources","Secondary sources"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","External links"],"downwards":[]}
{"upwards":["Term logic"],"downwards":["Aristotle's system","Basics","Term","Proposition","Singular terms","Syllogism in the first figure","Syllogism in the second figure","Syllogism in the third figure","Table of syllogisms","Decline of term logic","Revival","Boole\u2019s acceptance of Aristotle"]}
{"upwards":["Term logic","Aristotle's system"],"downwards":[]}
{"upwards":["Term logic","Basics"],"downwards":[]}
{"upwards":["Term logic","Term"],"downwards":[]}
{"upwards":["Term logic","Proposition"],"downwards":[]}
{"upwards":["Term logic","Singular terms"],"downwards":["The three figures","The fourth figure"]}
{"upwards":["Term logic","Singular terms","The three figures"],"downwards":[]}
{"upwards":["Term logic","Singular terms","The fourth figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the first figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the second figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the third figure"],"downwards":[]}
{"upwards":["Term logic","Table of syllogisms"],"downwards":[]}
{"upwards":["Term logic","Decline of term logic"],"downwards":[]}
{"upwards":["Term logic","Revival"],"downwards":[]}
{"upwards":["Term logic","Boole\u2019s acceptance of Aristotle"],"downwards":[]}
{"upwards":["Halting problem"],"downwards":["Background","History","Formalization","Computability theory","Generalization"]}
{"upwards":["Halting problem","Background"],"downwards":["Programming consequences","Common pitfalls"]}
{"upwards":["Halting problem","Background","Programming consequences"],"downwards":[]}
{"upwards":["Halting problem","Background","Common pitfalls"],"downwards":[]}
{"upwards":["Halting problem","History"],"downwards":["Timeline","Origin of the halting problem"]}
{"upwards":["Halting problem","History","Timeline"],"downwards":[]}
{"upwards":["Halting problem","History","Origin of the halting problem"],"downwards":[]}
{"upwards":["Halting problem","Formalization"],"downwards":["Representation as a set","Proof concept","Sketch of rigorous proof"]}
{"upwards":["Halting problem","Formalization","Representation as a set"],"downwards":[]}
{"upwards":["Halting problem","Formalization","Proof concept"],"downwards":[]}
{"upwards":["Halting problem","Formalization","Sketch of rigorous proof"],"downwards":[]}
{"upwards":["Halting problem","Computability theory"],"downwards":["Approximations","G\u00f6del's incompleteness theorems"]}
{"upwards":["Halting problem","Computability theory","Approximations"],"downwards":[]}
{"upwards":["Halting problem","Computability theory","G\u00f6del's incompleteness theorems"],"downwards":[]}
{"upwards":["Halting problem","Generalization"],"downwards":["Halting on all inputs","Recognizing partial solutions","Lossy computation","Oracle machines"]}
{"upwards":["Halting problem","Generalization","Halting on all inputs"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Recognizing partial solutions"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Lossy computation"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Oracle machines"],"downwards":[]}
{"upwards":["Classical logic"],"downwards":["Characteristics","History","Generalized semantics","References","Further reading"]}
{"upwards":["Classical logic","Characteristics"],"downwards":[]}
{"upwards":["Classical logic","History"],"downwards":[]}
{"upwards":["Classical logic","Generalized semantics"],"downwards":[]}
{"upwards":["Classical logic","References"],"downwards":[]}
{"upwards":["Classical logic","Further reading"],"downwards":[]}
{"upwards":["Compactness theorem"],"downwards":["History","Applications","Proofs"]}
{"upwards":["Compactness theorem","History"],"downwards":[]}
{"upwards":["Compactness theorem","Applications"],"downwards":["Robinson's principle","Upward L\u00f6wenheim\u2013Skolem theorem","Non-standard analysis"]}
{"upwards":["Compactness theorem","Applications","Robinson's principle"],"downwards":[]}
{"upwards":["Compactness theorem","Applications","Upward L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["Compactness theorem","Applications","Non-standard analysis"],"downwards":[]}
{"upwards":["Compactness theorem","Proofs"],"downwards":[]}
{"upwards":["Logical truth"],"downwards":["Logical truths and analytic truths","Truth values and tautologies","Logical truth and logical constants","Logical truth and rules of inference","Logical truth and logical positivism","Non-classical logics"]}
{"upwards":["Logical truth","Logical truths and analytic truths"],"downwards":[]}
{"upwards":["Logical truth","Truth values and tautologies"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and logical constants"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and rules of inference"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and logical positivism"],"downwards":[]}
{"upwards":["Logical truth","Non-classical logics"],"downwards":[]}
{"upwards":["Cantor's diagonal argument"],"downwards":["Uncountable set","Consequences","Version for Quine's New Foundations"]}
{"upwards":["Cantor's diagonal argument","Uncountable set"],"downwards":["Real numbers","General sets"]}
{"upwards":["Cantor's diagonal argument","Uncountable set","Real numbers"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Uncountable set","General sets"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences"],"downwards":["Ordering of cardinals","Open questions","Diagonalization in broader context"]}
{"upwards":["Cantor's diagonal argument","Consequences","Ordering of cardinals"],"downwards":["In the absence of excluded middle"]}
{"upwards":["Cantor's diagonal argument","Consequences","Ordering of cardinals","In the absence of excluded middle"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences","Open questions"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences","Diagonalization in broader context"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Version for Quine's New Foundations"],"downwards":[]}
{"upwards":["Tautology (logic)"],"downwards":["History","Background","Definition and examples","Verifying tautologies","Tautological implication","Substitution","Semantic completeness and soundness","Efficient verification and the Boolean satisfiability problem","Tautologies versus validities in first-order logic"]}
{"upwards":["Tautology (logic)","History"],"downwards":[]}
{"upwards":["Tautology (logic)","Background"],"downwards":[]}
{"upwards":["Tautology (logic)","Definition and examples"],"downwards":[]}
{"upwards":["Tautology (logic)","Verifying tautologies"],"downwards":[]}
{"upwards":["Tautology (logic)","Tautological implication"],"downwards":[]}
{"upwards":["Tautology (logic)","Substitution"],"downwards":[]}
{"upwards":["Tautology (logic)","Semantic completeness and soundness"],"downwards":[]}
{"upwards":["Tautology (logic)","Efficient verification and the Boolean satisfiability problem"],"downwards":[]}
{"upwards":["Tautology (logic)","Tautologies versus validities in first-order logic"],"downwards":[]}
{"upwards":["Cantor's theorem"],"downwards":["Proof","When A is countably infinite","Related paradoxes","History","Generalizations"]}
{"upwards":["Cantor's theorem","Proof"],"downwards":[]}
{"upwards":["Cantor's theorem","When A is countably infinite"],"downwards":[]}
{"upwards":["Cantor's theorem","Related paradoxes"],"downwards":[]}
{"upwards":["Cantor's theorem","History"],"downwards":[]}
{"upwards":["Cantor's theorem","Generalizations"],"downwards":[]}
{"upwards":["Proposition"],"downwards":["Historical usage","Relation to the mind","Treatment in logic","Objections to propositions"]}
{"upwards":["Proposition","Historical usage"],"downwards":["By Aristotle","By the logical positivists","By Russell"]}
{"upwards":["Proposition","Historical usage","By Aristotle"],"downwards":[]}
{"upwards":["Proposition","Historical usage","By the logical positivists"],"downwards":[]}
{"upwards":["Proposition","Historical usage","By Russell"],"downwards":[]}
{"upwards":["Proposition","Relation to the mind"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic"],"downwards":["Aristotelian logic","Syntactic characterization","Semantic characterization"]}
{"upwards":["Proposition","Treatment in logic","Aristotelian logic"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic","Syntactic characterization"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic","Semantic characterization"],"downwards":[]}
{"upwards":["Proposition","Objections to propositions"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox"],"downwards":["Banach and Tarski publication","Formal treatment","Connection with earlier work and the role of the axiom of choice","A sketch of the proof","Obtaining infinitely many balls from one","Von Neumann paradox in the Euclidean plane"]}
{"upwards":["Banach\u2013Tarski paradox","Banach and Tarski publication"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Formal treatment"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Connection with earlier work and the role of the axiom of choice"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof"],"downwards":["Step 1","Step 2","Step 3","Step 4","Some details, fleshed out"]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 1"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 2"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 3"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 4"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Some details, fleshed out"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Obtaining infinitely many balls from one"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Von Neumann paradox in the Euclidean plane"],"downwards":["Recent progress"]}
{"upwards":["Banach\u2013Tarski paradox","Von Neumann paradox in the Euclidean plane","Recent progress"],"downwards":[]}
{"upwards":["Inference"],"downwards":["Definition","Examples","Incorrect inference","Applications"]}
{"upwards":["Inference","Definition"],"downwards":[]}
{"upwards":["Inference","Examples"],"downwards":["Example for definition #1","Example for definition #2"]}
{"upwards":["Inference","Examples","Example for definition #1"],"downwards":[]}
{"upwards":["Inference","Examples","Example for definition #2"],"downwards":[]}
{"upwards":["Inference","Incorrect inference"],"downwards":[]}
{"upwards":["Inference","Applications"],"downwards":["Inference engines","Semantic web","Bayesian statistics and probability logic","Fuzzy logic","Non-monotonic logic"]}
{"upwards":["Inference","Applications","Inference engines"],"downwards":["Prolog engine"]}
{"upwards":["Inference","Applications","Inference engines","Prolog engine"],"downwards":[]}
{"upwards":["Inference","Applications","Semantic web"],"downwards":[]}
{"upwards":["Inference","Applications","Bayesian statistics and probability logic"],"downwards":[]}
{"upwards":["Inference","Applications","Fuzzy logic"],"downwards":[]}
{"upwards":["Inference","Applications","Non-monotonic logic"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem"],"downwards":["History","Statement","General form","Discussion"]}
{"upwards":["Tarski's undefinability theorem","History"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","Statement"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","General form"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","Discussion"],"downwards":[]}
{"upwards":["Logical equivalence"],"downwards":["Logical equivalences","Examples","Relation to material equivalence"]}
{"upwards":["Logical equivalence","Logical equivalences"],"downwards":["General logical equivalences","Logical equivalences involving conditional statements","Logical equivalences involving biconditionals"]}
{"upwards":["Logical equivalence","Logical equivalences","General logical equivalences"],"downwards":[]}
{"upwards":["Logical equivalence","Logical equivalences","Logical equivalences involving conditional statements"],"downwards":[]}
{"upwards":["Logical equivalence","Logical equivalences","Logical equivalences involving biconditionals"],"downwards":[]}
{"upwards":["Logical equivalence","Examples"],"downwards":["In logic"]}
{"upwards":["Logical equivalence","Examples","In logic"],"downwards":[]}
{"upwards":["Logical equivalence","Relation to material equivalence"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem"],"downwards":["Preliminaries","Statement","Consequences","Relationship to the incompleteness theorems","Relationship to the compactness theorem","Completeness in other logics","Proofs"]}
{"upwards":["G\u00f6del's completeness theorem","Preliminaries"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement"],"downwards":["G\u00f6del's original formulation","More general form","Model existence theorem","As a theorem of arithmetic"]}
{"upwards":["G\u00f6del's completeness theorem","Statement","G\u00f6del's original formulation"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","More general form"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","Model existence theorem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","As a theorem of arithmetic"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Consequences"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Relationship to the incompleteness theorems"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Relationship to the compactness theorem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Completeness in other logics"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Proofs"],"downwards":[]}
{"upwards":["Consistency"],"downwards":["Consistency and completeness in arithmetic and set theory","First-order logic","Model theory"]}
{"upwards":["Consistency","Consistency and completeness in arithmetic and set theory"],"downwards":[]}
{"upwards":["Consistency","First-order logic"],"downwards":["Notation","Definition","Basic results","Henkin's theorem","Sketch of proof"]}
{"upwards":["Consistency","First-order logic","Notation"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Definition"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Basic results"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Henkin's theorem"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Sketch of proof"],"downwards":[]}
{"upwards":["Consistency","Model theory"],"downwards":[]}
{"upwards":["Argument"],"downwards":["Etymology","Formal and informal","Standard logical account of argument types","Defeasible arguments and argumentation schemes","By analogy","Other kinds","Explanations","Fallacies and non-arguments","Elliptical or ethymematic arguments","Argument mining"]}
{"upwards":["Argument","Etymology"],"downwards":[]}
{"upwards":["Argument","Formal and informal"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types"],"downwards":["Deductive arguments","Inductive arguments"]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments"],"downwards":["Validity","Soundness"]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments","Validity"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments","Soundness"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types","Inductive arguments"],"downwards":[]}
{"upwards":["Argument","Defeasible arguments and argumentation schemes"],"downwards":[]}
{"upwards":["Argument","By analogy"],"downwards":[]}
{"upwards":["Argument","Other kinds"],"downwards":["World-disclosing"]}
{"upwards":["Argument","Other kinds","World-disclosing"],"downwards":[]}
{"upwards":["Argument","Explanations"],"downwards":[]}
{"upwards":["Argument","Fallacies and non-arguments"],"downwards":[]}
{"upwards":["Argument","Elliptical or ethymematic arguments"],"downwards":[]}
{"upwards":["Argument","Argument mining"],"downwards":[]}
{"upwards":["Equiconsistency"],"downwards":["Consistency","Consistency strength"]}
{"upwards":["Equiconsistency","Consistency"],"downwards":[]}
{"upwards":["Equiconsistency","Consistency strength"],"downwards":[]}
{"upwards":["Theory (mathematical logic)"],"downwards":["General theories (as expressed in formal language)","First-order theories","Examples"]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)"],"downwards":["Subtheories and extensions","Deductive theories","Consistency and completeness","Interpretation of a theory","Theories associated with a structure"]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Subtheories and extensions"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Deductive theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Consistency and completeness"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Interpretation of a theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Theories associated with a structure"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories"],"downwards":["Derivation in a first-order theory","Syntactic consequence in a first-order theory","Interpretation of a first-order theory","First-order theories with identity","Topics related to first-order theories"]}
{"upwards":["Theory (mathematical logic)","First-order theories","Derivation in a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Syntactic consequence in a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Interpretation of a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","First-order theories with identity"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Topics related to first-order theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","Examples"],"downwards":[]}
{"upwards":["Soundness"],"downwards":["Definition","Use in mathematical logic"]}
{"upwards":["Soundness","Definition"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic"],"downwards":["Logical systems","Relation to completeness"]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems"],"downwards":["Soundness","Strong soundness","Arithmetic soundness"]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Strong soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Arithmetic soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Relation to completeness"],"downwards":[]}
{"upwards":["Syllogism"],"downwards":["Early history","Modern history","Basic structure","Types","Terms in syllogism","Existential import","Syllogistic fallacies","Other types of syllogism"]}
{"upwards":["Syllogism","Early history"],"downwards":["Aristotle","Medieval syllogism"]}
{"upwards":["Syllogism","Early history","Aristotle"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism"],"downwards":["Boethius","Peter Abelard","Jean Buridan"]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Boethius"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Peter Abelard"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Jean Buridan"],"downwards":[]}
{"upwards":["Syllogism","Modern history"],"downwards":["Boole's acceptance of Aristotle"]}
{"upwards":["Syllogism","Modern history","Boole's acceptance of Aristotle"],"downwards":[]}
{"upwards":["Syllogism","Basic structure"],"downwards":["Polysyllogism"]}
{"upwards":["Syllogism","Basic structure","Polysyllogism"],"downwards":[]}
{"upwards":["Syllogism","Types"],"downwards":["Examples","Table of all syllogisms"]}
{"upwards":["Syllogism","Types","Examples"],"downwards":["Barbara (AAA-1)","Celarent (EAE-1)","Darii (AII-1)","Ferio (EIO-1)","Baroco (AOO-2)","Bocardo (OAO-3)","Barbari (AAI-1)","Celaront (EAO-1)","Camestros (AEO-2)","Felapton (EAO-3)","Darapti (AAI-3)"]}
{"upwards":["Syllogism","Types","Examples","Barbara (AAA-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Celarent (EAE-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Darii (AII-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Ferio (EIO-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Baroco (AOO-2)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Bocardo (OAO-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Barbari (AAI-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Celaront (EAO-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Camestros (AEO-2)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Felapton (EAO-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Darapti (AAI-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Table of all syllogisms"],"downwards":[]}
{"upwards":["Syllogism","Terms in syllogism"],"downwards":[]}
{"upwards":["Syllogism","Existential import"],"downwards":[]}
{"upwards":["Syllogism","Syllogistic fallacies"],"downwards":[]}
{"upwards":["Syllogism","Other types of syllogism"],"downwards":[]}
{"upwards":["Square of opposition"],"downwards":["Summary","The problem of existential import","Modern squares of opposition","Logical hexagons and other bi-simplexes","Square of opposition (or logical square) and modal logic","Set-theoretical interpretation of categorical statements"]}
{"upwards":["Square of opposition","Summary"],"downwards":[]}
{"upwards":["Square of opposition","The problem of existential import"],"downwards":[]}
{"upwards":["Square of opposition","Modern squares of opposition"],"downwards":[]}
{"upwards":["Square of opposition","Logical hexagons and other bi-simplexes"],"downwards":[]}
{"upwards":["Square of opposition","Square of opposition (or logical square) and modal logic"],"downwards":[]}
{"upwards":["Square of opposition","Set-theoretical interpretation of categorical statements"],"downwards":[]}
{"upwards":["Validity (logic)"],"downwards":["Arguments","Valid formula","Statements","Soundness","Satisfiability","Preservation"]}
{"upwards":["Validity (logic)","Arguments"],"downwards":[]}
{"upwards":["Validity (logic)","Valid formula"],"downwards":[]}
{"upwards":["Validity (logic)","Statements"],"downwards":[]}
{"upwards":["Validity (logic)","Soundness"],"downwards":[]}
{"upwards":["Validity (logic)","Satisfiability"],"downwards":[]}
{"upwards":["Validity (logic)","Preservation"],"downwards":[]}
{"upwards":["Propositional calculus"],"downwards":["Explanation","History","Terminology","Basic concepts","Generic description of a propositional calculus","Example 1. Simple axiom system","Example 2. Natural deduction system","Basic and derived argument forms","Proofs in propositional calculus","Soundness and completeness of the rules","Interpretation of a truth-functional propositional calculus","Alternative calculus","Equivalence to equational logics","Graphical calculi","Other logical calculi","Solvers"]}
{"upwards":["Propositional calculus","Explanation"],"downwards":[]}
{"upwards":["Propositional calculus","History"],"downwards":[]}
{"upwards":["Propositional calculus","Terminology"],"downwards":[]}
{"upwards":["Propositional calculus","Basic concepts"],"downwards":["Closure under operations","Argument"]}
{"upwards":["Propositional calculus","Basic concepts","Closure under operations"],"downwards":[]}
{"upwards":["Propositional calculus","Basic concepts","Argument"],"downwards":[]}
{"upwards":["Propositional calculus","Generic description of a propositional calculus"],"downwards":[]}
{"upwards":["Propositional calculus","Example 1. Simple axiom system"],"downwards":[]}
{"upwards":["Propositional calculus","Example 2. Natural deduction system"],"downwards":[]}
{"upwards":["Propositional calculus","Basic and derived argument forms"],"downwards":[]}
{"upwards":["Propositional calculus","Proofs in propositional calculus"],"downwards":["Example of a proof in natural deduction system","Example of a proof in a classical propositional calculus system"]}
{"upwards":["Propositional calculus","Proofs in propositional calculus","Example of a proof in natural deduction system"],"downwards":[]}
{"upwards":["Propositional calculus","Proofs in propositional calculus","Example of a proof in a classical propositional calculus system"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules"],"downwards":["Sketch of a soundness proof","Sketch of completeness proof","Another outline for a completeness proof"]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of a soundness proof"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof"],"downwards":["Example","Verifying completeness for the classical propositional calculus system"]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof","Example"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof","Verifying completeness for the classical propositional calculus system"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Another outline for a completeness proof"],"downwards":[]}
{"upwards":["Propositional calculus","Interpretation of a truth-functional propositional calculus"],"downwards":["Interpretation of a sentence of truth-functional propositional logic"]}
{"upwards":["Propositional calculus","Interpretation of a truth-functional propositional calculus","Interpretation of a sentence of truth-functional propositional logic"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus"],"downwards":["Axioms","Inference rule","Meta-inference rule","Example of a proof"]}
{"upwards":["Propositional calculus","Alternative calculus","Axioms"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Inference rule"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Meta-inference rule"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Example of a proof"],"downwards":[]}
{"upwards":["Propositional calculus","Equivalence to equational logics"],"downwards":[]}
{"upwards":["Propositional calculus","Graphical calculi"],"downwards":[]}
{"upwards":["Propositional calculus","Other logical calculi"],"downwards":[]}
{"upwards":["Propositional calculus","Solvers"],"downwards":[]}
{"upwards":["Information theory"],"downwards":["Overview","Historical background","Quantities of information","Coding theory","Applications to other fields"]}
{"upwards":["Information theory","Overview"],"downwards":[]}
{"upwards":["Information theory","Historical background"],"downwards":[]}
{"upwards":["Information theory","Quantities of information"],"downwards":["Entropy of an information source","Joint entropy","Conditional entropy (equivocation)","Mutual information (transinformation)","Kullback\u2013Leibler divergence (information gain)","Directed Information","Other quantities"]}
{"upwards":["Information theory","Quantities of information","Entropy of an information source"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Joint entropy"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Conditional entropy (equivocation)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Mutual information (transinformation)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Kullback\u2013Leibler divergence (information gain)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Directed Information"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Other quantities"],"downwards":[]}
{"upwards":["Information theory","Coding theory"],"downwards":["Source theory","Channel capacity"]}
{"upwards":["Information theory","Coding theory","Source theory"],"downwards":["Rate"]}
{"upwards":["Information theory","Coding theory","Source theory","Rate"],"downwards":[]}
{"upwards":["Information theory","Coding theory","Channel capacity"],"downwards":["Capacity of particular channel models","Channels with memory and directed information"]}
{"upwards":["Information theory","Coding theory","Channel capacity","Capacity of particular channel models"],"downwards":[]}
{"upwards":["Information theory","Coding theory","Channel capacity","Channels with memory and directed information"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields"],"downwards":["Intelligence uses and secrecy applications","Pseudorandom number generation","Seismic exploration","Semiotics","Integrated process organization of neural information","Miscellaneous applications"]}
{"upwards":["Information theory","Applications to other fields","Intelligence uses and secrecy applications"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Pseudorandom number generation"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Seismic exploration"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Semiotics"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Integrated process organization of neural information"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Miscellaneous applications"],"downwards":[]}
{"upwards":["First-order logic"],"downwards":["Introduction","Syntax","Semantics","Deductive systems","Equality and its axioms","Metalogical properties","Limitations","Restrictions, extensions, and variations","Automated theorem proving and formal methods"]}
{"upwards":["First-order logic","Introduction"],"downwards":[]}
{"upwards":["First-order logic","Syntax"],"downwards":["Alphabet","Formation rules","Free and bound variables","Example: ordered abelian groups"]}
{"upwards":["First-order logic","Syntax","Alphabet"],"downwards":["Logical symbols","Non-logical symbols"]}
{"upwards":["First-order logic","Syntax","Alphabet","Logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Alphabet","Non-logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules"],"downwards":["Terms","Formulas","Notational conventions"]}
{"upwards":["First-order logic","Syntax","Formation rules","Terms"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Formulas"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Notational conventions"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Free and bound variables"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Example: ordered abelian groups"],"downwards":[]}
{"upwards":["First-order logic","Semantics"],"downwards":["First-order structures","Evaluation of truth values","Validity, satisfiability, and logical consequence","Algebraizations","First-order theories, models, and elementary classes","Empty domains"]}
{"upwards":["First-order logic","Semantics","First-order structures"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Evaluation of truth values"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Validity, satisfiability, and logical consequence"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Algebraizations"],"downwards":[]}
{"upwards":["First-order logic","Semantics","First-order theories, models, and elementary classes"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Empty domains"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems"],"downwards":["Rules of inference","Hilbert-style systems and natural deduction","Sequent calculus","Tableaux method","Resolution","Provable identities"]}
{"upwards":["First-order logic","Deductive systems","Rules of inference"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Hilbert-style systems and natural deduction"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Sequent calculus"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Tableaux method"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Resolution"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Provable identities"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms"],"downwards":["First-order logic without equality","Defining equality within a theory"]}
{"upwards":["First-order logic","Equality and its axioms","First-order logic without equality"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms","Defining equality within a theory"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties"],"downwards":["Completeness and undecidability","The L\u00f6wenheim\u2013Skolem theorem","The compactness theorem","Lindstr\u00f6m's theorem"]}
{"upwards":["First-order logic","Metalogical properties","Completeness and undecidability"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The compactness theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","Lindstr\u00f6m's theorem"],"downwards":[]}
{"upwards":["First-order logic","Limitations"],"downwards":["Expressiveness","Formalizing natural languages"]}
{"upwards":["First-order logic","Limitations","Expressiveness"],"downwards":[]}
{"upwards":["First-order logic","Limitations","Formalizing natural languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations"],"downwards":["Restricted languages","Many-sorted logic","Additional quantifiers","Infinitary logics","Non-classical and modal logics","Fixpoint logic","Higher-order logics"]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Restricted languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Many-sorted logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Additional quantifiers"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Infinitary logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Non-classical and modal logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Fixpoint logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Higher-order logics"],"downwards":[]}
{"upwards":["First-order logic","Automated theorem proving and formal methods"],"downwards":[]}
{"upwards":["Monadic second-order logic"],"downwards":["Variants","Computational complexity of evaluation","Decidability and complexity of satisfiability"]}
{"upwards":["Monadic second-order logic","Variants"],"downwards":[]}
{"upwards":["Monadic second-order logic","Computational complexity of evaluation"],"downwards":[]}
{"upwards":["Monadic second-order logic","Decidability and complexity of satisfiability"],"downwards":["Use of satisfiability of MSO on trees in verification"]}
{"upwards":["Monadic second-order logic","Decidability and complexity of satisfiability","Use of satisfiability of MSO on trees in verification"],"downwards":[]}
{"upwards":["Equivalence relation"],"downwards":["Notation","Definition","Examples","Connections to other relations","Well-definedness under an equivalence relation","Related important definitions","Fundamental theorem of equivalence relations","Comparing equivalence relations","Generating equivalence relations","Algebraic structure","Equivalence relations and mathematical logic"]}
{"upwards":["Equivalence relation","Notation"],"downwards":[]}
{"upwards":["Equivalence relation","Definition"],"downwards":["Alternative definition using relational algebra"]}
{"upwards":["Equivalence relation","Definition","Alternative definition using relational algebra"],"downwards":[]}
{"upwards":["Equivalence relation","Examples"],"downwards":["Simple example","Equivalence relations","Relations that are not equivalences"]}
{"upwards":["Equivalence relation","Examples","Simple example"],"downwards":[]}
{"upwards":["Equivalence relation","Examples","Equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Examples","Relations that are not equivalences"],"downwards":[]}
{"upwards":["Equivalence relation","Connections to other relations"],"downwards":[]}
{"upwards":["Equivalence relation","Well-definedness under an equivalence relation"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions"],"downwards":["Equivalence class","Quotient set","Projection","Equivalence kernel","Partition"]}
{"upwards":["Equivalence relation","Related important definitions","Equivalence class"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Quotient set"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Projection"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Equivalence kernel"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Partition"],"downwards":["Counting partitions"]}
{"upwards":["Equivalence relation","Related important definitions","Partition","Counting partitions"],"downwards":[]}
{"upwards":["Equivalence relation","Fundamental theorem of equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Comparing equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Generating equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure"],"downwards":["Group theory","Categories and groupoids","Lattices"]}
{"upwards":["Equivalence relation","Algebraic structure","Group theory"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure","Categories and groupoids"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure","Lattices"],"downwards":[]}
{"upwards":["Equivalence relation","Equivalence relations and mathematical logic"],"downwards":[]}
{"upwards":["Ordinal number"],"downwards":["Ordinals extend the natural numbers","Definitions","Transfinite sequence","Transfinite induction","Arithmetic of ordinals","Ordinals and cardinals","Some \"large\" countable ordinals","Topology and ordinals","History"]}
{"upwards":["Ordinal number","Ordinals extend the natural numbers"],"downwards":[]}
{"upwards":["Ordinal number","Definitions"],"downwards":["Well-ordered sets","Definition of an ordinal as an equivalence class","Von Neumann definition of ordinals","Other definitions"]}
{"upwards":["Ordinal number","Definitions","Well-ordered sets"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Definition of an ordinal as an equivalence class"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Von Neumann definition of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Other definitions"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite sequence"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction"],"downwards":["Transfinite recursion","Successor and limit ordinals","Indexing classes of ordinals","Closed unbounded sets and classes"]}
{"upwards":["Ordinal number","Transfinite induction","Transfinite recursion"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Successor and limit ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Indexing classes of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Closed unbounded sets and classes"],"downwards":[]}
{"upwards":["Ordinal number","Arithmetic of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals"],"downwards":["Initial ordinal of a cardinal","Cofinality"]}
{"upwards":["Ordinal number","Ordinals and cardinals","Initial ordinal of a cardinal"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals","Cofinality"],"downwards":[]}
{"upwards":["Ordinal number","Some \"large\" countable ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Topology and ordinals"],"downwards":[]}
{"upwards":["Ordinal number","History"],"downwards":[]}
{"upwards":["Extensionality"],"downwards":["Example","In mathematics"]}
{"upwards":["Extensionality","Example"],"downwards":[]}
{"upwards":["Extensionality","In mathematics"],"downwards":[]}
{"upwards":["Higher-order logic"],"downwards":["Quantification scope","Semantics","Properties"]}
{"upwards":["Higher-order logic","Quantification scope"],"downwards":[]}
{"upwards":["Higher-order logic","Semantics"],"downwards":[]}
{"upwards":["Higher-order logic","Properties"],"downwards":[]}
{"upwards":["Finite set"],"downwards":["Definition and terminology","Basic properties","Necessary and sufficient conditions for finiteness","Foundational issues","Set-theoretic definitions of finiteness"]}
{"upwards":["Finite set","Definition and terminology"],"downwards":[]}
{"upwards":["Finite set","Basic properties"],"downwards":[]}
{"upwards":["Finite set","Necessary and sufficient conditions for finiteness"],"downwards":[]}
{"upwards":["Finite set","Foundational issues"],"downwards":[]}
{"upwards":["Finite set","Set-theoretic definitions of finiteness"],"downwards":["Other concepts of finiteness"]}
{"upwards":["Finite set","Set-theoretic definitions of finiteness","Other concepts of finiteness"],"downwards":[]}
{"upwards":["Predicate (mathematical logic)"],"downwards":["Predicates in different systems"]}
{"upwards":["Predicate (mathematical logic)","Predicates in different systems"],"downwards":[]}
{"upwards":["Ordered pair"],"downwards":["Generalities","Informal and formal definitions","Defining the ordered pair using set theory","Category theory"]}
{"upwards":["Ordered pair","Generalities"],"downwards":[]}
{"upwards":["Ordered pair","Informal and formal definitions"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory"],"downwards":["Wiener's definition","Hausdorff's definition","Kuratowski's definition","Quine\u2013Rosser definition","Cantor\u2013Frege definition","Morse definition","Axiomatic definition"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Wiener's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Hausdorff's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition"],"downwards":["Variants","Proving that definitions satisfy the characteristic property"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Variants"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Proving that definitions satisfy the characteristic property"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Quine\u2013Rosser definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Cantor\u2013Frege definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Morse definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Axiomatic definition"],"downwards":[]}
{"upwards":["Ordered pair","Category theory"],"downwards":[]}
{"upwards":["Transitive set"],"downwards":["Examples","Properties","Transitive closure","Transitive models of set theory"]}
{"upwards":["Transitive set","Examples"],"downwards":[]}
{"upwards":["Transitive set","Properties"],"downwards":[]}
{"upwards":["Transitive set","Transitive closure"],"downwards":[]}
{"upwards":["Transitive set","Transitive models of set theory"],"downwards":[]}
{"upwards":["Forcing (mathematics)"],"downwards":["Intuition","Forcing conditions and forcing posets","Generic filters","P-names and interpretations","Forcing","Consistency","Cohen forcing","The countable chain condition","Easton forcing","Random reals","Boolean-valued models","Meta-mathematical explanation","Logical explanation"]}
{"upwards":["Forcing (mathematics)","Intuition"],"downwards":["The role of the model"]}
{"upwards":["Forcing (mathematics)","Intuition","The role of the model"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Forcing conditions and forcing posets"],"downwards":["Examples"]}
{"upwards":["Forcing (mathematics)","Forcing conditions and forcing posets","Examples"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Generic filters"],"downwards":[]}
{"upwards":["Forcing (mathematics)","P-names and interpretations"],"downwards":["Rigorous definitions"]}
{"upwards":["Forcing (mathematics)","P-names and interpretations","Rigorous definitions"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Forcing"],"downwards":["Internal definition"]}
{"upwards":["Forcing (mathematics)","Forcing","Internal definition"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Consistency"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Cohen forcing"],"downwards":[]}
{"upwards":["Forcing (mathematics)","The countable chain condition"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Easton forcing"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Random reals"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Boolean-valued models"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Meta-mathematical explanation"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Logical explanation"],"downwards":[]}
{"upwards":["Urelement"],"downwards":["Theory","Urelements in set theory","Quine atoms","References","External links"]}
{"upwards":["Urelement","Theory"],"downwards":[]}
{"upwards":["Urelement","Urelements in set theory"],"downwards":[]}
{"upwards":["Urelement","Quine atoms"],"downwards":[]}
{"upwards":["Urelement","References"],"downwards":[]}
{"upwards":["Urelement","External links"],"downwards":[]}
{"upwards":["Inhabited set"],"downwards":["Comparison with nonempty sets","Example"]}
{"upwards":["Inhabited set","Comparison with nonempty sets"],"downwards":[]}
{"upwards":["Inhabited set","Example"],"downwards":[]}
{"upwards":["Computable set"],"downwards":["Formal definition","Examples and non-examples","Properties"]}
{"upwards":["Computable set","Formal definition"],"downwards":[]}
{"upwards":["Computable set","Examples and non-examples"],"downwards":[]}
{"upwards":["Computable set","Properties"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)"],"downwards":["Definitions","Generalization to ultra prefilters","Characterizations","Examples, properties, and sufficient conditions","The ultrafilter lemma","Completeness","Ordering on ultrafilters","Ultrafilters on \u2118(\u03c9)"]}
{"upwards":["Ultrafilter (set theory)","Definitions"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Generalization to ultra prefilters"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Characterizations"],"downwards":["Grills and filter-grills","Free or principal"]}
{"upwards":["Ultrafilter (set theory)","Characterizations","Grills and filter-grills"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Characterizations","Free or principal"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Examples, properties, and sufficient conditions"],"downwards":["Monad structure"]}
{"upwards":["Ultrafilter (set theory)","Examples, properties, and sufficient conditions","Monad structure"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma"],"downwards":["Relationships to other statements under ZF"]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF"],"downwards":["Statements that cannot be deduced","Equivalent statements","Weaker statements"]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Statements that cannot be deduced"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Equivalent statements"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Weaker statements"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Completeness"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Ordering on ultrafilters"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Ultrafilters on \u2118(\u03c9)"],"downwards":[]}
{"upwards":["Free logic"],"downwards":["Explanation","Interpretation"]}
{"upwards":["Free logic","Explanation"],"downwards":[]}
{"upwards":["Free logic","Interpretation"],"downwards":[]}
{"upwards":["Quantifier (logic)"],"downwards":["Relations to logical conjunction and disjunction","Algebraic approaches to quantification","Notation","Order of quantifiers (nesting)","Equivalent expressions","Range of quantification","Formal semantics","Paucal, multal and other degree quantifiers","Other quantifiers","History"]}
{"upwards":["Quantifier (logic)","Relations to logical conjunction and disjunction"],"downwards":["Infinite domain of discourse"]}
{"upwards":["Quantifier (logic)","Relations to logical conjunction and disjunction","Infinite domain of discourse"],"downwards":[]}
{"upwards":["Quantifier (logic)","Algebraic approaches to quantification"],"downwards":[]}
{"upwards":["Quantifier (logic)","Notation"],"downwards":[]}
{"upwards":["Quantifier (logic)","Order of quantifiers (nesting)"],"downwards":[]}
{"upwards":["Quantifier (logic)","Equivalent expressions"],"downwards":["Quantifier elimination"]}
{"upwards":["Quantifier (logic)","Equivalent expressions","Quantifier elimination"],"downwards":[]}
{"upwards":["Quantifier (logic)","Range of quantification"],"downwards":[]}
{"upwards":["Quantifier (logic)","Formal semantics"],"downwards":[]}
{"upwards":["Quantifier (logic)","Paucal, multal and other degree quantifiers"],"downwards":[]}
{"upwards":["Quantifier (logic)","Other quantifiers"],"downwards":[]}
{"upwards":["Quantifier (logic)","History"],"downwards":[]}
{"upwards":["Hereditary set"],"downwards":["Examples","In formulations of set theory","Assumptions"]}
{"upwards":["Hereditary set","Examples"],"downwards":[]}
{"upwards":["Hereditary set","In formulations of set theory"],"downwards":[]}
{"upwards":["Hereditary set","Assumptions"],"downwards":[]}
{"upwards":["Monadic predicate calculus"],"downwards":["Expressiveness","Relationship with term logic","Variants","Footnotes"]}
{"upwards":["Monadic predicate calculus","Expressiveness"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Relationship with term logic"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Variants"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Footnotes"],"downwards":[]}
{"upwards":["Operation (mathematics)"],"downwards":["Types of operation","Definition"]}
{"upwards":["Operation (mathematics)","Types of operation"],"downwards":[]}
{"upwards":["Operation (mathematics)","Definition"],"downwards":[]}
{"upwards":["General set theory"],"downwards":["Ontology","Axioms","Discussion","History","Footnotes","References","External links"]}
{"upwards":["General set theory","Ontology"],"downwards":[]}
{"upwards":["General set theory","Axioms"],"downwards":[]}
{"upwards":["General set theory","Discussion"],"downwards":["Metamathematics","Peano arithmetic","Infinite sets"]}
{"upwards":["General set theory","Discussion","Metamathematics"],"downwards":[]}
{"upwards":["General set theory","Discussion","Peano arithmetic"],"downwards":[]}
{"upwards":["General set theory","Discussion","Infinite sets"],"downwards":[]}
{"upwards":["General set theory","History"],"downwards":[]}
{"upwards":["General set theory","Footnotes"],"downwards":[]}
{"upwards":["General set theory","References"],"downwards":[]}
{"upwards":["General set theory","External links"],"downwards":[]}
{"upwards":["Binary operation"],"downwards":["Terminology","Properties and examples","Notation","Binary operations as ternary relations","External binary operations"]}
{"upwards":["Binary operation","Terminology"],"downwards":[]}
{"upwards":["Binary operation","Properties and examples"],"downwards":[]}
{"upwards":["Binary operation","Notation"],"downwards":[]}
{"upwards":["Binary operation","Binary operations as ternary relations"],"downwards":[]}
{"upwards":["Binary operation","External binary operations"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory"],"downwards":["Axioms","Related definitions","Theorems","Metalogic"]}
{"upwards":["Kripke\u2013Platek set theory","Axioms"],"downwards":["Empty set","Comparison with Zermelo-Fraenkel set theory"]}
{"upwards":["Kripke\u2013Platek set theory","Axioms","Empty set"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Axioms","Comparison with Zermelo-Fraenkel set theory"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Related definitions"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Theorems"],"downwards":["Admissible sets","Cartesian products exist"]}
{"upwards":["Kripke\u2013Platek set theory","Theorems","Admissible sets"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Theorems","Cartesian products exist"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory"],"downwards":["Introduction","Subtheories of ZF","Sorted theories"]}
{"upwards":["Constructive set theory","Introduction"],"downwards":["Constructive outlook","History and overview"]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook"],"downwards":["Preliminary on the use of intuitionistic logic","Imposed restrictions on a set theory","Metalogic","Anti-classical principles"]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Preliminary on the use of intuitionistic logic"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Imposed restrictions on a set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Anti-classical principles"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","History and overview"],"downwards":["Models"]}
{"upwards":["Constructive set theory","Introduction","History and overview","Models"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF"],"downwards":["Notation","Equality","Merging sets","Set existence","BCST","ECST","Functions","Choice functions","Arithmetic","Exponentiation","Analysis","Non-constructive principles","Induction","Strong Collection","Constructive Zermelo\u2013Fraenkel","Metalogic of CZF","Intuitionistic Zermelo\u2013Fraenkel","Intuitionistic Z","Intuitionistic KP"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation"],"downwards":["Language","Variables","Classes"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Language"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Variables"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Classes"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Equality"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Merging sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Set existence"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST"],"downwards":["Separation","No universal set","Predicativity","Replacement"]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Separation"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","No universal set"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Predicativity"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Replacement"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST"],"downwards":["Weaker formulations of infinity","Number bounds"]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST","Weaker formulations of infinity"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST","Number bounds"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions"],"downwards":["Finiteness","General note on programs and functions","Total functional relations","Characteristic functions","Computable sets","Boundedness criteria"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Finiteness"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","General note on programs and functions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Total functional relations"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Characteristic functions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Computable sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Boundedness criteria"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions"],"downwards":["Diaconescu's theorem","Analysis of Diaconescu's theorem","Regularity implies PEM"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Diaconescu's theorem"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Analysis of Diaconescu's theorem"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Regularity implies PEM"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic"],"downwards":["Recursion","Moderate induction in ECST","Induction without infinite sets"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Recursion"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Moderate induction in ECST"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Induction without infinite sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation"],"downwards":["On countable sets","The class of subsets of a set","Metalogic","Category and type theoretic notions"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","On countable sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","The class of subsets of a set"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","Category and type theoretic notions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis"],"downwards":["Cauchy sequences","Cauchy reals","Towards the Dedekind reals","Constructive schools"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Cauchy sequences"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Cauchy reals"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Towards the Dedekind reals"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Constructive schools"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Non-constructive principles"],"downwards":["Infinite trees"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Non-constructive principles","Infinite trees"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction"],"downwards":["Mathematical induction","Set Induction","Metalogic","Relation to ZF"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Mathematical induction"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Set Induction"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Relation to ZF"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Strong Collection"],"downwards":["Metalogic"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Strong Collection","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel"],"downwards":["Binary refinement","Subset Collection","Fullness"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Binary refinement"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Subset Collection"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Fullness"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF"],"downwards":["Unprovable claims","Ordinal analysis","Models","Indecomposability","Breaking with ZF"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Unprovable claims"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Ordinal analysis"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Models"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Indecomposability"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Breaking with ZF"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Zermelo\u2013Fraenkel"],"downwards":["Metalogic"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Zermelo\u2013Fraenkel","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Z"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic KP"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories"],"downwards":["Constructive set theory","Bishop style set theory","Category theories"]}
{"upwards":["Constructive set theory","Sorted theories","Constructive set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories","Bishop style set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories","Category theories"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory"],"downwards":["Classes in set theory","Axiomatization of NBG","History","NBG, ZFC, and MK","Category theory","Notes","References","Bibliography","External links"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory"],"downwards":["The uses of classes","Axiom schema versus class existence theorem"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory","The uses of classes"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory","Axiom schema versus class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG"],"downwards":["Classes and sets","Definitions and axioms of extensionality and pairing","Class existence axioms and axiom of regularity","Class existence theorem","Extending the class existence theorem","Set axioms","Axiom of global choice"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Classes and sets"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Definitions and axioms of extensionality and pairing"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Class existence axioms and axiom of regularity"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Extending the class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Set axioms"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Axiom of global choice"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History"],"downwards":["Von Neumann's 1925 axiom system","Von Neumann's 1929 axiom system","Bernays' axiom system","G\u00f6del's axiom system (NBG)"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Von Neumann's 1925 axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Von Neumann's 1929 axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Bernays' axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","G\u00f6del's axiom system (NBG)"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","NBG, ZFC, and MK"],"downwards":["Models"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","NBG, ZFC, and MK","Models"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Category theory"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Notes"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","References"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Bibliography"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","External links"],"downwards":[]}
{"upwards":["Ackermann set theory"],"downwards":["The language","The axioms","Relation to Zermelo\u2013Fraenkel set theory","AST and category theory"]}
{"upwards":["Ackermann set theory","The language"],"downwards":[]}
{"upwards":["Ackermann set theory","The axioms"],"downwards":[]}
{"upwards":["Ackermann set theory","Relation to Zermelo\u2013Fraenkel set theory"],"downwards":[]}
{"upwards":["Ackermann set theory","AST and category theory"],"downwards":[]}
{"upwards":["New Foundations"],"downwards":["The Type Theory TST","Quinean set theory","Consequences","The consistency problem and related partial results","Models of NFU","Strong axioms of infinity","The system ML (Mathematical Logic)"]}
{"upwards":["New Foundations","The Type Theory TST"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory"],"downwards":["Axioms and stratification","Finite axiomatization","Ordered pairs","Natural numbers and the axiom of infinity"]}
{"upwards":["New Foundations","Quinean set theory","Axioms and stratification"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Finite axiomatization"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Ordered pairs"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Natural numbers and the axiom of infinity"],"downwards":[]}
{"upwards":["New Foundations","Consequences"],"downwards":["Admissibility of useful large sets","Resolution of set-theoretic paradoxes","Cartesian closure"]}
{"upwards":["New Foundations","Consequences","Admissibility of useful large sets"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes"],"downwards":["Cantor's paradox and Cantorian sets","Burali-Forti paradox and the T operation"]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes","Cantor's paradox and Cantorian sets"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes","Burali-Forti paradox and the T operation"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Cartesian closure"],"downwards":[]}
{"upwards":["New Foundations","The consistency problem and related partial results"],"downwards":[]}
{"upwards":["New Foundations","Models of NFU"],"downwards":["Self-sufficiency of mathematical foundations in NFU","Facts about the automorphism j"]}
{"upwards":["New Foundations","Models of NFU","Self-sufficiency of mathematical foundations in NFU"],"downwards":[]}
{"upwards":["New Foundations","Models of NFU","Facts about the automorphism j"],"downwards":[]}
{"upwards":["New Foundations","Strong axioms of infinity"],"downwards":[]}
{"upwards":["New Foundations","The system ML (Mathematical Logic)"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory"],"downwards":["MK axioms and ontology","Discussion","The axioms in Kelley's General Topology","Notes","References","External links"]}
{"upwards":["Morse\u2013Kelley set theory","MK axioms and ontology"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Discussion"],"downwards":["Model theory","History"]}
{"upwards":["Morse\u2013Kelley set theory","Discussion","Model theory"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Discussion","History"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","The axioms in Kelley's General Topology"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Notes"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","References"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","External links"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory"],"downwards":["Axioms","Implementation in the Mizar system","Implementation in Metamath"]}
{"upwards":["Tarski\u2013Grothendieck set theory","Axioms"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory","Implementation in the Mizar system"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory","Implementation in Metamath"],"downwards":[]}
{"upwards":["Formal language"],"downwards":["History","Words over an alphabet","Definition","Examples","Language-specification formalisms","Operations on languages","Applications"]}
{"upwards":["Formal language","History"],"downwards":[]}
{"upwards":["Formal language","Words over an alphabet"],"downwards":[]}
{"upwards":["Formal language","Definition"],"downwards":[]}
{"upwards":["Formal language","Examples"],"downwards":["Constructions"]}
{"upwards":["Formal language","Examples","Constructions"],"downwards":[]}
{"upwards":["Formal language","Language-specification formalisms"],"downwards":[]}
{"upwards":["Formal language","Operations on languages"],"downwards":[]}
{"upwards":["Formal language","Applications"],"downwards":["Programming languages","Formal theories, systems, and proofs"]}
{"upwards":["Formal language","Applications","Programming languages"],"downwards":[]}
{"upwards":["Formal language","Applications","Formal theories, systems, and proofs"],"downwards":["Interpretations and models"]}
{"upwards":["Formal language","Applications","Formal theories, systems, and proofs","Interpretations and models"],"downwards":[]}
{"upwards":["Logical conjunction"],"downwards":["Notation","Definition","Introduction and elimination rules","Negation","Properties","Applications in computer engineering","Set-theoretic correspondence","Natural language"]}
{"upwards":["Logical conjunction","Notation"],"downwards":[]}
{"upwards":["Logical conjunction","Definition"],"downwards":["Truth table","Defined by other operators"]}
{"upwards":["Logical conjunction","Definition","Truth table"],"downwards":[]}
{"upwards":["Logical conjunction","Definition","Defined by other operators"],"downwards":[]}
{"upwards":["Logical conjunction","Introduction and elimination rules"],"downwards":[]}
{"upwards":["Logical conjunction","Negation"],"downwards":["Definition","Other proof strategies"]}
{"upwards":["Logical conjunction","Negation","Definition"],"downwards":[]}
{"upwards":["Logical conjunction","Negation","Other proof strategies"],"downwards":[]}
{"upwards":["Logical conjunction","Properties"],"downwards":[]}
{"upwards":["Logical conjunction","Applications in computer engineering"],"downwards":[]}
{"upwards":["Logical conjunction","Set-theoretic correspondence"],"downwards":[]}
{"upwards":["Logical conjunction","Natural language"],"downwards":[]}
{"upwards":["Ground expression"],"downwards":["Examples","Formal definitions"]}
{"upwards":["Ground expression","Examples"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions"],"downwards":["Ground term","Ground atom","Ground formula"]}
{"upwards":["Ground expression","Formal definitions","Ground term"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground atom"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground formula"],"downwards":[]}
{"upwards":["Negation"],"downwards":["Definition","Notation","Properties","Rules of inference","Programming language and ordinary language","Kripke semantics"]}
{"upwards":["Negation","Definition"],"downwards":[]}
{"upwards":["Negation","Notation"],"downwards":[]}
{"upwards":["Negation","Properties"],"downwards":["Double negation","Distributivity","Linearity","Self dual","Negations of quantifiers"]}
{"upwards":["Negation","Properties","Double negation"],"downwards":[]}
{"upwards":["Negation","Properties","Distributivity"],"downwards":[]}
{"upwards":["Negation","Properties","Linearity"],"downwards":[]}
{"upwards":["Negation","Properties","Self dual"],"downwards":[]}
{"upwards":["Negation","Properties","Negations of quantifiers"],"downwards":[]}
{"upwards":["Negation","Rules of inference"],"downwards":[]}
{"upwards":["Negation","Programming language and ordinary language"],"downwards":[]}
{"upwards":["Negation","Kripke semantics"],"downwards":[]}
{"upwards":["Metalanguage"],"downwards":["Types of metalanguage","In natural language","Types of expressions","Role in metaphor","Metaprogramming"]}
{"upwards":["Metalanguage","Types of metalanguage"],"downwards":["Embedded","Ordered","Nested"]}
{"upwards":["Metalanguage","Types of metalanguage","Embedded"],"downwards":[]}
{"upwards":["Metalanguage","Types of metalanguage","Ordered"],"downwards":[]}
{"upwards":["Metalanguage","Types of metalanguage","Nested"],"downwards":[]}
{"upwards":["Metalanguage","In natural language"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions"],"downwards":["Deductive systems","Metavariables","Metatheories and metatheorems","Interpretations"]}
{"upwards":["Metalanguage","Types of expressions","Deductive systems"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Metavariables"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Metatheories and metatheorems"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Interpretations"],"downwards":[]}
{"upwards":["Metalanguage","Role in metaphor"],"downwards":[]}
{"upwards":["Metalanguage","Metaprogramming"],"downwards":[]}
{"upwards":["Sentence (mathematical logic)"],"downwards":["Example"]}
{"upwards":["Sentence (mathematical logic)","Example"],"downwards":[]}
{"upwards":["Expression (mathematics)"],"downwards":["Examples","Syntax versus semantics","Variables"]}
{"upwards":["Expression (mathematics)","Examples"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics"],"downwards":["Syntax","Semantics","Formal languages and lambda calculus"]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Syntax"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Semantics"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Formal languages and lambda calculus"],"downwards":[]}
{"upwards":["Expression (mathematics)","Variables"],"downwards":[]}
{"upwards":["Logical constant"],"downwards":[]}
{"upwards":["Variable (mathematics)"],"downwards":["History","Notation","Specific kinds of variables","Moduli spaces","Conventional variable names"]}
{"upwards":["Variable (mathematics)","History"],"downwards":[]}
{"upwards":["Variable (mathematics)","Notation"],"downwards":[]}
{"upwards":["Variable (mathematics)","Specific kinds of variables"],"downwards":["Dependent and independent variables","Examples"]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Dependent and independent variables"],"downwards":[]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Examples"],"downwards":["Example: the ideal gas law"]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Examples","Example: the ideal gas law"],"downwards":[]}
{"upwards":["Variable (mathematics)","Moduli spaces"],"downwards":[]}
{"upwards":["Variable (mathematics)","Conventional variable names"],"downwards":[]}
{"upwards":["Uninterpreted function"],"downwards":["Example","Discussion"]}
{"upwards":["Uninterpreted function","Example"],"downwards":[]}
{"upwards":["Uninterpreted function","Discussion"],"downwards":[]}
{"upwards":["Predicate variable"],"downwards":["Notation","Usage"]}
{"upwards":["Predicate variable","Notation"],"downwards":[]}
{"upwards":["Predicate variable","Usage"],"downwards":[]}
{"upwards":["Universal quantification"],"downwards":["Basics","Properties","Universal closure","As adjoint"]}
{"upwards":["Universal quantification","Basics"],"downwards":["Notation"]}
{"upwards":["Universal quantification","Basics","Notation"],"downwards":[]}
{"upwards":["Universal quantification","Properties"],"downwards":["Negation","Other connectives","Rules of inference","The empty set"]}
{"upwards":["Universal quantification","Properties","Negation"],"downwards":[]}
{"upwards":["Universal quantification","Properties","Other connectives"],"downwards":[]}
{"upwards":["Universal quantification","Properties","Rules of inference"],"downwards":[]}
{"upwards":["Universal quantification","Properties","The empty set"],"downwards":[]}
{"upwards":["Universal quantification","Universal closure"],"downwards":[]}
{"upwards":["Universal quantification","As adjoint"],"downwards":[]}
{"upwards":["Material conditional"],"downwards":["Notation","Definitions","Formal properties","Discrepancies with natural language"]}
{"upwards":["Material conditional","Notation"],"downwards":["History"]}
{"upwards":["Material conditional","Notation","History"],"downwards":[]}
{"upwards":["Material conditional","Definitions"],"downwards":["Semantics","Truth table","Deductive definition"]}
{"upwards":["Material conditional","Definitions","Semantics"],"downwards":[]}
{"upwards":["Material conditional","Definitions","Truth table"],"downwards":[]}
{"upwards":["Material conditional","Definitions","Deductive definition"],"downwards":[]}
{"upwards":["Material conditional","Formal properties"],"downwards":[]}
{"upwards":["Material conditional","Discrepancies with natural language"],"downwards":[]}
{"upwards":["Existential quantification"],"downwards":["Basics","Properties","As adjoint","Encoding","Origin"]}
{"upwards":["Existential quantification","Basics"],"downwards":[]}
{"upwards":["Existential quantification","Properties"],"downwards":["Negation","Rules of inference","The empty set"]}
{"upwards":["Existential quantification","Properties","Negation"],"downwards":[]}
{"upwards":["Existential quantification","Properties","Rules of inference"],"downwards":[]}
{"upwards":["Existential quantification","Properties","The empty set"],"downwards":[]}
{"upwards":["Existential quantification","As adjoint"],"downwards":[]}
{"upwards":["Existential quantification","Encoding"],"downwards":[]}
{"upwards":["Existential quantification","Origin"],"downwards":[]}
{"upwards":["Arity"],"downwards":["Examples","Terminology"]}
{"upwards":["Arity","Examples"],"downwards":["Nullary","Unary","Binary","Ternary","n-ary","Varying arity"]}
{"upwards":["Arity","Examples","Nullary"],"downwards":[]}
{"upwards":["Arity","Examples","Unary"],"downwards":[]}
{"upwards":["Arity","Examples","Binary"],"downwards":[]}
{"upwards":["Arity","Examples","Ternary"],"downwards":[]}
{"upwards":["Arity","Examples","n-ary"],"downwards":[]}
{"upwards":["Arity","Examples","Varying arity"],"downwards":[]}
{"upwards":["Arity","Terminology"],"downwards":[]}
{"upwards":["Logical biconditional"],"downwards":["Definition","Properties","Rules of inference","Colloquial usage"]}
{"upwards":["Logical biconditional","Definition"],"downwards":["Truth table","Venn diagrams"]}
{"upwards":["Logical biconditional","Definition","Truth table"],"downwards":[]}
{"upwards":["Logical biconditional","Definition","Venn diagrams"],"downwards":[]}
{"upwards":["Logical biconditional","Properties"],"downwards":[]}
{"upwards":["Logical biconditional","Rules of inference"],"downwards":["Biconditional introduction","Biconditional elimination"]}
{"upwards":["Logical biconditional","Rules of inference","Biconditional introduction"],"downwards":[]}
{"upwards":["Logical biconditional","Rules of inference","Biconditional elimination"],"downwards":[]}
{"upwards":["Logical biconditional","Colloquial usage"],"downwards":[]}
{"upwards":["Automata theory"],"downwards":["History","Automata","Variant definitions of automata","Types of automata","Hierarchy in terms of powers","Applications","Automata simulators","Connection to category theory"]}
{"upwards":["Automata theory","History"],"downwards":[]}
{"upwards":["Automata theory","Automata"],"downwards":["Informal description","Formal definition"]}
{"upwards":["Automata theory","Automata","Informal description"],"downwards":[]}
{"upwards":["Automata theory","Automata","Formal definition"],"downwards":[]}
{"upwards":["Automata theory","Variant definitions of automata"],"downwards":[]}
{"upwards":["Automata theory","Types of automata"],"downwards":["Discrete, continuous, and hybrid automata"]}
{"upwards":["Automata theory","Types of automata","Discrete, continuous, and hybrid automata"],"downwards":[]}
{"upwards":["Automata theory","Hierarchy in terms of powers"],"downwards":[]}
{"upwards":["Automata theory","Applications"],"downwards":[]}
{"upwards":["Automata theory","Automata simulators"],"downwards":[]}
{"upwards":["Automata theory","Connection to category theory"],"downwards":[]}
{"upwards":["List of formal systems"],"downwards":["Mathematical","Other formal systems"]}
{"upwards":["List of formal systems","Mathematical"],"downwards":[]}
{"upwards":["List of formal systems","Other formal systems"],"downwards":[]}
{"upwards":["Logical disjunction"],"downwards":["Inclusive and exclusive disjunction","Notation","Classical disjunction","Applications in computer science","Set theory","Natural language"]}
{"upwards":["Logical disjunction","Inclusive and exclusive disjunction"],"downwards":[]}
{"upwards":["Logical disjunction","Notation"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction"],"downwards":["Semantics","Defined by other operators","Properties"]}
{"upwards":["Logical disjunction","Classical disjunction","Semantics"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction","Defined by other operators"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction","Properties"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science"],"downwards":["Bitwise operation","Logical operation","Constructive disjunction"]}
{"upwards":["Logical disjunction","Applications in computer science","Bitwise operation"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science","Logical operation"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science","Constructive disjunction"],"downwards":[]}
{"upwards":["Logical disjunction","Set theory"],"downwards":[]}
{"upwards":["Logical disjunction","Natural language"],"downwards":[]}
{"upwards":["Free variables and bound variables"],"downwards":["Examples","Formal explanation","Natural language"]}
{"upwards":["Free variables and bound variables","Examples"],"downwards":["Variable-binding operators"]}
{"upwards":["Free variables and bound variables","Examples","Variable-binding operators"],"downwards":[]}
{"upwards":["Free variables and bound variables","Formal explanation"],"downwards":["Function expressions"]}
{"upwards":["Free variables and bound variables","Formal explanation","Function expressions"],"downwards":[]}
{"upwards":["Free variables and bound variables","Natural language"],"downwards":[]}
{"upwards":["Axiom schema"],"downwards":["Formal definition","Finite axiomatization","Examples","Finitely axiomatized theories","In higher-order logic"]}
{"upwards":["Axiom schema","Formal definition"],"downwards":[]}
{"upwards":["Axiom schema","Finite axiomatization"],"downwards":[]}
{"upwards":["Axiom schema","Examples"],"downwards":[]}
{"upwards":["Axiom schema","Finitely axiomatized theories"],"downwards":[]}
{"upwards":["Axiom schema","In higher-order logic"],"downwards":[]}
{"upwards":["Uniqueness quantification"],"downwards":["Proving uniqueness","Reduction to ordinary existential and universal quantification","Generalizations"]}
{"upwards":["Uniqueness quantification","Proving uniqueness"],"downwards":[]}
{"upwards":["Uniqueness quantification","Reduction to ordinary existential and universal quantification"],"downwards":[]}
{"upwards":["Uniqueness quantification","Generalizations"],"downwards":[]}
{"upwards":["Atomic formula"],"downwards":["Atomic formula in first-order logic"]}
{"upwards":["Atomic formula","Atomic formula in first-order logic"],"downwards":[]}
{"upwards":["Well-formed formula"],"downwards":["Introduction","Propositional calculus","Predicate logic","Atomic and open formulas","Closed formulas","Properties applicable to formulas","Usage of the terminology"]}
{"upwards":["Well-formed formula","Introduction"],"downwards":[]}
{"upwards":["Well-formed formula","Propositional calculus"],"downwards":[]}
{"upwards":["Well-formed formula","Predicate logic"],"downwards":[]}
{"upwards":["Well-formed formula","Atomic and open formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Closed formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Properties applicable to formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Usage of the terminology"],"downwards":[]}
{"upwards":["Formal grammar"],"downwards":["History","Introductory example","Formal definition","The Chomsky hierarchy","Analytic grammars"]}
{"upwards":["Formal grammar","History"],"downwards":[]}
{"upwards":["Formal grammar","Introductory example"],"downwards":["Example 1","Examples 2 and 3"]}
{"upwards":["Formal grammar","Introductory example","Example 1"],"downwards":[]}
{"upwards":["Formal grammar","Introductory example","Examples 2 and 3"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition"],"downwards":["The syntax of grammars","Some mathematical constructs regarding formal grammars","Example"]}
{"upwards":["Formal grammar","Formal definition","The syntax of grammars"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition","Some mathematical constructs regarding formal grammars"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition","Example"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy"],"downwards":["Context-free grammars","Regular grammars","Other forms of generative grammars","Recursive grammars"]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Context-free grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Regular grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Other forms of generative grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Recursive grammars"],"downwards":[]}
{"upwards":["Formal grammar","Analytic grammars"],"downwards":[]}
{"upwards":["Ground expression"],"downwards":["Examples","Formal definitions"]}
{"upwards":["Ground expression","Examples"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions"],"downwards":["Ground term","Ground atom","Ground formula"]}
{"upwards":["Ground expression","Formal definitions","Ground term"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground atom"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground formula"],"downwards":[]}
{"upwards":["Symbol (formal)"],"downwards":["Overview","Can words be modeled as formal symbols?","References"]}
{"upwards":["Symbol (formal)","Overview"],"downwards":[]}
{"upwards":["Symbol (formal)","Can words be modeled as formal symbols?"],"downwards":[]}
{"upwards":["Symbol (formal)","References"],"downwards":[]}
{"upwards":["Atomic sentence"],"downwards":["Examples","Interpretations","Translating sentences from a natural language into an artificial language","Philosophical significance"]}
{"upwards":["Atomic sentence","Examples"],"downwards":["Assumptions","Atomic sentences","Atomic formulae","Compound sentences","Compound formulae"]}
{"upwards":["Atomic sentence","Examples","Assumptions"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Atomic sentences"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Atomic formulae"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Compound sentences"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Compound formulae"],"downwards":[]}
{"upwards":["Atomic sentence","Interpretations"],"downwards":[]}
{"upwards":["Atomic sentence","Translating sentences from a natural language into an artificial language"],"downwards":[]}
{"upwards":["Atomic sentence","Philosophical significance"],"downwards":[]}
{"upwards":["Substitution (logic)"],"downwards":["Propositional logic","First-order logic","Algebra"]}
{"upwards":["Substitution (logic)","Propositional logic"],"downwards":["Definition","Tautologies"]}
{"upwards":["Substitution (logic)","Propositional logic","Definition"],"downwards":[]}
{"upwards":["Substitution (logic)","Propositional logic","Tautologies"],"downwards":[]}
{"upwards":["Substitution (logic)","First-order logic"],"downwards":[]}
{"upwards":["Substitution (logic)","Algebra"],"downwards":[]}
{"upwards":["Formation rule"],"downwards":["Formal language","Formal systems","Propositional and predicate logic"]}
{"upwards":["Formation rule","Formal language"],"downwards":[]}
{"upwards":["Formation rule","Formal systems"],"downwards":[]}
{"upwards":["Formation rule","Propositional and predicate logic"],"downwards":[]}
{"upwards":["Signature (logic)"],"downwards":["Definition","Other conventions","Use of signatures in logic and algebra","Many-sorted signatures"]}
{"upwards":["Signature (logic)","Definition"],"downwards":[]}
{"upwards":["Signature (logic)","Other conventions"],"downwards":[]}
{"upwards":["Signature (logic)","Use of signatures in logic and algebra"],"downwards":[]}
{"upwards":["Signature (logic)","Many-sorted signatures"],"downwards":["Symbol types","Signature"]}
{"upwards":["Signature (logic)","Many-sorted signatures","Symbol types"],"downwards":[]}
{"upwards":["Signature (logic)","Many-sorted signatures","Signature"],"downwards":[]}
{"upwards":["Open formula"],"downwards":[]}
{"upwards":["Conservative extension"],"downwards":["Examples","Model-theoretic conservative extension"]}
{"upwards":["Conservative extension","Examples"],"downwards":[]}
{"upwards":["Conservative extension","Model-theoretic conservative extension"],"downwards":[]}
{"upwards":["Logical equality"],"downwards":["Definition","Alternative descriptions"]}
{"upwards":["Logical equality","Definition"],"downwards":[]}
{"upwards":["Logical equality","Alternative descriptions"],"downwards":[]}
{"upwards":["Functional predicate"],"downwards":["Introducing new function symbols","Doing without functional predicates"]}
{"upwards":["Functional predicate","Introducing new function symbols"],"downwards":[]}
{"upwards":["Functional predicate","Doing without functional predicates"],"downwards":[]}
{"upwards":["Quantifier rank"],"downwards":["Definition","Examples"]}
{"upwards":["Quantifier rank","Definition"],"downwards":[]}
{"upwards":["Quantifier rank","Examples"],"downwards":[]}
{"upwards":["String (computer science)"],"downwards":["Purpose","History","String datatypes","Literal strings","Non-text strings","String processing algorithms","Character string-oriented languages and utilities","Character string functions","Formal theory"]}
{"upwards":["String (computer science)","Purpose"],"downwards":[]}
{"upwards":["String (computer science)","History"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes"],"downwards":["String length","Character encoding","Implementations","Representations","Security concerns"]}
{"upwards":["String (computer science)","String datatypes","String length"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Character encoding"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Implementations"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations"],"downwards":["Null-terminated","Byte- and bit-terminated","Length-prefixed","Strings as records","Other representations"]}
{"upwards":["String (computer science)","String datatypes","Representations","Null-terminated"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Byte- and bit-terminated"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Length-prefixed"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Strings as records"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Other representations"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Security concerns"],"downwards":[]}
{"upwards":["String (computer science)","Literal strings"],"downwards":[]}
{"upwards":["String (computer science)","Non-text strings"],"downwards":[]}
{"upwards":["String (computer science)","String processing algorithms"],"downwards":[]}
{"upwards":["String (computer science)","Character string-oriented languages and utilities"],"downwards":[]}
{"upwards":["String (computer science)","Character string functions"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory"],"downwards":["Concatenation and substrings","Prefixes and suffixes","Reversal","Rotations","Lexicographical ordering","String operations","Topology"]}
{"upwards":["String (computer science)","Formal theory","Concatenation and substrings"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Prefixes and suffixes"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Reversal"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Rotations"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Lexicographical ordering"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","String operations"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Topology"],"downwards":[]}
{"upwards":["Propositional variable"],"downwards":["Uses","Predicate logic"]}
{"upwards":["Propositional variable","Uses"],"downwards":[]}
{"upwards":["Propositional variable","Predicate logic"],"downwards":[]}
{"upwards":["Alphabet (formal languages)"],"downwards":["Notation","Applications"]}
{"upwards":["Alphabet (formal languages)","Notation"],"downwards":[]}
{"upwards":["Alphabet (formal languages)","Applications"],"downwards":[]}
{"upwards":["Extension by definitions"],"downwards":["Definition of relation symbols","Definition of function symbols","Extensions by definitions","Examples"]}
{"upwards":["Extension by definitions","Definition of relation symbols"],"downwards":[]}
{"upwards":["Extension by definitions","Definition of function symbols"],"downwards":[]}
{"upwards":["Extension by definitions","Extensions by definitions"],"downwards":[]}
{"upwards":["Extension by definitions","Examples"],"downwards":[]}
{"upwards":["Non-logical symbol"],"downwards":["Signatures","Models","Informal semantics","Descriptive signs"]}
{"upwards":["Non-logical symbol","Signatures"],"downwards":[]}
{"upwards":["Non-logical symbol","Models"],"downwards":[]}
{"upwards":["Non-logical symbol","Informal semantics"],"downwards":[]}
{"upwards":["Non-logical symbol","Descriptive signs"],"downwards":[]}
{"upwards":["Extension by new constant and function names"],"downwards":[]}
{"upwards":["Spectrum of a sentence"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Spectrum of a sentence","Definition"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Examples"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Properties"],"downwards":["Equivalence to Turing machines","Other properties"]}
{"upwards":["Spectrum of a sentence","Properties","Equivalence to Turing machines"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Properties","Other properties"],"downwards":[]}
{"upwards":["Finitary relation"],"downwards":["Example","Definitions","History"]}
{"upwards":["Finitary relation","Example"],"downwards":[]}
{"upwards":["Finitary relation","Definitions"],"downwards":[]}
{"upwards":["Finitary relation","History"],"downwards":[]}
{"upwards":["Syntax (logic)"],"downwards":["Syntactic entities"]}
{"upwards":["Syntax (logic)","Syntactic entities"],"downwards":["Symbols","Formal language","Formation rules","Propositions","Formal theories","Formal systems","Interpretations"]}
{"upwards":["Syntax (logic)","Syntactic entities","Symbols"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal language"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formation rules"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Propositions"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal theories"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems"],"downwards":["Syntactic consequence within a formal system","Syntactic completeness of a formal system"]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems","Syntactic consequence within a formal system"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems","Syntactic completeness of a formal system"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Interpretations"],"downwards":[]}
{"upwards":["List of set identities and relations"],"downwards":["Notation","One subset involved","Two sets involved","Three sets involved","Cartesian products \u2a2f of finitely many sets","Arbitrary families of sets","Functions and sets","Sequences and collections of families of sets"]}
{"upwards":["List of set identities and relations","Notation"],"downwards":["Elementary set operations"]}
{"upwards":["List of set identities and relations","Notation","Elementary set operations"],"downwards":[]}
{"upwards":["List of set identities and relations","One subset involved"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved"],"downwards":["Formulas for binary set operations \u22c2, \u22c3, \\, and \u2206","De Morgan's laws","Commutativity","Other identities involving two sets","Subsets \u2286 and supersets \u2287"]}
{"upwards":["List of set identities and relations","Two sets involved","Formulas for binary set operations \u22c2, \u22c3, \\, and \u2206"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","De Morgan's laws"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Commutativity"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Other identities involving two sets"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Subsets \u2286 and supersets \u2287"],"downwards":["Set equality","Meets, Joins, and lattice properties"]}
{"upwards":["List of set identities and relations","Two sets involved","Subsets \u2286 and supersets \u2287","Set equality"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Subsets \u2286 and supersets \u2287","Meets, Joins, and lattice properties"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved"],"downwards":["Associativity","Distributivity","Two set subtractions","One set subtraction","Three operations on three sets","Other simplifications"]}
{"upwards":["List of set identities and relations","Three sets involved","Associativity"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Distributivity"],"downwards":["Distributivity and symmetric difference \u2206","Distributivity and set subtraction \\"]}
{"upwards":["List of set identities and relations","Three sets involved","Distributivity","Distributivity and symmetric difference \u2206"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Distributivity","Distributivity and set subtraction \\"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Two set subtractions"],"downwards":["(L\\M)\\R","L\\(M\\R)"]}
{"upwards":["List of set identities and relations","Three sets involved","Two set subtractions","(L\\M)\\R"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Two set subtractions","L\\(M\\R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction"],"downwards":["(L\\M) \u204e R","L\\(M \u204e R)","(L \u204e M)\\R","L \u204e (M\\R)"]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","(L\\M) \u204e R"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","L\\(M \u204e R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","(L \u204e M)\\R"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","L \u204e (M\\R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets"],"downwards":["(L \u2022 M) \u204e (M \u2022 R)","(L \u2022 M) \u204e (R\\M)","(L\\M) \u204e (L\\R)"]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets","(L \u2022 M) \u204e (M \u2022 R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets","(L \u2022 M) \u204e (R\\M)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets","(L\\M) \u204e (L\\R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Other simplifications"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets"],"downwards":["Binary \u22c2 of finite \u2a2f","Binary \u22c3 of finite \u2a2f","Difference \\ of finite \u2a2f","Finite \u2a2f of differences \\","Symmetric difference \u2206 and finite \u2a2f"]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Binary \u22c2 of finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Binary \u22c3 of finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Difference \\ of finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Finite \u2a2f of differences \\"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Symmetric difference \u2206 and finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets"],"downwards":["Definitions","Distributing unions and intersections","Distributing subtraction over \u22c3 and \u22c2","Commutativity and associativity of \u22c3 and \u22c2","Cartesian products \u03a0 of arbitrarily many sets"]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Definitions"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections"],"downwards":["Binary \u22c2 of arbitrary \u22c3's","Binary \u22c3 of arbitrary \u22c2's","Arbitrary \u22c2's and arbitrary \u22c3's"]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections","Binary \u22c2 of arbitrary \u22c3's"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections","Binary \u22c3 of arbitrary \u22c2's"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections","Arbitrary \u22c2's and arbitrary \u22c3's"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing subtraction over \u22c3 and \u22c2"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Commutativity and associativity of \u22c3 and \u22c2"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets"],"downwards":["Intersections \u22c2 of \u03a0","Unions \u22c3 of \u03a0","Difference \\ of \u03a0","Symmetric difference \u2206 of \u03a0"]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Intersections \u22c2 of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Unions \u22c3 of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Difference \\ of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Symmetric difference \u2206 of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets"],"downwards":["Definitions","(Pre)Images of arbitrary unions \u22c3's and intersections \u22c2's","(Pre)Images of binary set operations","(Pre)Images of set operations on (pre)images","(Pre)Images and Cartesian products \u03a0","(Pre)Image of a single set"]}
{"upwards":["List of set identities and relations","Functions and sets","Definitions"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of arbitrary unions \u22c3's and intersections \u22c2's"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations"],"downwards":["Counter-examples: images of operations not distributing","Conditions guaranteeing that images distribute over set operations","Exact formulas\/equalities for images of set operations"]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations","Counter-examples: images of operations not distributing"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations","Conditions guaranteeing that images distribute over set operations"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations","Exact formulas\/equalities for images of set operations"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of set operations on (pre)images"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images and Cartesian products \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Image of a single set"],"downwards":["Containments \u2286 and intersections \u22c2 of images and preimages"]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Image of a single set","Containments \u2286 and intersections \u22c2 of images and preimages"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets"],"downwards":["Definitions","Power set","Sequences of sets"]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Definitions"],"downwards":["Definitions of categories of families of sets","Elementwise operations on families"]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Definitions","Definitions of categories of families of sets"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Definitions","Elementwise operations on families"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Power set"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Sequences of sets"],"downwards":["Partitions"]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Sequences of sets","Partitions"],"downwards":[]}
{"upwards":["True arithmetic"],"downwards":["Definition","Arithmetic undefinability","Computability properties","Model-theoretic properties","True theory of second-order arithmetic","Notes","References","External links"]}
{"upwards":["True arithmetic","Definition"],"downwards":[]}
{"upwards":["True arithmetic","Arithmetic undefinability"],"downwards":[]}
{"upwards":["True arithmetic","Computability properties"],"downwards":[]}
{"upwards":["True arithmetic","Model-theoretic properties"],"downwards":[]}
{"upwards":["True arithmetic","True theory of second-order arithmetic"],"downwards":[]}
{"upwards":["True arithmetic","Notes"],"downwards":[]}
{"upwards":["True arithmetic","References"],"downwards":[]}
{"upwards":["True arithmetic","External links"],"downwards":[]}
{"upwards":["Peano axioms"],"downwards":["Historical second-order formulation","Peano arithmetic as first-order theory"]}
{"upwards":["Peano axioms","Historical second-order formulation"],"downwards":["Defining arithmetic operations and relations","Models","Consistency"]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations"],"downwards":["Addition","Multiplication","Inequalities"]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Addition"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Multiplication"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Inequalities"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Models"],"downwards":["Set-theoretic models","Interpretation in category theory"]}
{"upwards":["Peano axioms","Historical second-order formulation","Models","Set-theoretic models"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Models","Interpretation in category theory"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Consistency"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory"],"downwards":["Equivalent axiomatizations","Undecidability and incompleteness","Nonstandard models"]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Equivalent axiomatizations"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Undecidability and incompleteness"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Nonstandard models"],"downwards":["Overspill"]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Nonstandard models","Overspill"],"downwards":[]}
{"upwards":["Axiomatic system"],"downwards":["Properties","Relative consistency","Models","Axiomatic method"]}
{"upwards":["Axiomatic system","Properties"],"downwards":[]}
{"upwards":["Axiomatic system","Relative consistency"],"downwards":[]}
{"upwards":["Axiomatic system","Models"],"downwards":["Example"]}
{"upwards":["Axiomatic system","Models","Example"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method"],"downwards":["History","Issues","Example: The Peano axiomatization of natural numbers","Axiomatization"]}
{"upwards":["Axiomatic system","Axiomatic method","History"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Issues"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Example: The Peano axiomatization of natural numbers"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Axiomatization"],"downwards":[]}
{"upwards":["Primitive recursive arithmetic"],"downwards":["Language and axioms","Logic-free calculus"]}
{"upwards":["Primitive recursive arithmetic","Language and axioms"],"downwards":[]}
{"upwards":["Primitive recursive arithmetic","Logic-free calculus"],"downwards":[]}
{"upwards":["Second-order arithmetic"],"downwards":["Definition","Models","Subsystems","Projective determinacy","Coding mathematics"]}
{"upwards":["Second-order arithmetic","Definition"],"downwards":["Syntax","Semantics","Axioms","The full system"]}
{"upwards":["Second-order arithmetic","Definition","Syntax"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Semantics"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Axioms"],"downwards":["Basic","Induction and comprehension schema"]}
{"upwards":["Second-order arithmetic","Definition","Axioms","Basic"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Axioms","Induction and comprehension schema"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","The full system"],"downwards":[]}
{"upwards":["Second-order arithmetic","Models"],"downwards":["Definable functions","More types of models"]}
{"upwards":["Second-order arithmetic","Models","Definable functions"],"downwards":[]}
{"upwards":["Second-order arithmetic","Models","More types of models"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems"],"downwards":["Arithmetical comprehension","The arithmetical hierarchy for formulas","Recursive comprehension","Weaker systems","Stronger systems"]}
{"upwards":["Second-order arithmetic","Subsystems","Arithmetical comprehension"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","The arithmetical hierarchy for formulas"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Recursive comprehension"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Weaker systems"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Stronger systems"],"downwards":[]}
{"upwards":["Second-order arithmetic","Projective determinacy"],"downwards":[]}
{"upwards":["Second-order arithmetic","Coding mathematics"],"downwards":[]}
{"upwards":["Euclidean geometry"],"downwards":["The Elements","Notation and terminology","Some important or well known results","System of measurement and arithmetic","Applications","Later history","As a description of the structure of space","Treatment of infinity","Logical basis"]}
{"upwards":["Euclidean geometry","The Elements"],"downwards":["Axioms","Parallel postulate","Methods of proof"]}
{"upwards":["Euclidean geometry","The Elements","Axioms"],"downwards":[]}
{"upwards":["Euclidean geometry","The Elements","Parallel postulate"],"downwards":[]}
{"upwards":["Euclidean geometry","The Elements","Methods of proof"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology"],"downwards":["Naming of points and figures","Complementary and supplementary angles","Modern versions of Euclid's notation"]}
{"upwards":["Euclidean geometry","Notation and terminology","Naming of points and figures"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology","Complementary and supplementary angles"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology","Modern versions of Euclid's notation"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results"],"downwards":["Pons asinorum","Congruence of triangles","Triangle angle sum","Pythagorean theorem","Thales' theorem","Scaling of area and volume"]}
{"upwards":["Euclidean geometry","Some important or well known results","Pons asinorum"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Congruence of triangles"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Triangle angle sum"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Pythagorean theorem"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Thales' theorem"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Scaling of area and volume"],"downwards":[]}
{"upwards":["Euclidean geometry","System of measurement and arithmetic"],"downwards":[]}
{"upwards":["Euclidean geometry","Applications"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history"],"downwards":["Archimedes and Apollonius","17th century: Descartes","18th century","19th century","20th century and relativity"]}
{"upwards":["Euclidean geometry","Later history","Archimedes and Apollonius"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","17th century: Descartes"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","18th century"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","19th century"],"downwards":["Higher dimensions","Non-Euclidean geometry"]}
{"upwards":["Euclidean geometry","Later history","19th century","Higher dimensions"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","19th century","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","20th century and relativity"],"downwards":[]}
{"upwards":["Euclidean geometry","As a description of the structure of space"],"downwards":[]}
{"upwards":["Euclidean geometry","Treatment of infinity"],"downwards":["Infinite objects","Infinite processes"]}
{"upwards":["Euclidean geometry","Treatment of infinity","Infinite objects"],"downwards":[]}
{"upwards":["Euclidean geometry","Treatment of infinity","Infinite processes"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis"],"downwards":["Classical logic","Modern standards of rigor","Axiomatic formulations"]}
{"upwards":["Euclidean geometry","Logical basis","Classical logic"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis","Modern standards of rigor"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis","Axiomatic formulations"],"downwards":[]}
{"upwards":["Minimal axioms for Boolean algebra"],"downwards":["References"]}
{"upwards":["Minimal axioms for Boolean algebra","References"],"downwards":[]}
{"upwards":["Construction of the real numbers"],"downwards":["Axiomatic definitions","Explicit constructions of models"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions"],"downwards":["Axioms","Tarski's axiomatization of the reals"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms"],"downwards":["On the least upper bound property","On models"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms","On the least upper bound property"],"downwards":[]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms","On models"],"downwards":[]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Tarski's axiomatization of the reals"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models"],"downwards":["Construction from Cauchy sequences","Construction by Dedekind cuts","Construction using hyperreal numbers","Construction from surreal numbers","Construction from integers (Eudoxus reals)","Other constructions"]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from Cauchy sequences"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction by Dedekind cuts"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction using hyperreal numbers"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from surreal numbers"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from integers (Eudoxus reals)"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Other constructions"],"downwards":[]}
{"upwards":["Foundations of geometry"],"downwards":["Axiomatic systems","Euclidean geometry","Non-Euclidean geometry","Other geometries"]}
{"upwards":["Foundations of geometry","Axiomatic systems"],"downwards":["Properties of axiomatic systems"]}
{"upwards":["Foundations of geometry","Axiomatic systems","Properties of axiomatic systems"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry"],"downwards":["Euclid's Elements","A critique of Euclid","Pasch and Peano","Pieri and the Italian school of geometers","Hilbert's axioms","Birkhoff's axioms","School geometry","Other systems"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Euclid's Elements"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","A critique of Euclid"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Pasch and Peano"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Pieri and the Italian school of geometers"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms"],"downwards":["Changes in Hilbert's axioms","Consistency and independence"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms","Changes in Hilbert's axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms","Consistency and independence"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Birkhoff's axioms"],"downwards":["Postulates"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Birkhoff's axioms","Postulates"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry"],"downwards":["Mac Lane's axioms","SMSG (School Mathematics Study Group) axioms","UCSMP (University of Chicago School Mathematics Project) axioms"]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","Mac Lane's axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","SMSG (School Mathematics Study Group) axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","UCSMP (University of Chicago School Mathematics Project) axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Other systems"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry"],"downwards":["Euclid's parallel postulate","Neutral (or absolute) geometry","Hyperbolic geometry","Elliptic geometry"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Euclid's parallel postulate"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry"],"downwards":["Relation to other geometries","Incompleteness"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry","Relation to other geometries"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry","Incompleteness"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Hyperbolic geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Elliptic geometry"],"downwards":["Spherical geometry"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Elliptic geometry","Spherical geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries"],"downwards":["Projective geometry","Affine geometry","Ordered geometry","Finite geometry"]}
{"upwards":["Foundations of geometry","Other geometries","Projective geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Affine geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Ordered geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Finite geometry"],"downwards":[]}
{"upwards":["Euclid's Elements"],"downwards":["History","Influence","Contents","Euclid's method and style of presentation","Criticism","Apocrypha","Editions","References","External links"]}
{"upwards":["Euclid's Elements","History"],"downwards":["Basis in earlier work","Transmission of the text"]}
{"upwards":["Euclid's Elements","History","Basis in earlier work"],"downwards":[]}
{"upwards":["Euclid's Elements","History","Transmission of the text"],"downwards":[]}
{"upwards":["Euclid's Elements","Influence"],"downwards":["In modern mathematics"]}
{"upwards":["Euclid's Elements","Influence","In modern mathematics"],"downwards":[]}
{"upwards":["Euclid's Elements","Contents"],"downwards":[]}
{"upwards":["Euclid's Elements","Euclid's method and style of presentation"],"downwards":[]}
{"upwards":["Euclid's Elements","Criticism"],"downwards":[]}
{"upwards":["Euclid's Elements","Apocrypha"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions"],"downwards":["Translations","Currently in print","Free versions"]}
{"upwards":["Euclid's Elements","Editions","Translations"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions","Currently in print"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions","Free versions"],"downwards":[]}
{"upwards":["Euclid's Elements","References"],"downwards":["Notes","Citations","Sources"]}
{"upwards":["Euclid's Elements","References","Notes"],"downwards":[]}
{"upwards":["Euclid's Elements","References","Citations"],"downwards":[]}
{"upwards":["Euclid's Elements","References","Sources"],"downwards":[]}
{"upwards":["Euclid's Elements","External links"],"downwards":[]}
{"upwards":["Elementary function arithmetic"],"downwards":["Definition","Friedman's grand conjecture","Related systems"]}
{"upwards":["Elementary function arithmetic","Definition"],"downwards":[]}
{"upwards":["Elementary function arithmetic","Friedman's grand conjecture"],"downwards":[]}
{"upwards":["Elementary function arithmetic","Related systems"],"downwards":[]}
{"upwards":["Skolem arithmetic"],"downwards":["Expressive power","Idea of decidability","Complexity","Decidable extensions","Undecidable extensions"]}
{"upwards":["Skolem arithmetic","Expressive power"],"downwards":[]}
{"upwards":["Skolem arithmetic","Idea of decidability"],"downwards":[]}
{"upwards":["Skolem arithmetic","Complexity"],"downwards":[]}
{"upwards":["Skolem arithmetic","Decidable extensions"],"downwards":[]}
{"upwards":["Skolem arithmetic","Undecidable extensions"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined"],"downwards":["Definition","Basis","Truth tables","Examples","Boolean algebras of Boolean operations","Axiomatizing Boolean algebras","Underlying lattice structure","Boolean homomorphisms","Infinitary extensions","Other definitions of Boolean algebra"]}
{"upwards":["Boolean algebras canonically defined","Definition"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Basis"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Truth tables"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples"],"downwards":["Bit vectors","Power set algebra","Representation theorems","Other examples"]}
{"upwards":["Boolean algebras canonically defined","Examples","Bit vectors"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Power set algebra"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Representation theorems"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Other examples"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Boolean algebras of Boolean operations"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Axiomatizing Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Underlying lattice structure"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Boolean homomorphisms"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Infinitary extensions"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Other definitions of Boolean algebra"],"downwards":[]}
{"upwards":["Robinson arithmetic"],"downwards":["Axioms","Metamathematics"]}
{"upwards":["Robinson arithmetic","Axioms"],"downwards":["Variant axiomatizations"]}
{"upwards":["Robinson arithmetic","Axioms","Variant axiomatizations"],"downwards":[]}
{"upwards":["Robinson arithmetic","Metamathematics"],"downwards":[]}
{"upwards":["Tarski's axiomatization of the reals"],"downwards":["The axioms","How these axioms imply a field"]}
{"upwards":["Tarski's axiomatization of the reals","The axioms"],"downwards":[]}
{"upwards":["Tarski's axiomatization of the reals","How these axioms imply a field"],"downwards":[]}
{"upwards":["Boolean algebra (structure)"],"downwards":["History","Definition","Examples","Homomorphisms and isomorphisms","Boolean rings","Ideals and filters","Representations","Axiomatics","Generalizations"]}
{"upwards":["Boolean algebra (structure)","History"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Definition"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Examples"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Homomorphisms and isomorphisms"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Boolean rings"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Ideals and filters"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Representations"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Axiomatics"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Generalizations"],"downwards":[]}
{"upwards":["List of mathematical theories"],"downwards":[]}
{"upwards":["Term (logic)"],"downwards":["Formal definition","Operations with terms","Related concepts"]}
{"upwards":["Term (logic)","Formal definition"],"downwards":["Term structure vs. representation","Structural equality","Ground and linear terms","Building formulas from terms"]}
{"upwards":["Term (logic)","Formal definition","Term structure vs. representation"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Structural equality"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Ground and linear terms"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Building formulas from terms"],"downwards":[]}
{"upwards":["Term (logic)","Operations with terms"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts"],"downwards":["Sorted terms","Lambda terms"]}
{"upwards":["Term (logic)","Related concepts","Sorted terms"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts","Lambda terms"],"downwards":["Motivation","Definition"]}
{"upwards":["Term (logic)","Related concepts","Lambda terms","Motivation"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts","Lambda terms","Definition"],"downwards":[]}
{"upwards":["Non-Euclidean geometry"],"downwards":["Principles","History","Axiomatic basis of non-Euclidean geometry","Models","Uncommon properties","Importance","Planar algebras","Kinematic geometries","Fiction"]}
{"upwards":["Non-Euclidean geometry","Principles"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History"],"downwards":["Background","Discovery of non-Euclidean geometry","Terminology"]}
{"upwards":["Non-Euclidean geometry","History","Background"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History","Discovery of non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History","Terminology"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Axiomatic basis of non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models"],"downwards":["Elliptic geometry","Hyperbolic geometry","Three-dimensional non-Euclidean geometry"]}
{"upwards":["Non-Euclidean geometry","Models","Elliptic geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models","Hyperbolic geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models","Three-dimensional non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Uncommon properties"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Importance"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Planar algebras"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Kinematic geometries"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Fiction"],"downwards":[]}
{"upwards":["Sequent calculus"],"downwards":["Overview","Proving logical formulas","The system LK","Variants"]}
{"upwards":["Sequent calculus","Overview"],"downwards":["Hilbert-style deduction systems","Natural deduction systems","Sequent calculus systems","Distinction between natural deduction and sequent calculus","Origin of word \"sequent\""]}
{"upwards":["Sequent calculus","Overview","Hilbert-style deduction systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Natural deduction systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Sequent calculus systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Distinction between natural deduction and sequent calculus"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Origin of word \"sequent\""],"downwards":[]}
{"upwards":["Sequent calculus","Proving logical formulas"],"downwards":["Reduction trees","Relation to standard axiomatizations"]}
{"upwards":["Sequent calculus","Proving logical formulas","Reduction trees"],"downwards":[]}
{"upwards":["Sequent calculus","Proving logical formulas","Relation to standard axiomatizations"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK"],"downwards":["Inference rules","An intuitive explanation","Example derivations","Relation to analytic tableaux","Structural rules","Properties of the system LK"]}
{"upwards":["Sequent calculus","The system LK","Inference rules"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","An intuitive explanation"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Example derivations"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Relation to analytic tableaux"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Structural rules"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Properties of the system LK"],"downwards":[]}
{"upwards":["Sequent calculus","Variants"],"downwards":["Minor structural alternatives","Absurdity","Substructural logics","Intuitionistic sequent calculus: System LJ"]}
{"upwards":["Sequent calculus","Variants","Minor structural alternatives"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Absurdity"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Substructural logics"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Intuitionistic sequent calculus: System LJ"],"downwards":[]}
{"upwards":["Rule of inference"],"downwards":["Standard form","Example: Hilbert systems for two propositional logics","Admissibility and derivability"]}
{"upwards":["Rule of inference","Standard form"],"downwards":[]}
{"upwards":["Rule of inference","Example: Hilbert systems for two propositional logics"],"downwards":[]}
{"upwards":["Rule of inference","Admissibility and derivability"],"downwards":[]}
{"upwards":["Natural deduction"],"downwards":["Motivation","Judgments and propositions","Introduction and elimination","Hypothetical derivations","Consistency, completeness, and normal forms","First and higher-order extensions","Different presentations of natural deduction","Proofs and type theory","Classical and modal logics","Comparison with other foundational approaches"]}
{"upwards":["Natural deduction","Motivation"],"downwards":[]}
{"upwards":["Natural deduction","Judgments and propositions"],"downwards":[]}
{"upwards":["Natural deduction","Introduction and elimination"],"downwards":[]}
{"upwards":["Natural deduction","Hypothetical derivations"],"downwards":[]}
{"upwards":["Natural deduction","Consistency, completeness, and normal forms"],"downwards":[]}
{"upwards":["Natural deduction","First and higher-order extensions"],"downwards":[]}
{"upwards":["Natural deduction","Different presentations of natural deduction"],"downwards":["Tree-like presentations","Sequential presentations"]}
{"upwards":["Natural deduction","Different presentations of natural deduction","Tree-like presentations"],"downwards":[]}
{"upwards":["Natural deduction","Different presentations of natural deduction","Sequential presentations"],"downwards":[]}
{"upwards":["Natural deduction","Proofs and type theory"],"downwards":["Example: Dependent Type Theory"]}
{"upwards":["Natural deduction","Proofs and type theory","Example: Dependent Type Theory"],"downwards":[]}
{"upwards":["Natural deduction","Classical and modal logics"],"downwards":[]}
{"upwards":["Natural deduction","Comparison with other foundational approaches"],"downwards":["Sequent calculus"]}
{"upwards":["Natural deduction","Comparison with other foundational approaches","Sequent calculus"],"downwards":[]}
{"upwards":["Tarski's axioms"],"downwards":["Overview","The axioms","Discussion","Comparison with Hilbert"]}
{"upwards":["Tarski's axioms","Overview"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms"],"downwards":["Fundamental relations","Congruence axioms","Betweenness axioms","Congruence and betweenness"]}
{"upwards":["Tarski's axioms","The axioms","Fundamental relations"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Congruence axioms"],"downwards":["Commentary"]}
{"upwards":["Tarski's axioms","The axioms","Congruence axioms","Commentary"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Betweenness axioms"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Congruence and betweenness"],"downwards":[]}
{"upwards":["Tarski's axioms","Discussion"],"downwards":[]}
{"upwards":["Tarski's axioms","Comparison with Hilbert"],"downwards":[]}
{"upwards":["Proof theory"],"downwards":["History","Structural proof theory","Ordinal analysis","Provability logic","Reverse mathematics","Functional interpretations","Formal and informal proof","Proof-theoretic semantics"]}
{"upwards":["Proof theory","History"],"downwards":[]}
{"upwards":["Proof theory","Structural proof theory"],"downwards":[]}
{"upwards":["Proof theory","Ordinal analysis"],"downwards":[]}
{"upwards":["Proof theory","Provability logic"],"downwards":[]}
{"upwards":["Proof theory","Reverse mathematics"],"downwards":[]}
{"upwards":["Proof theory","Functional interpretations"],"downwards":[]}
{"upwards":["Proof theory","Formal and informal proof"],"downwards":[]}
{"upwards":["Proof theory","Proof-theoretic semantics"],"downwards":[]}
{"upwards":["Formal system"],"downwards":["Background","History","Examples","Variants"]}
{"upwards":["Formal system","Background"],"downwards":["Recursive","Inference and entailment"]}
{"upwards":["Formal system","Background","Recursive"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment"],"downwards":["Formal language","Deductive system","Logical system"]}
{"upwards":["Formal system","Background","Inference and entailment","Formal language"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Deductive system"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Logical system"],"downwards":[]}
{"upwards":["Formal system","History"],"downwards":["Formalism","QED manifesto"]}
{"upwards":["Formal system","History","Formalism"],"downwards":["Hilbert's program"]}
{"upwards":["Formal system","History","Formalism","Hilbert's program"],"downwards":[]}
{"upwards":["Formal system","History","QED manifesto"],"downwards":[]}
{"upwards":["Formal system","Examples"],"downwards":[]}
{"upwards":["Formal system","Variants"],"downwards":["Proof system"]}
{"upwards":["Formal system","Variants","Proof system"],"downwards":[]}
{"upwards":["Hilbert system"],"downwards":["Formal deductions","Conservative extensions","Metatheorems","Some useful theorems and their proofs","Alternative axiomatizations","Further connections"]}
{"upwards":["Hilbert system","Formal deductions"],"downwards":["Logical axioms"]}
{"upwards":["Hilbert system","Formal deductions","Logical axioms"],"downwards":[]}
{"upwards":["Hilbert system","Conservative extensions"],"downwards":["Existential quantification","Conjunction and disjunction"]}
{"upwards":["Hilbert system","Conservative extensions","Existential quantification"],"downwards":[]}
{"upwards":["Hilbert system","Conservative extensions","Conjunction and disjunction"],"downwards":[]}
{"upwards":["Hilbert system","Metatheorems"],"downwards":[]}
{"upwards":["Hilbert system","Some useful theorems and their proofs"],"downwards":[]}
{"upwards":["Hilbert system","Alternative axiomatizations"],"downwards":[]}
{"upwards":["Hilbert system","Further connections"],"downwards":[]}
{"upwards":["Hilbert's axioms"],"downwards":["The axioms","Hilbert's discarded axiom","Editions and translations of Grundlagen der Geometrie","Application"]}
{"upwards":["Hilbert's axioms","The axioms"],"downwards":["I. Incidence","II. Order","III. Congruence","IV. Parallels","V. Continuity"]}
{"upwards":["Hilbert's axioms","The axioms","I. Incidence"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","II. Order"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","III. Congruence"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","IV. Parallels"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","V. Continuity"],"downwards":[]}
{"upwards":["Hilbert's axioms","Hilbert's discarded axiom"],"downwards":[]}
{"upwards":["Hilbert's axioms","Editions and translations of Grundlagen der Geometrie"],"downwards":[]}
{"upwards":["Hilbert's axioms","Application"],"downwards":[]}
{"upwards":["List of Hilbert systems"],"downwards":["Classical propositional calculus systems","Implicational propositional calculus","Intuitionistic and intermediate logics","Positive implicational calculus","Positive propositional calculus","Equivalential calculus"]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems"],"downwards":["Implication and negation","Implication and falsum","Negation and disjunction","Sheffer's stroke"]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Implication and negation"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Implication and falsum"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Negation and disjunction"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Sheffer's stroke"],"downwards":[]}
{"upwards":["List of Hilbert systems","Implicational propositional calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Intuitionistic and intermediate logics"],"downwards":[]}
{"upwards":["List of Hilbert systems","Positive implicational calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Positive propositional calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Equivalential calculus"],"downwards":[]}
{"upwards":["Ordinal analysis"],"downwards":["History","Definition","Upper bound","Examples","Table of ordinal analyses"]}
{"upwards":["Ordinal analysis","History"],"downwards":[]}
{"upwards":["Ordinal analysis","Definition"],"downwards":["Ordinal notations"]}
{"upwards":["Ordinal analysis","Definition","Ordinal notations"],"downwards":[]}
{"upwards":["Ordinal analysis","Upper bound"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples"],"downwards":["Theories with proof-theoretic ordinal \u03c9","Theories with proof-theoretic ordinal \u03c92","Theories with proof-theoretic ordinal \u03c93","Theories with proof-theoretic ordinal \u03c9n (for n = 2, 3, ... \u03c9)","Theories with proof-theoretic ordinal \u03c9\u03c9","Theories with proof-theoretic ordinal \u03b50","Theories with proof-theoretic ordinal the Feferman\u2013Sch\u00fctte ordinal \u03930","Theories with proof-theoretic ordinal the Bachmann\u2013Howard ordinal","Theories with larger proof-theoretic ordinals"]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c92"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c93"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9n (for n = 2, 3, ... \u03c9)"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9\u03c9"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03b50"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal the Feferman\u2013Sch\u00fctte ordinal \u03930"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal the Bachmann\u2013Howard ordinal"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with larger proof-theoretic ordinals"],"downwards":[]}
{"upwards":["Ordinal analysis","Table of ordinal analyses"],"downwards":["Key"]}
{"upwards":["Ordinal analysis","Table of ordinal analyses","Key"],"downwards":[]}
{"upwards":["Proof of impossibility"],"downwards":["Types of proof","Types of disproof","Pythagorean proof of the existence of irrational numbers","Impossible constructions sought by the ancient Greeks","Euclid's parallel postulate","Fermat's Last Theorem","Richard's paradox","Incompleteness of axiomatic systems: G\u00f6del's proof","Halting problem: Turing's first proof","String compressibility: Chaitin's proof","Integer solutions of Diophantine equations: Hilbert's tenth problem","In social science","In natural science"]}
{"upwards":["Proof of impossibility","Types of proof"],"downwards":["By contradiction"]}
{"upwards":["Proof of impossibility","Types of proof","By contradiction"],"downwards":["By descent"]}
{"upwards":["Proof of impossibility","Types of proof","By contradiction","By descent"],"downwards":[]}
{"upwards":["Proof of impossibility","Types of disproof"],"downwards":[]}
{"upwards":["Proof of impossibility","Pythagorean proof of the existence of irrational numbers"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks"],"downwards":["Angle trisection and doubling the cube","Squaring the circle","Constructing an equilateral n-gon"]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Angle trisection and doubling the cube"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Squaring the circle"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Constructing an equilateral n-gon"],"downwards":[]}
{"upwards":["Proof of impossibility","Euclid's parallel postulate"],"downwards":[]}
{"upwards":["Proof of impossibility","Fermat's Last Theorem"],"downwards":[]}
{"upwards":["Proof of impossibility","Richard's paradox"],"downwards":[]}
{"upwards":["Proof of impossibility","Incompleteness of axiomatic systems: G\u00f6del's proof"],"downwards":[]}
{"upwards":["Proof of impossibility","Halting problem: Turing's first proof"],"downwards":[]}
{"upwards":["Proof of impossibility","String compressibility: Chaitin's proof"],"downwards":[]}
{"upwards":["Proof of impossibility","Integer solutions of Diophantine equations: Hilbert's tenth problem"],"downwards":[]}
{"upwards":["Proof of impossibility","In social science"],"downwards":[]}
{"upwards":["Proof of impossibility","In natural science"],"downwards":[]}
{"upwards":["Reverse mathematics"],"downwards":["General principles","The big five subsystems of second-order arithmetic","Additional systems","\u03c9-models and \u03b2-models"]}
{"upwards":["Reverse mathematics","General principles"],"downwards":["Use of second-order arithmetic","Use of higher-order arithmetic"]}
{"upwards":["Reverse mathematics","General principles","Use of second-order arithmetic"],"downwards":[]}
{"upwards":["Reverse mathematics","General principles","Use of higher-order arithmetic"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic"],"downwards":["The base system RCA0","Weak K\u0151nig's lemma WKL0","Arithmetical comprehension ACA0","Arithmetical transfinite recursion ATR0","\u03a011 comprehension \u03a011-CA0"]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","The base system RCA0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Weak K\u0151nig's lemma WKL0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Arithmetical comprehension ACA0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Arithmetical transfinite recursion ATR0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","\u03a011 comprehension \u03a011-CA0"],"downwards":[]}
{"upwards":["Reverse mathematics","Additional systems"],"downwards":["Stronger Systems"]}
{"upwards":["Reverse mathematics","Additional systems","Stronger Systems"],"downwards":[]}
{"upwards":["Reverse mathematics","\u03c9-models and \u03b2-models"],"downwards":[]}
{"upwards":["Church encoding"],"downwards":["Use","Church numerals","Church Booleans","Predicates","Church pairs","List encodings"]}
{"upwards":["Church encoding","Use"],"downwards":[]}
{"upwards":["Church encoding","Church numerals"],"downwards":["Calculation with Church numerals","Table of functions on Church numerals","Derivation of predecessor function","Division","Signed numbers","Plus and minus","Multiply and  divide","Rational and real numbers","Translation with other representations"]}
{"upwards":["Church encoding","Church numerals","Calculation with Church numerals"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Table of functions on Church numerals"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Derivation of predecessor function"],"downwards":["Another way of defining pred"]}
{"upwards":["Church encoding","Church numerals","Derivation of predecessor function","Another way of defining pred"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Division"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Signed numbers"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Plus and minus"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Multiply and  divide"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Rational and real numbers"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Translation with other representations"],"downwards":[]}
{"upwards":["Church encoding","Church Booleans"],"downwards":[]}
{"upwards":["Church encoding","Predicates"],"downwards":[]}
{"upwards":["Church encoding","Church pairs"],"downwards":[]}
{"upwards":["Church encoding","List encodings"],"downwards":["Two pairs as a list node","One pair as a list node","Represent the list using right fold","Represent the list using Scott encoding"]}
{"upwards":["Church encoding","List encodings","Two pairs as a list node"],"downwards":[]}
{"upwards":["Church encoding","List encodings","One pair as a list node"],"downwards":[]}
{"upwards":["Church encoding","List encodings","Represent the list using right fold"],"downwards":[]}
{"upwards":["Church encoding","List encodings","Represent the list using Scott encoding"],"downwards":[]}
{"upwards":["Computability theory"],"downwards":["Introduction","Turing computability","Areas of research","Relationships between definability, proof and computability","Name","Professional organizations"]}
{"upwards":["Computability theory","Introduction"],"downwards":[]}
{"upwards":["Computability theory","Turing computability"],"downwards":[]}
{"upwards":["Computability theory","Areas of research"],"downwards":["Relative computability and the Turing degrees","Other reducibilities","Rice's theorem and the arithmetical hierarchy","Reverse mathematics","Numberings","The priority method","The lattice of computably enumerable sets","Automorphism problems","Kolmogorov complexity","Frequency computation","Inductive inference","Generalizations of Turing computability","Continuous computability theory"]}
{"upwards":["Computability theory","Areas of research","Relative computability and the Turing degrees"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Other reducibilities"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Rice's theorem and the arithmetical hierarchy"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Reverse mathematics"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Numberings"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","The priority method"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","The lattice of computably enumerable sets"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Automorphism problems"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Kolmogorov complexity"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Frequency computation"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Inductive inference"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Generalizations of Turing computability"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Continuous computability theory"],"downwards":[]}
{"upwards":["Computability theory","Relationships between definability, proof and computability"],"downwards":[]}
{"upwards":["Computability theory","Name"],"downwards":[]}
{"upwards":["Computability theory","Professional organizations"],"downwards":[]}
{"upwards":["Satisfiability"],"downwards":["Reduction of validity to satisfiability","Propositional satisfiability for classical logic","Satisfiability in first-order logic","Satisfiability in model theory","Finite satisfiability","Numerical constraints"]}
{"upwards":["Satisfiability","Reduction of validity to satisfiability"],"downwards":[]}
{"upwards":["Satisfiability","Propositional satisfiability for classical logic"],"downwards":[]}
{"upwards":["Satisfiability","Satisfiability in first-order logic"],"downwards":[]}
{"upwards":["Satisfiability","Satisfiability in model theory"],"downwards":[]}
{"upwards":["Satisfiability","Finite satisfiability"],"downwards":[]}
{"upwards":["Satisfiability","Numerical constraints"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Computably enumerable set"],"downwards":["Formal definition","Equivalent formulations","Examples","Properties","Remarks"]}
{"upwards":["Computably enumerable set","Formal definition"],"downwards":[]}
{"upwards":["Computably enumerable set","Equivalent formulations"],"downwards":[]}
{"upwards":["Computably enumerable set","Examples"],"downwards":[]}
{"upwards":["Computably enumerable set","Properties"],"downwards":[]}
{"upwards":["Computably enumerable set","Remarks"],"downwards":[]}
{"upwards":["Truth value"],"downwards":["Computing","Classical logic","Intuitionistic and constructive logic","Multi-valued logic","Algebraic semantics","In other theories"]}
{"upwards":["Truth value","Computing"],"downwards":[]}
{"upwards":["Truth value","Classical logic"],"downwards":[]}
{"upwards":["Truth value","Intuitionistic and constructive logic"],"downwards":[]}
{"upwards":["Truth value","Multi-valued logic"],"downwards":[]}
{"upwards":["Truth value","Algebraic semantics"],"downwards":[]}
{"upwards":["Truth value","In other theories"],"downwards":[]}
{"upwards":["Computable function"],"downwards":["Definition","Characteristics of computable functions","Computable sets and relations","Formal languages","Examples","Church\u2013Turing thesis","Provability","Uncomputable functions and unsolvable problems","Extensions of computability"]}
{"upwards":["Computable function","Definition"],"downwards":[]}
{"upwards":["Computable function","Characteristics of computable functions"],"downwards":[]}
{"upwards":["Computable function","Computable sets and relations"],"downwards":[]}
{"upwards":["Computable function","Formal languages"],"downwards":[]}
{"upwards":["Computable function","Examples"],"downwards":[]}
{"upwards":["Computable function","Church\u2013Turing thesis"],"downwards":[]}
{"upwards":["Computable function","Provability"],"downwards":["Relation to recursively defined functions","Total functions that are not provably total"]}
{"upwards":["Computable function","Provability","Relation to recursively defined functions"],"downwards":[]}
{"upwards":["Computable function","Provability","Total functions that are not provably total"],"downwards":[]}
{"upwards":["Computable function","Uncomputable functions and unsolvable problems"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability"],"downwards":["Relative computability","Higher recursion theory","Hyper-computation"]}
{"upwards":["Computable function","Extensions of computability","Relative computability"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability","Higher recursion theory"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability","Hyper-computation"],"downwards":[]}
{"upwards":["Finite model theory"],"downwards":["Axiomatisability","Zero-one laws","Descriptive complexity theory","Applications","History","Citations","References","Further reading","External links"]}
{"upwards":["Finite model theory","Axiomatisability"],"downwards":["Characterisation of a single structure","Characterisation of a class of structures"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure"],"downwards":["Problem","Approach","Extension to a fixed number of structures","Extension to an infinite structure"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Problem"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Approach"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Extension to a fixed number of structures"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Extension to an infinite structure"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures"],"downwards":["Problem","Approach","Example"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Problem"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Approach"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Example"],"downwards":[]}
{"upwards":["Finite model theory","Zero-one laws"],"downwards":[]}
{"upwards":["Finite model theory","Descriptive complexity theory"],"downwards":[]}
{"upwards":["Finite model theory","Applications"],"downwards":["Database theory","Querying and search"]}
{"upwards":["Finite model theory","Applications","Database theory"],"downwards":[]}
{"upwards":["Finite model theory","Applications","Querying and search"],"downwards":[]}
{"upwards":["Finite model theory","History"],"downwards":[]}
{"upwards":["Finite model theory","Citations"],"downwards":[]}
{"upwards":["Finite model theory","References"],"downwards":[]}
{"upwards":["Finite model theory","Further reading"],"downwards":[]}
{"upwards":["Finite model theory","External links"],"downwards":[]}
{"upwards":["Non-standard model"],"downwards":["Existence","Importance"]}
{"upwards":["Non-standard model","Existence"],"downwards":[]}
{"upwards":["Non-standard model","Importance"],"downwards":[]}
{"upwards":["Interpretation (model theory)"],"downwards":["Definition","Bi-interpretability","Example","References"]}
{"upwards":["Interpretation (model theory)","Definition"],"downwards":[]}
{"upwards":["Interpretation (model theory)","Bi-interpretability"],"downwards":[]}
{"upwards":["Interpretation (model theory)","Example"],"downwards":[]}
{"upwards":["Interpretation (model theory)","References"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis"],"downwards":["Statement in Church's and Turing's words","History","Success of the thesis","Informal usage in proofs","Variations","Philosophical implications","Non-computable functions"]}
{"upwards":["Church\u2013Turing thesis","Statement in Church's and Turing's words"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History"],"downwards":["Circa 1930\u20131952","Later developments","The thesis as a definition"]}
{"upwards":["Church\u2013Turing thesis","History","Circa 1930\u20131952"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History","Later developments"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History","The thesis as a definition"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Success of the thesis"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Informal usage in proofs"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Variations"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Philosophical implications"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Non-computable functions"],"downwards":[]}
{"upwards":["Ultraproduct"],"downwards":["Definition","Examples","\u0141o\u015b's theorem","Direct limits of ultrapowers (ultralimits)","Ultraproduct monad"]}
{"upwards":["Ultraproduct","Definition"],"downwards":["Ultrapower"]}
{"upwards":["Ultraproduct","Definition","Ultrapower"],"downwards":[]}
{"upwards":["Ultraproduct","Examples"],"downwards":[]}
{"upwards":["Ultraproduct","\u0141o\u015b's theorem"],"downwards":["Examples"]}
{"upwards":["Ultraproduct","\u0141o\u015b's theorem","Examples"],"downwards":[]}
{"upwards":["Ultraproduct","Direct limits of ultrapowers (ultralimits)"],"downwards":[]}
{"upwards":["Ultraproduct","Ultraproduct monad"],"downwards":[]}
{"upwards":["Elementary diagram"],"downwards":["Definition"]}
{"upwards":["Elementary diagram","Definition"],"downwards":[]}
{"upwards":["T-schema"],"downwards":["The inductive definition","Natural languages"]}
{"upwards":["T-schema","The inductive definition"],"downwards":[]}
{"upwards":["T-schema","Natural languages"],"downwards":[]}
{"upwards":["Categorical theory"],"downwards":["History and motivation","Examples","Properties"]}
{"upwards":["Categorical theory","History and motivation"],"downwards":[]}
{"upwards":["Categorical theory","Examples"],"downwards":[]}
{"upwards":["Categorical theory","Properties"],"downwards":[]}
{"upwards":["Truth predicate"],"downwards":["Languages which allow a truth predicate"]}
{"upwards":["Truth predicate","Languages which allow a truth predicate"],"downwards":[]}
{"upwards":["Elementary equivalence"],"downwards":["Elementarily equivalent structures","Elementary substructures and elementary extensions","Tarski\u2013Vaught test","Elementary embeddings","References"]}
{"upwards":["Elementary equivalence","Elementarily equivalent structures"],"downwards":[]}
{"upwards":["Elementary equivalence","Elementary substructures and elementary extensions"],"downwards":[]}
{"upwards":["Elementary equivalence","Tarski\u2013Vaught test"],"downwards":[]}
{"upwards":["Elementary equivalence","Elementary embeddings"],"downwards":[]}
{"upwards":["Elementary equivalence","References"],"downwards":[]}
{"upwards":["Interpretation (logic)"],"downwards":["Formal languages","General properties of truth-functional interpretations","Interpretation of a theory","Interpretations for propositional logic","First-order logic","Higher-order predicate logics","Non-classical interpretations","Intended interpretations","Other concepts of interpretation"]}
{"upwards":["Interpretation (logic)","Formal languages"],"downwards":["Example","Logical constants"]}
{"upwards":["Interpretation (logic)","Formal languages","Example"],"downwards":[]}
{"upwards":["Interpretation (logic)","Formal languages","Logical constants"],"downwards":[]}
{"upwards":["Interpretation (logic)","General properties of truth-functional interpretations"],"downwards":["Logical connectives"]}
{"upwards":["Interpretation (logic)","General properties of truth-functional interpretations","Logical connectives"],"downwards":[]}
{"upwards":["Interpretation (logic)","Interpretation of a theory"],"downwards":[]}
{"upwards":["Interpretation (logic)","Interpretations for propositional logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic"],"downwards":["Formal languages for first-order logic","Interpretations of a first-order language","Example of a first-order interpretation","Non-empty domain requirement","Interpreting equality","Many-sorted first-order logic"]}
{"upwards":["Interpretation (logic)","First-order logic","Formal languages for first-order logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Interpretations of a first-order language"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Example of a first-order interpretation"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Non-empty domain requirement"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Interpreting equality"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Many-sorted first-order logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","Higher-order predicate logics"],"downwards":[]}
{"upwards":["Interpretation (logic)","Non-classical interpretations"],"downwards":[]}
{"upwards":["Interpretation (logic)","Intended interpretations"],"downwards":["Example"]}
{"upwards":["Interpretation (logic)","Intended interpretations","Example"],"downwards":[]}
{"upwards":["Interpretation (logic)","Other concepts of interpretation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)"],"downwards":["Definition","Induced substructures and closed subsets","Homomorphisms and embeddings","Structures and first-order logic","Many-sorted structures","Other generalizations"]}
{"upwards":["Structure (mathematical logic)","Definition"],"downwards":["Domain","Signature","Interpretation function","Examples"]}
{"upwards":["Structure (mathematical logic)","Definition","Domain"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Signature"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Interpretation function"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets"],"downwards":["Examples"]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings"],"downwards":["Homomorphisms","Embeddings","Example","Homomorphism problem"]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphisms"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Embeddings"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Example"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphism problem"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic"],"downwards":["Satisfaction relation","Definable relations"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Satisfaction relation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations"],"downwards":["Definability with parameters","Implicit definability"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Definability with parameters"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Implicit definability"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Many-sorted structures"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations"],"downwards":["Partial algebras","Structures for typed languages","Higher-order languages","Structures that are proper classes"]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Partial algebras"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures for typed languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Higher-order languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures that are proper classes"],"downwards":[]}
{"upwards":["Strength (mathematical logic)"],"downwards":[]}
{"upwards":["Computable set"],"downwards":["Formal definition","Examples and non-examples","Properties"]}
{"upwards":["Computable set","Formal definition"],"downwards":[]}
{"upwards":["Computable set","Examples and non-examples"],"downwards":[]}
{"upwards":["Computable set","Properties"],"downwards":[]}
{"upwards":["Model theory"],"downwards":["Overview","Fundamental notions of first-order model theory","Definability","Types","Constructing models","Categoricity","Stability theory","Non-elementary model theory","Selected applications","History","Connections to related branches of mathematical logic"]}
{"upwards":["Model theory","Overview"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory"],"downwards":["First-order logic","Basic model-theoretic concepts","Compactness and the L\u00f6wenheim-Skolem theorem"]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","First-order logic"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","Basic model-theoretic concepts"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","Compactness and the L\u00f6wenheim-Skolem theorem"],"downwards":[]}
{"upwards":["Model theory","Definability"],"downwards":["Definable sets","Eliminating quantifiers","Minimality","Definable and interpretable structures"]}
{"upwards":["Model theory","Definability","Definable sets"],"downwards":[]}
{"upwards":["Model theory","Definability","Eliminating quantifiers"],"downwards":[]}
{"upwards":["Model theory","Definability","Minimality"],"downwards":[]}
{"upwards":["Model theory","Definability","Definable and interpretable structures"],"downwards":[]}
{"upwards":["Model theory","Types"],"downwards":["Basic notions","Structures and types","Stone spaces"]}
{"upwards":["Model theory","Types","Basic notions"],"downwards":[]}
{"upwards":["Model theory","Types","Structures and types"],"downwards":[]}
{"upwards":["Model theory","Types","Stone spaces"],"downwards":[]}
{"upwards":["Model theory","Constructing models"],"downwards":["Realising and omitting types","Ultraproducts"]}
{"upwards":["Model theory","Constructing models","Realising and omitting types"],"downwards":[]}
{"upwards":["Model theory","Constructing models","Ultraproducts"],"downwards":[]}
{"upwards":["Model theory","Categoricity"],"downwards":["\u03c9-categoricity","Uncountable categoricity"]}
{"upwards":["Model theory","Categoricity","\u03c9-categoricity"],"downwards":[]}
{"upwards":["Model theory","Categoricity","Uncountable categoricity"],"downwards":[]}
{"upwards":["Model theory","Stability theory"],"downwards":["The stability hierarchy","Geometric stability theory"]}
{"upwards":["Model theory","Stability theory","The stability hierarchy"],"downwards":[]}
{"upwards":["Model theory","Stability theory","Geometric stability theory"],"downwards":[]}
{"upwards":["Model theory","Non-elementary model theory"],"downwards":[]}
{"upwards":["Model theory","Selected applications"],"downwards":[]}
{"upwards":["Model theory","History"],"downwards":[]}
{"upwards":["Model theory","Connections to related branches of mathematical logic"],"downwards":["Finite model theory","Set theory"]}
{"upwards":["Model theory","Connections to related branches of mathematical logic","Finite model theory"],"downwards":[]}
{"upwards":["Model theory","Connections to related branches of mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Truth"],"downwards":["Definition and etymology","Major theories","Formal theories","Folk beliefs","Notable views"]}
{"upwards":["Truth","Definition and etymology"],"downwards":[]}
{"upwards":["Truth","Major theories"],"downwards":["Substantive","Minimalist","Pluralist"]}
{"upwards":["Truth","Major theories","Substantive"],"downwards":["Correspondence","Coherence","Pragmatic","Constructivist","Consensus"]}
{"upwards":["Truth","Major theories","Substantive","Correspondence"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Coherence"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Pragmatic"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Constructivist"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Consensus"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist"],"downwards":["Deflationary","Performative","Redundancy and related","Philosophical skepticism"]}
{"upwards":["Truth","Major theories","Minimalist","Deflationary"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Performative"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Redundancy and related"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Philosophical skepticism"],"downwards":[]}
{"upwards":["Truth","Major theories","Pluralist"],"downwards":[]}
{"upwards":["Truth","Formal theories"],"downwards":["Logic","Mathematics","Tarski's semantics","Kripke's semantics"]}
{"upwards":["Truth","Formal theories","Logic"],"downwards":[]}
{"upwards":["Truth","Formal theories","Mathematics"],"downwards":[]}
{"upwards":["Truth","Formal theories","Tarski's semantics"],"downwards":[]}
{"upwards":["Truth","Formal theories","Kripke's semantics"],"downwards":[]}
{"upwards":["Truth","Folk beliefs"],"downwards":[]}
{"upwards":["Truth","Notable views"],"downwards":["Ancient Greek philosophy","Medieval philosophy","Modern philosophy","Theological views"]}
{"upwards":["Truth","Notable views","Ancient Greek philosophy"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy"],"downwards":["Avicenna (980\u20131037)","Aquinas (1225\u20131274)","Changing concepts of truth in the Middle Ages"]}
{"upwards":["Truth","Notable views","Medieval philosophy","Avicenna (980\u20131037)"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy","Aquinas (1225\u20131274)"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy","Changing concepts of truth in the Middle Ages"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy"],"downwards":["Kant (1724\u20131804)","Hegel (1770\u20131831)","Schopenhauer (1788\u20131860)","Kierkegaard (1813\u20131855)","Nietzsche (1844\u20131900)","Heidegger (1889\u20131976)","Sartre (1905\u20131980)","Camus (1913\u20131960)","Whitehead (1861\u20131947)","Peirce (1839\u20131914)","Nishida (1870\u20131945)","Fromm (1900\u20131980)","Foucault (1926\u20131984)","Baudrillard (1929\u20132007)","Other contemporary positions"]}
{"upwards":["Truth","Notable views","Modern philosophy","Kant (1724\u20131804)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Hegel (1770\u20131831)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Schopenhauer (1788\u20131860)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Kierkegaard (1813\u20131855)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Nietzsche (1844\u20131900)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Heidegger (1889\u20131976)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Sartre (1905\u20131980)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Camus (1913\u20131960)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Whitehead (1861\u20131947)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Peirce (1839\u20131914)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Nishida (1870\u20131945)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Fromm (1900\u20131980)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Foucault (1926\u20131984)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Baudrillard (1929\u20132007)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Other contemporary positions"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views"],"downwards":["Hinduism","Buddhism","Christianity"]}
{"upwards":["Truth","Notable views","Theological views","Hinduism"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views","Buddhism"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views","Christianity"],"downwards":[]}
{"upwards":["Transfer principle"],"downwards":["History","Transfer principle for the hyperreals","Generalizations of the concept of number","Differences between R and *R","Constructions of the hyperreals","Statement","Three examples"]}
{"upwards":["Transfer principle","History"],"downwards":[]}
{"upwards":["Transfer principle","Transfer principle for the hyperreals"],"downwards":["Example"]}
{"upwards":["Transfer principle","Transfer principle for the hyperreals","Example"],"downwards":[]}
{"upwards":["Transfer principle","Generalizations of the concept of number"],"downwards":[]}
{"upwards":["Transfer principle","Differences between R and *R"],"downwards":[]}
{"upwards":["Transfer principle","Constructions of the hyperreals"],"downwards":[]}
{"upwards":["Transfer principle","Statement"],"downwards":[]}
{"upwards":["Transfer principle","Three examples"],"downwards":[]}
{"upwards":["List of statements independent of ZFC"],"downwards":["Axiomatic set theory","Set theory of the real line","Order theory","Abstract algebra","Number theory","Measure theory","Topology","Functional analysis","Model theory","Computability theory","References","External links"]}
{"upwards":["List of statements independent of ZFC","Axiomatic set theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Set theory of the real line"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Order theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Abstract algebra"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Number theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Measure theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Topology"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Functional analysis"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Model theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Computability theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","References"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","External links"],"downwards":[]}
{"upwards":["Model complete theory"],"downwards":["Model companion and model completion","Examples","Non-examples","Sufficient condition for completeness of model-complete theories","Notes","References"]}
{"upwards":["Model complete theory","Model companion and model completion"],"downwards":[]}
{"upwards":["Model complete theory","Examples"],"downwards":[]}
{"upwards":["Model complete theory","Non-examples"],"downwards":[]}
{"upwards":["Model complete theory","Sufficient condition for completeness of model-complete theories"],"downwards":[]}
{"upwards":["Model complete theory","Notes"],"downwards":[]}
{"upwards":["Model complete theory","References"],"downwards":[]}
{"upwards":["Self-verifying theories"],"downwards":["References","External links"]}
{"upwards":["Self-verifying theories","References"],"downwards":[]}
{"upwards":["Self-verifying theories","External links"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic"],"downwards":["Existence","Structure of countable non-standard models","References"]}
{"upwards":["Non-standard model of arithmetic","Existence"],"downwards":["From the compactness theorem","From the incompleteness theorems","From an ultraproduct"]}
{"upwards":["Non-standard model of arithmetic","Existence","From the compactness theorem"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Existence","From the incompleteness theorems"],"downwards":["Arithmetic unsoundness for models with ~G true"]}
{"upwards":["Non-standard model of arithmetic","Existence","From the incompleteness theorems","Arithmetic unsoundness for models with ~G true"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Existence","From an ultraproduct"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Structure of countable non-standard models"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","References"],"downwards":["Citations","Sources"]}
{"upwards":["Non-standard model of arithmetic","References","Citations"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","References","Sources"],"downwards":[]}
{"upwards":["Complete theory"],"downwards":["Examples"]}
{"upwards":["Complete theory","Examples"],"downwards":[]}
{"upwards":["Type (model theory)"],"downwards":["Formal definition","Examples of types","Stone spaces","Omitting types theorem","References"]}
{"upwards":["Type (model theory)","Formal definition"],"downwards":[]}
{"upwards":["Type (model theory)","Examples of types"],"downwards":[]}
{"upwards":["Type (model theory)","Stone spaces"],"downwards":[]}
{"upwards":["Type (model theory)","Omitting types theorem"],"downwards":[]}
{"upwards":["Type (model theory)","References"],"downwards":[]}
{"upwards":["Spectrum of a theory"],"downwards":["Early results","List of possible spectra of a countable theory"]}
{"upwards":["Spectrum of a theory","Early results"],"downwards":[]}
{"upwards":["Spectrum of a theory","List of possible spectra of a countable theory"],"downwards":[]}
{"upwards":["Saturated model"],"downwards":["Definition","Motivation","Examples","Relationship to prime models","Notes","References"]}
{"upwards":["Saturated model","Definition"],"downwards":[]}
{"upwards":["Saturated model","Motivation"],"downwards":[]}
{"upwards":["Saturated model","Examples"],"downwards":[]}
{"upwards":["Saturated model","Relationship to prime models"],"downwards":[]}
{"upwards":["Saturated model","Notes"],"downwards":[]}
{"upwards":["Saturated model","References"],"downwards":[]}
{"upwards":["Substructure (mathematics)"],"downwards":["Definition","Example","As subobjects","Submodel"]}
{"upwards":["Substructure (mathematics)","Definition"],"downwards":[]}
{"upwards":["Substructure (mathematics)","Example"],"downwards":[]}
{"upwards":["Substructure (mathematics)","As subobjects"],"downwards":[]}
{"upwards":["Substructure (mathematics)","Submodel"],"downwards":[]}
{"upwards":["Diagram (mathematical logic)"],"downwards":["Definition"]}
{"upwards":["Diagram (mathematical logic)","Definition"],"downwards":[]}
{"upwards":["Semantics of logic"],"downwards":["Overview"]}
{"upwards":["Semantics of logic","Overview"],"downwards":[]}
{"upwards":["Blackboard bold"],"downwards":["History","Encoding","Usage"]}
{"upwards":["Blackboard bold","History"],"downwards":["Origin","Use in textbooks"]}
{"upwards":["Blackboard bold","History","Origin"],"downwards":[]}
{"upwards":["Blackboard bold","History","Use in textbooks"],"downwards":[]}
{"upwards":["Blackboard bold","Encoding"],"downwards":[]}
{"upwards":["Blackboard bold","Usage"],"downwards":[]}
{"upwards":["Wikipedia, the free encyclopedia"],"downwards":[]}
{"upwards":["Well-defined expression"],"downwards":["Example","\"Definition\" as anticipation of definition","Independence of representative","Well-defined notation","Other uses of the term"]}
{"upwards":["Well-defined expression","Example"],"downwards":[]}
{"upwards":["Well-defined expression","\"Definition\" as anticipation of definition"],"downwards":[]}
{"upwards":["Well-defined expression","Independence of representative"],"downwards":["Functions with one argument","Operations"]}
{"upwards":["Well-defined expression","Independence of representative","Functions with one argument"],"downwards":[]}
{"upwards":["Well-defined expression","Independence of representative","Operations"],"downwards":[]}
{"upwards":["Well-defined expression","Well-defined notation"],"downwards":[]}
{"upwards":["Well-defined expression","Other uses of the term"],"downwards":[]}
{"upwards":["Axiom"],"downwards":["Etymology","Historical development","Mathematical logic"]}
{"upwards":["Axiom","Etymology"],"downwards":[]}
{"upwards":["Axiom","Historical development"],"downwards":["Early Greeks","Modern development","Other sciences"]}
{"upwards":["Axiom","Historical development","Early Greeks"],"downwards":[]}
{"upwards":["Axiom","Historical development","Modern development"],"downwards":[]}
{"upwards":["Axiom","Historical development","Other sciences"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic"],"downwards":["Logical axioms","Non-logical axioms","Role in mathematical logic","Further discussion"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic"],"downwards":["Deductive systems and completeness"]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic","Deductive systems and completeness"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Further discussion"],"downwards":[]}
{"upwards":["Map (mathematics)"],"downwards":["Maps as functions","As morphisms"]}
{"upwards":["Map (mathematics)","Maps as functions"],"downwards":[]}
{"upwards":["Map (mathematics)","As morphisms"],"downwards":[]}
{"upwards":["Injective function"],"downwards":["Definition","Examples","Injections can be undone","Injections may be made invertible","Other properties","Proving that functions are injective","Gallery"]}
{"upwards":["Injective function","Definition"],"downwards":[]}
{"upwards":["Injective function","Examples"],"downwards":[]}
{"upwards":["Injective function","Injections can be undone"],"downwards":[]}
{"upwards":["Injective function","Injections may be made invertible"],"downwards":[]}
{"upwards":["Injective function","Other properties"],"downwards":[]}
{"upwards":["Injective function","Proving that functions are injective"],"downwards":[]}
{"upwards":["Injective function","Gallery"],"downwards":[]}
{"upwards":["Primitive notion"],"downwards":["Details","Examples","Russell's primitives"]}
{"upwards":["Primitive notion","Details"],"downwards":[]}
{"upwards":["Primitive notion","Examples"],"downwards":[]}
{"upwards":["Primitive notion","Russell's primitives"],"downwards":[]}
{"upwards":["Russell's paradox"],"downwards":["Informal presentation","Formal presentation","Philosophical implications","Set-theoretic responses","History","Applied versions","Applications and related topics","Related paradoxes"]}
{"upwards":["Russell's paradox","Informal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Formal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Philosophical implications"],"downwards":[]}
{"upwards":["Russell's paradox","Set-theoretic responses"],"downwards":[]}
{"upwards":["Russell's paradox","History"],"downwards":[]}
{"upwards":["Russell's paradox","Applied versions"],"downwards":[]}
{"upwards":["Russell's paradox","Applications and related topics"],"downwards":["Russell-like paradoxes"]}
{"upwards":["Russell's paradox","Applications and related topics","Russell-like paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox","Related paradoxes"],"downwards":[]}
{"upwards":["e (mathematical constant)"],"downwards":["History","Applications","In calculus","Properties","Representations","Computing the digits","In computer culture","References","Further reading","External links"]}
{"upwards":["e (mathematical constant)","History"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications"],"downwards":["Compound interest","Bernoulli trials","Standard normal distribution","Derangements","Optimal planning problems","Asymptotics"]}
{"upwards":["e (mathematical constant)","Applications","Compound interest"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Bernoulli trials"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Standard normal distribution"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Derangements"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Optimal planning problems"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Asymptotics"],"downwards":[]}
{"upwards":["e (mathematical constant)","In calculus"],"downwards":["Alternative characterizations"]}
{"upwards":["e (mathematical constant)","In calculus","Alternative characterizations"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties"],"downwards":["Calculus","Inequalities","Exponential-like functions","Number theory","Complex numbers"]}
{"upwards":["e (mathematical constant)","Properties","Calculus"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Inequalities"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Exponential-like functions"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Number theory"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Complex numbers"],"downwards":[]}
{"upwards":["e (mathematical constant)","Representations"],"downwards":["Stochastic representations","Known digits"]}
{"upwards":["e (mathematical constant)","Representations","Stochastic representations"],"downwards":[]}
{"upwards":["e (mathematical constant)","Representations","Known digits"],"downwards":[]}
{"upwards":["e (mathematical constant)","Computing the digits"],"downwards":[]}
{"upwards":["e (mathematical constant)","In computer culture"],"downwards":[]}
{"upwards":["e (mathematical constant)","References"],"downwards":[]}
{"upwards":["e (mathematical constant)","Further reading"],"downwards":[]}
{"upwards":["e (mathematical constant)","External links"],"downwards":[]}
{"upwards":["Tuple"],"downwards":["Etymology","Properties","Definitions","n-tuples of m-sets","Type theory"]}
{"upwards":["Tuple","Etymology"],"downwards":["Names for tuples of specific lengths"]}
{"upwards":["Tuple","Etymology","Names for tuples of specific lengths"],"downwards":[]}
{"upwards":["Tuple","Properties"],"downwards":[]}
{"upwards":["Tuple","Definitions"],"downwards":["Tuples as functions","Tuples as nested ordered pairs","Tuples as nested sets"]}
{"upwards":["Tuple","Definitions","Tuples as functions"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested ordered pairs"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested sets"],"downwards":[]}
{"upwards":["Tuple","n-tuples of m-sets"],"downwards":[]}
{"upwards":["Tuple","Type theory"],"downwards":[]}
{"upwards":["Theorem"],"downwards":["Theoremhood and truth","Epistemological considerations","Informal account of theorems","Relation with scientific theories","Terminology","Layout","Lore","Theorems in logic"]}
{"upwards":["Theorem","Theoremhood and truth"],"downwards":[]}
{"upwards":["Theorem","Epistemological considerations"],"downwards":[]}
{"upwards":["Theorem","Informal account of theorems"],"downwards":[]}
{"upwards":["Theorem","Relation with scientific theories"],"downwards":[]}
{"upwards":["Theorem","Terminology"],"downwards":[]}
{"upwards":["Theorem","Layout"],"downwards":[]}
{"upwards":["Theorem","Lore"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic"],"downwards":["Syntax and semantics","Interpretation of a formal theorem","Theorems and theories"]}
{"upwards":["Theorem","Theorems in logic","Syntax and semantics"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Interpretation of a formal theorem"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Theorems and theories"],"downwards":[]}
{"upwards":["Italic type"],"downwards":["History","Examples","Usage","Oblique type compared to italics","More complex usage"]}
{"upwards":["Italic type","History"],"downwards":[]}
{"upwards":["Italic type","Examples"],"downwards":[]}
{"upwards":["Italic type","Usage"],"downwards":[]}
{"upwards":["Italic type","Oblique type compared to italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage"],"downwards":["Italics within italics","Left-leaning italics","Upright italics","Parentheses","Substitutes","OpenType","Web pages","Unicode"]}
{"upwards":["Italic type","More complex usage","Italics within italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Left-leaning italics"],"downwards":["Iranic font style"]}
{"upwards":["Italic type","More complex usage","Left-leaning italics","Iranic font style"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Upright italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Parentheses"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Substitutes"],"downwards":[]}
{"upwards":["Italic type","More complex usage","OpenType"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Web pages"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Unicode"],"downwards":[]}
{"upwards":["Real number"],"downwards":["Characterizing properties","Arithmetic","Dedekind completeness","Decimal representation","Topological completeness","Cardinality","Other properties","History","Formal definitions","Applications and connections","Vocabulary and notation","Generalizations and extensions"]}
{"upwards":["Real number","Characterizing properties"],"downwards":[]}
{"upwards":["Real number","Arithmetic"],"downwards":["Auxiliary operations","Auxiliary order relations","Integers and fractions as real numbers"]}
{"upwards":["Real number","Arithmetic","Auxiliary operations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Auxiliary order relations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Integers and fractions as real numbers"],"downwards":[]}
{"upwards":["Real number","Dedekind completeness"],"downwards":[]}
{"upwards":["Real number","Decimal representation"],"downwards":[]}
{"upwards":["Real number","Topological completeness"],"downwards":["\"The complete ordered field\""]}
{"upwards":["Real number","Topological completeness","\"The complete ordered field\""],"downwards":[]}
{"upwards":["Real number","Cardinality"],"downwards":[]}
{"upwards":["Real number","Other properties"],"downwards":[]}
{"upwards":["Real number","History"],"downwards":[]}
{"upwards":["Real number","Formal definitions"],"downwards":["Axiomatic approach","Construction from the rational numbers"]}
{"upwards":["Real number","Formal definitions","Axiomatic approach"],"downwards":[]}
{"upwards":["Real number","Formal definitions","Construction from the rational numbers"],"downwards":[]}
{"upwards":["Real number","Applications and connections"],"downwards":["Physics","Logic","Computation","Set theory"]}
{"upwards":["Real number","Applications and connections","Physics"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Logic"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Computation"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Set theory"],"downwards":[]}
{"upwards":["Real number","Vocabulary and notation"],"downwards":[]}
{"upwards":["Real number","Generalizations and extensions"],"downwards":[]}
{"upwards":["Bijection"],"downwards":["Definition","Examples","More mathematical examples","Inverses","Composition","Cardinality","Properties","Category theory","Generalization to partial functions","Gallery"]}
{"upwards":["Bijection","Definition"],"downwards":[]}
{"upwards":["Bijection","Examples"],"downwards":["Batting line-up of a baseball or cricket team","Seats and students of a classroom"]}
{"upwards":["Bijection","Examples","Batting line-up of a baseball or cricket team"],"downwards":[]}
{"upwards":["Bijection","Examples","Seats and students of a classroom"],"downwards":[]}
{"upwards":["Bijection","More mathematical examples"],"downwards":[]}
{"upwards":["Bijection","Inverses"],"downwards":[]}
{"upwards":["Bijection","Composition"],"downwards":[]}
{"upwards":["Bijection","Cardinality"],"downwards":[]}
{"upwards":["Bijection","Properties"],"downwards":[]}
{"upwards":["Bijection","Category theory"],"downwards":[]}
{"upwards":["Bijection","Generalization to partial functions"],"downwards":[]}
{"upwards":["Bijection","Gallery"],"downwards":[]}
{"upwards":["Bernard Bolzano"],"downwards":["Family","Career","Mathematical work","Philosophical work","Philosophical legacy","Writings"]}
{"upwards":["Bernard Bolzano","Family"],"downwards":[]}
{"upwards":["Bernard Bolzano","Career"],"downwards":[]}
{"upwards":["Bernard Bolzano","Mathematical work"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work"],"downwards":["Wissenschaftslehre (Theory of Science)"]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)"],"downwards":["Introduction to Wissenschaftslehre","Metaphysics","Satz an Sich (proposition in itself)","Ideas and objects","Sensation and simple ideas","Logic","Truth","Judgments and cognitions"]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Introduction to Wissenschaftslehre"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Metaphysics"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Satz an Sich (proposition in itself)"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Ideas and objects"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Sensation and simple ideas"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Logic"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Truth"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Judgments and cognitions"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical legacy"],"downwards":[]}
{"upwards":["Bernard Bolzano","Writings"],"downwards":["Translations and compilations"]}
{"upwards":["Bernard Bolzano","Writings","Translations and compilations"],"downwards":[]}
{"upwards":["Foundations of mathematics"],"downwards":["Historical context","Foundational crisis","Toward resolution of the crisis"]}
{"upwards":["Foundations of mathematics","Historical context"],"downwards":["Ancient Greek mathematics","Platonism as a philosophy of mathematics","Aristotelian realism","Middle Ages and Renaissance","19th century"]}
{"upwards":["Foundations of mathematics","Historical context","Ancient Greek mathematics"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Platonism as a philosophy of mathematics"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Aristotelian realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Middle Ages and Renaissance"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century"],"downwards":["Real analysis","Group theory","Non-Euclidean geometries","Projective geometry","Boolean algebra and logic","Peano arithmetic"]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Real analysis"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Group theory"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Non-Euclidean geometries"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Projective geometry"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Boolean algebra and logic"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Peano arithmetic"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis"],"downwards":["Philosophical views","More paradoxes"]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views"],"downwards":["Formalism","Intuitionism","Logicism","Set-theoretic Platonism","Indispensability argument for realism","Rough-and-ready realism","Philosophical consequences of G\u00f6del's completeness theorem"]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Formalism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Intuitionism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Logicism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Set-theoretic Platonism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Indispensability argument for realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Rough-and-ready realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Philosophical consequences of G\u00f6del's completeness theorem"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","More paradoxes"],"downwards":[]}
{"upwards":["Foundations of mathematics","Toward resolution of the crisis"],"downwards":[]}
{"upwards":["Complex number"],"downwards":["Definition","Notation","Visualization","History","Relations and operations","Properties","Formal construction","Complex analysis","Applications","Generalizations and related notions"]}
{"upwards":["Complex number","Definition"],"downwards":[]}
{"upwards":["Complex number","Notation"],"downwards":[]}
{"upwards":["Complex number","Visualization"],"downwards":["Cartesian complex plane","Polar complex plane","Complex graphs"]}
{"upwards":["Complex number","Visualization","Cartesian complex plane"],"downwards":[]}
{"upwards":["Complex number","Visualization","Polar complex plane"],"downwards":["Modulus and argument"]}
{"upwards":["Complex number","Visualization","Polar complex plane","Modulus and argument"],"downwards":[]}
{"upwards":["Complex number","Visualization","Complex graphs"],"downwards":[]}
{"upwards":["Complex number","History"],"downwards":[]}
{"upwards":["Complex number","Relations and operations"],"downwards":["Equality","Ordering","Conjugate","Addition and subtraction","Multiplication and square","Reciprocal and division","Multiplication and division in polar form","Square root","Exponential function","Complex logarithm","Exponentiation"]}
{"upwards":["Complex number","Relations and operations","Equality"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Ordering"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Conjugate"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and square"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Reciprocal and division"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and division in polar form"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Square root"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function"],"downwards":["Functional equation","Euler's formula"]}
{"upwards":["Complex number","Relations and operations","Exponential function","Functional equation"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function","Euler's formula"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Complex logarithm"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponentiation"],"downwards":["Integer and fractional exponents"]}
{"upwards":["Complex number","Relations and operations","Exponentiation","Integer and fractional exponents"],"downwards":[]}
{"upwards":["Complex number","Properties"],"downwards":["Field structure","Solutions of polynomial equations","Algebraic characterization","Characterization as a topological field"]}
{"upwards":["Complex number","Properties","Field structure"],"downwards":[]}
{"upwards":["Complex number","Properties","Solutions of polynomial equations"],"downwards":[]}
{"upwards":["Complex number","Properties","Algebraic characterization"],"downwards":[]}
{"upwards":["Complex number","Properties","Characterization as a topological field"],"downwards":[]}
{"upwards":["Complex number","Formal construction"],"downwards":["Construction as ordered pairs","Construction as a quotient field","Matrix representation of complex numbers"]}
{"upwards":["Complex number","Formal construction","Construction as ordered pairs"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Construction as a quotient field"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Matrix representation of complex numbers"],"downwards":[]}
{"upwards":["Complex number","Complex analysis"],"downwards":["Complex exponential and related functions","Holomorphic functions"]}
{"upwards":["Complex number","Complex analysis","Complex exponential and related functions"],"downwards":[]}
{"upwards":["Complex number","Complex analysis","Holomorphic functions"],"downwards":[]}
{"upwards":["Complex number","Applications"],"downwards":["Geometry","Algebraic number theory","Analytic number theory","Improper integrals","Dynamic equations","Linear algebra","In applied mathematics","In physics"]}
{"upwards":["Complex number","Applications","Geometry"],"downwards":["Shapes","Fractal geometry","Triangles"]}
{"upwards":["Complex number","Applications","Geometry","Shapes"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Fractal geometry"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Triangles"],"downwards":[]}
{"upwards":["Complex number","Applications","Algebraic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Analytic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Improper integrals"],"downwards":[]}
{"upwards":["Complex number","Applications","Dynamic equations"],"downwards":[]}
{"upwards":["Complex number","Applications","Linear algebra"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics"],"downwards":["Control theory","Signal analysis"]}
{"upwards":["Complex number","Applications","In applied mathematics","Control theory"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics","Signal analysis"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics"],"downwards":["Electromagnetism and electrical engineering","Fluid dynamics","Quantum mechanics","Relativity"]}
{"upwards":["Complex number","Applications","In physics","Electromagnetism and electrical engineering"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Fluid dynamics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Quantum mechanics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Relativity"],"downwards":[]}
{"upwards":["Complex number","Generalizations and related notions"],"downwards":[]}
{"upwards":["Russell's paradox"],"downwards":["Informal presentation","Formal presentation","Philosophical implications","Set-theoretic responses","History","Applied versions","Applications and related topics","Related paradoxes"]}
{"upwards":["Russell's paradox","Informal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Formal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Philosophical implications"],"downwards":[]}
{"upwards":["Russell's paradox","Set-theoretic responses"],"downwards":[]}
{"upwards":["Russell's paradox","History"],"downwards":[]}
{"upwards":["Russell's paradox","Applied versions"],"downwards":[]}
{"upwards":["Russell's paradox","Applications and related topics"],"downwards":["Russell-like paradoxes"]}
{"upwards":["Russell's paradox","Applications and related topics","Russell-like paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox","Related paradoxes"],"downwards":[]}
{"upwards":["Element (mathematics)"],"downwards":["Sets","Notation and terminology","Cardinality of sets","Examples","Formal relation"]}
{"upwards":["Element (mathematics)","Sets"],"downwards":[]}
{"upwards":["Element (mathematics)","Notation and terminology"],"downwards":[]}
{"upwards":["Element (mathematics)","Cardinality of sets"],"downwards":[]}
{"upwards":["Element (mathematics)","Examples"],"downwards":[]}
{"upwards":["Element (mathematics)","Formal relation"],"downwards":[]}
{"upwards":["Set (mathematics)"],"downwards":["History","How sets are defined and set notation","Membership","The empty set","Singleton sets","Subsets","Euler and Venn diagrams","Special sets of numbers in mathematics","Functions","Cardinality","Power sets","Partitions","Basic operations","Applications","Principle of inclusion and exclusion"]}
{"upwards":["Set (mathematics)","History"],"downwards":["Naive set theory","Axiomatic set theory"]}
{"upwards":["Set (mathematics)","History","Naive set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","History","Axiomatic set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation"],"downwards":["Roster notation","Semantic definition","Set-builder notation","Classifying methods of definition"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation"],"downwards":["Infinite sets in roster notation"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation","Infinite sets in roster notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Semantic definition"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Set-builder notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Classifying methods of definition"],"downwards":[]}
{"upwards":["Set (mathematics)","Membership"],"downwards":[]}
{"upwards":["Set (mathematics)","The empty set"],"downwards":[]}
{"upwards":["Set (mathematics)","Singleton sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Subsets"],"downwards":[]}
{"upwards":["Set (mathematics)","Euler and Venn diagrams"],"downwards":[]}
{"upwards":["Set (mathematics)","Special sets of numbers in mathematics"],"downwards":[]}
{"upwards":["Set (mathematics)","Functions"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality"],"downwards":["Infinite sets and infinite cardinality","The continuum hypothesis"]}
{"upwards":["Set (mathematics)","Cardinality","Infinite sets and infinite cardinality"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality","The continuum hypothesis"],"downwards":[]}
{"upwards":["Set (mathematics)","Power sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Partitions"],"downwards":[]}
{"upwards":["Set (mathematics)","Basic operations"],"downwards":[]}
{"upwards":["Set (mathematics)","Applications"],"downwards":[]}
{"upwards":["Set (mathematics)","Principle of inclusion and exclusion"],"downwards":[]}
{"upwards":["Algebraic number"],"downwards":["Examples","Properties","Field","Related fields","Algebraic integers","Special classes","Notes","References"]}
{"upwards":["Algebraic number","Examples"],"downwards":[]}
{"upwards":["Algebraic number","Properties"],"downwards":[]}
{"upwards":["Algebraic number","Field"],"downwards":[]}
{"upwards":["Algebraic number","Related fields"],"downwards":["Numbers defined by radicals","Closed-form number"]}
{"upwards":["Algebraic number","Related fields","Numbers defined by radicals"],"downwards":[]}
{"upwards":["Algebraic number","Related fields","Closed-form number"],"downwards":[]}
{"upwards":["Algebraic number","Algebraic integers"],"downwards":[]}
{"upwards":["Algebraic number","Special classes"],"downwards":[]}
{"upwards":["Algebraic number","Notes"],"downwards":[]}
{"upwards":["Algebraic number","References"],"downwards":[]}
{"upwards":["Mathematical object"],"downwards":["List of mathematical objects by branch"]}
{"upwards":["Mathematical object","List of mathematical objects by branch"],"downwards":[]}
{"upwards":["Euler diagram"],"downwards":["History","Relation between Euler and Venn diagrams","Gallery"]}
{"upwards":["Euler diagram","History"],"downwards":[]}
{"upwards":["Euler diagram","Relation between Euler and Venn diagrams"],"downwards":["Example: Euler- to Venn-diagram and Karnaugh map"]}
{"upwards":["Euler diagram","Relation between Euler and Venn diagrams","Example: Euler- to Venn-diagram and Karnaugh map"],"downwards":[]}
{"upwards":["Euler diagram","Gallery"],"downwards":[]}
{"upwards":["Singleton (mathematics)"],"downwards":["Properties","In category theory","Definition by indicator functions","Definition in Principia Mathematica"]}
{"upwards":["Singleton (mathematics)","Properties"],"downwards":[]}
{"upwards":["Singleton (mathematics)","In category theory"],"downwards":[]}
{"upwards":["Singleton (mathematics)","Definition by indicator functions"],"downwards":[]}
{"upwards":["Singleton (mathematics)","Definition in Principia Mathematica"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Equality (mathematics)"],"downwards":["Etymology","Basic properties","Equality as predicate","Identities","Equations","Approximate equality","Relation with equivalence, congruence, and isomorphism","Logical definitions","Equality in set theory"]}
{"upwards":["Equality (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Equality (mathematics)","Basic properties"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality as predicate"],"downwards":[]}
{"upwards":["Equality (mathematics)","Identities"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equations"],"downwards":[]}
{"upwards":["Equality (mathematics)","Approximate equality"],"downwards":[]}
{"upwards":["Equality (mathematics)","Relation with equivalence, congruence, and isomorphism"],"downwards":[]}
{"upwards":["Equality (mathematics)","Logical definitions"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality in set theory"],"downwards":["Set equality based on first-order logic with equality","Set equality based on first-order logic without equality"]}
{"upwards":["Equality (mathematics)","Equality in set theory","Set equality based on first-order logic with equality"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality in set theory","Set equality based on first-order logic without equality"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Cantor's paradox"],"downwards":["Statements and proofs","Discussion and consequences","Historical notes","References","External links"]}
{"upwards":["Cantor's paradox","Statements and proofs"],"downwards":[]}
{"upwards":["Cantor's paradox","Discussion and consequences"],"downwards":[]}
{"upwards":["Cantor's paradox","Historical notes"],"downwards":[]}
{"upwards":["Cantor's paradox","References"],"downwards":[]}
{"upwards":["Cantor's paradox","External links"],"downwards":[]}
{"upwards":["Infinite set"],"downwards":["Properties","Examples"]}
{"upwards":["Infinite set","Properties"],"downwards":[]}
{"upwards":["Infinite set","Examples"],"downwards":["Countably infinite sets","Uncountably infinite sets"]}
{"upwards":["Infinite set","Examples","Countably infinite sets"],"downwards":[]}
{"upwards":["Infinite set","Examples","Uncountably infinite sets"],"downwards":[]}
{"upwards":["Permutation"],"downwards":["History","Permutations without repetitions","Definition","Notations","Other uses of the term permutation","Properties","Permutations of totally ordered sets","Permutations in computing"]}
{"upwards":["Permutation","History"],"downwards":[]}
{"upwards":["Permutation","Permutations without repetitions"],"downwards":[]}
{"upwards":["Permutation","Definition"],"downwards":[]}
{"upwards":["Permutation","Notations"],"downwards":["Two-line notation","One-line notation","Cycle notation","Canonical cycle notation","Composition of permutations"]}
{"upwards":["Permutation","Notations","Two-line notation"],"downwards":[]}
{"upwards":["Permutation","Notations","One-line notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Cycle notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Canonical cycle notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Composition of permutations"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation"],"downwards":["k-permutations of n","Permutations with repetition","Permutations of multisets","Circular permutations"]}
{"upwards":["Permutation","Other uses of the term permutation","k-permutations of n"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Permutations with repetition"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Permutations of multisets"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Circular permutations"],"downwards":[]}
{"upwards":["Permutation","Properties"],"downwards":["Cycle type","Conjugating permutations","Permutation order","Parity of a permutation","Matrix representation"]}
{"upwards":["Permutation","Properties","Cycle type"],"downwards":[]}
{"upwards":["Permutation","Properties","Conjugating permutations"],"downwards":[]}
{"upwards":["Permutation","Properties","Permutation order"],"downwards":[]}
{"upwards":["Permutation","Properties","Parity of a permutation"],"downwards":[]}
{"upwards":["Permutation","Properties","Matrix representation"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets"],"downwards":["Ascents, descents, runs and exceedances","Foata's transition lemma","Inversions"]}
{"upwards":["Permutation","Permutations of totally ordered sets","Ascents, descents, runs and exceedances"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets","Foata's transition lemma"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets","Inversions"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing"],"downwards":["Numbering permutations","Algorithms to generate permutations","Applications"]}
{"upwards":["Permutation","Permutations in computing","Numbering permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations"],"downwards":["Random generation of permutations","Generation in lexicographic order","Generation with minimal changes","Meandric permutations"]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Random generation of permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Generation in lexicographic order"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Generation with minimal changes"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Meandric permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Applications"],"downwards":[]}
{"upwards":["Surjective function"],"downwards":["Definition","Examples","Properties","Space of surjections","Gallery"]}
{"upwards":["Surjective function","Definition"],"downwards":[]}
{"upwards":["Surjective function","Examples"],"downwards":[]}
{"upwards":["Surjective function","Properties"],"downwards":["Surjections as right invertible functions","Surjections as epimorphisms","Surjections as binary relations","Cardinality of the domain of a surjection","Composition and decomposition","Induced surjection and induced bijection"]}
{"upwards":["Surjective function","Properties","Surjections as right invertible functions"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as epimorphisms"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as binary relations"],"downwards":[]}
{"upwards":["Surjective function","Properties","Cardinality of the domain of a surjection"],"downwards":[]}
{"upwards":["Surjective function","Properties","Composition and decomposition"],"downwards":[]}
{"upwards":["Surjective function","Properties","Induced surjection and induced bijection"],"downwards":[]}
{"upwards":["Surjective function","Space of surjections"],"downwards":[]}
{"upwards":["Surjective function","Gallery"],"downwards":[]}
{"upwards":["First-order logic"],"downwards":["Introduction","Syntax","Semantics","Deductive systems","Equality and its axioms","Metalogical properties","Limitations","Restrictions, extensions, and variations","Automated theorem proving and formal methods"]}
{"upwards":["First-order logic","Introduction"],"downwards":[]}
{"upwards":["First-order logic","Syntax"],"downwards":["Alphabet","Formation rules","Free and bound variables","Example: ordered abelian groups"]}
{"upwards":["First-order logic","Syntax","Alphabet"],"downwards":["Logical symbols","Non-logical symbols"]}
{"upwards":["First-order logic","Syntax","Alphabet","Logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Alphabet","Non-logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules"],"downwards":["Terms","Formulas","Notational conventions"]}
{"upwards":["First-order logic","Syntax","Formation rules","Terms"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Formulas"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Notational conventions"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Free and bound variables"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Example: ordered abelian groups"],"downwards":[]}
{"upwards":["First-order logic","Semantics"],"downwards":["First-order structures","Evaluation of truth values","Validity, satisfiability, and logical consequence","Algebraizations","First-order theories, models, and elementary classes","Empty domains"]}
{"upwards":["First-order logic","Semantics","First-order structures"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Evaluation of truth values"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Validity, satisfiability, and logical consequence"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Algebraizations"],"downwards":[]}
{"upwards":["First-order logic","Semantics","First-order theories, models, and elementary classes"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Empty domains"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems"],"downwards":["Rules of inference","Hilbert-style systems and natural deduction","Sequent calculus","Tableaux method","Resolution","Provable identities"]}
{"upwards":["First-order logic","Deductive systems","Rules of inference"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Hilbert-style systems and natural deduction"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Sequent calculus"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Tableaux method"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Resolution"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Provable identities"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms"],"downwards":["First-order logic without equality","Defining equality within a theory"]}
{"upwards":["First-order logic","Equality and its axioms","First-order logic without equality"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms","Defining equality within a theory"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties"],"downwards":["Completeness and undecidability","The L\u00f6wenheim\u2013Skolem theorem","The compactness theorem","Lindstr\u00f6m's theorem"]}
{"upwards":["First-order logic","Metalogical properties","Completeness and undecidability"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The compactness theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","Lindstr\u00f6m's theorem"],"downwards":[]}
{"upwards":["First-order logic","Limitations"],"downwards":["Expressiveness","Formalizing natural languages"]}
{"upwards":["First-order logic","Limitations","Expressiveness"],"downwards":[]}
{"upwards":["First-order logic","Limitations","Formalizing natural languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations"],"downwards":["Restricted languages","Many-sorted logic","Additional quantifiers","Infinitary logics","Non-classical and modal logics","Fixpoint logic","Higher-order logics"]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Restricted languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Many-sorted logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Additional quantifiers"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Infinitary logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Non-classical and modal logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Fixpoint logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Higher-order logics"],"downwards":[]}
{"upwards":["First-order logic","Automated theorem proving and formal methods"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Colon (punctuation)"],"downwards":["History","Usage in English","Usage in other languages","Mathematics and logic","Computing"]}
{"upwards":["Colon (punctuation)","History"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English"],"downwards":["Syntactical-deductive","Syntactical-descriptive","Appositive","Segmental","Use of capitals","Spacing and parentheses"]}
{"upwards":["Colon (punctuation)","Usage in English","Syntactical-deductive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Syntactical-descriptive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Appositive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Segmental"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Use of capitals"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Spacing and parentheses"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages"],"downwards":["Suffix separator","Abbreviation mark","Word separator","End of sentence or verse","Score divider"]}
{"upwards":["Colon (punctuation)","Usage in other languages","Suffix separator"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Abbreviation mark"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Word separator"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","End of sentence or verse"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Score divider"],"downwards":[]}
{"upwards":["Colon (punctuation)","Mathematics and logic"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing"],"downwards":["Programming languages","Addresses","Text markup"]}
{"upwards":["Colon (punctuation)","Computing","Programming languages"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing","Addresses"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing","Text markup"],"downwards":[]}
{"upwards":["Cardinality"],"downwards":["History","Comparing sets","Cardinal numbers","Finite, countable and uncountable sets","Infinite sets","Examples and properties","Union and intersection"]}
{"upwards":["Cardinality","History"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets"],"downwards":["Definition 1: |A| = |B|","Definition 2: |A| \u2264 |B|","Definition 3: |A| < |B|"]}
{"upwards":["Cardinality","Comparing sets","Definition 1: |A| = |B|"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets","Definition 2: |A| \u2264 |B|"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets","Definition 3: |A| < |B|"],"downwards":[]}
{"upwards":["Cardinality","Cardinal numbers"],"downwards":[]}
{"upwards":["Cardinality","Finite, countable and uncountable sets"],"downwards":[]}
{"upwards":["Cardinality","Infinite sets"],"downwards":["Cardinality of the continuum"]}
{"upwards":["Cardinality","Infinite sets","Cardinality of the continuum"],"downwards":[]}
{"upwards":["Cardinality","Examples and properties"],"downwards":[]}
{"upwards":["Cardinality","Union and intersection"],"downwards":[]}
{"upwards":["Set-builder notation"],"downwards":["Sets defined by enumeration","Sets defined by a predicate","More complex expressions on the left side of the notation","Equivalent predicates yield equal sets","Set existence axiom","In programming languages"]}
{"upwards":["Set-builder notation","Sets defined by enumeration"],"downwards":[]}
{"upwards":["Set-builder notation","Sets defined by a predicate"],"downwards":["Specifying the domain","Examples"]}
{"upwards":["Set-builder notation","Sets defined by a predicate","Specifying the domain"],"downwards":[]}
{"upwards":["Set-builder notation","Sets defined by a predicate","Examples"],"downwards":[]}
{"upwards":["Set-builder notation","More complex expressions on the left side of the notation"],"downwards":[]}
{"upwards":["Set-builder notation","Equivalent predicates yield equal sets"],"downwards":[]}
{"upwards":["Set-builder notation","Set existence axiom"],"downwards":[]}
{"upwards":["Set-builder notation","In programming languages"],"downwards":[]}
{"upwards":["Integer"],"downwards":["History","Algebraic properties","Order-theoretic properties","Construction","Computer science","Cardinality"]}
{"upwards":["Integer","History"],"downwards":[]}
{"upwards":["Integer","Algebraic properties"],"downwards":[]}
{"upwards":["Integer","Order-theoretic properties"],"downwards":[]}
{"upwards":["Integer","Construction"],"downwards":["Traditional development","Equivalence classes of ordered pairs","Other approaches"]}
{"upwards":["Integer","Construction","Traditional development"],"downwards":[]}
{"upwards":["Integer","Construction","Equivalence classes of ordered pairs"],"downwards":[]}
{"upwards":["Integer","Construction","Other approaches"],"downwards":[]}
{"upwards":["Integer","Computer science"],"downwards":[]}
{"upwards":["Integer","Cardinality"],"downwards":[]}
{"upwards":["Extensional and intensional definitions"],"downwards":["Intensional definition","Extensional definition","History"]}
{"upwards":["Extensional and intensional definitions","Intensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","Extensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","History"],"downwards":[]}
{"upwards":["Line (geometry)"],"downwards":["Properties","Definition","Other representations","Generalizations of the Euclidean line","Extensions"]}
{"upwards":["Line (geometry)","Properties"],"downwards":["In higher dimensions","Types","In axiomatic systems"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions"],"downwards":["Collinear points"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions","Collinear points"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","Types"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","In axiomatic systems"],"downwards":[]}
{"upwards":["Line (geometry)","Definition"],"downwards":["Linear equation","Parametric equation","Hesse normal form"]}
{"upwards":["Line (geometry)","Definition","Linear equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Parametric equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Hesse normal form"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations"],"downwards":["Vectors","Polar coordinates"]}
{"upwards":["Line (geometry)","Other representations","Vectors"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations","Polar coordinates"],"downwards":[]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line"],"downwards":["Projective geometry"]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line","Projective geometry"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions"],"downwards":["Ray","Line segment","Number line"]}
{"upwards":["Line (geometry)","Extensions","Ray"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Line segment"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Number line"],"downwards":[]}
{"upwards":["Extensional and intensional definitions"],"downwards":["Intensional definition","Extensional definition","History"]}
{"upwards":["Extensional and intensional definitions","Intensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","Extensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","History"],"downwards":[]}
{"upwards":["Uncountable set"],"downwards":["Characterizations","Properties","Examples","Without the axiom of choice"]}
{"upwards":["Uncountable set","Characterizations"],"downwards":[]}
{"upwards":["Uncountable set","Properties"],"downwards":[]}
{"upwards":["Uncountable set","Examples"],"downwards":[]}
{"upwards":["Uncountable set","Without the axiom of choice"],"downwards":[]}
{"upwards":["Relation (mathematics)"],"downwards":["Definition","Representation of relations","Properties of relations","Operations on relations","Examples","Generalizations"]}
{"upwards":["Relation (mathematics)","Definition"],"downwards":[]}
{"upwards":["Relation (mathematics)","Representation of relations"],"downwards":[]}
{"upwards":["Relation (mathematics)","Properties of relations"],"downwards":["Combinations of properties"]}
{"upwards":["Relation (mathematics)","Properties of relations","Combinations of properties"],"downwards":[]}
{"upwards":["Relation (mathematics)","Operations on relations"],"downwards":[]}
{"upwards":["Relation (mathematics)","Examples"],"downwards":[]}
{"upwards":["Relation (mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Integer"],"downwards":["History","Algebraic properties","Order-theoretic properties","Construction","Computer science","Cardinality"]}
{"upwards":["Integer","History"],"downwards":[]}
{"upwards":["Integer","Algebraic properties"],"downwards":[]}
{"upwards":["Integer","Order-theoretic properties"],"downwards":[]}
{"upwards":["Integer","Construction"],"downwards":["Traditional development","Equivalence classes of ordered pairs","Other approaches"]}
{"upwards":["Integer","Construction","Traditional development"],"downwards":[]}
{"upwards":["Integer","Construction","Equivalence classes of ordered pairs"],"downwards":[]}
{"upwards":["Integer","Construction","Other approaches"],"downwards":[]}
{"upwards":["Integer","Computer science"],"downwards":[]}
{"upwards":["Integer","Cardinality"],"downwards":[]}
{"upwards":["Enumerative definition"],"downwards":[]}
{"upwards":["Rational number"],"downwards":["Terminology","Arithmetic","Continued fraction representation","Other representations","Formal construction","Properties","Real numbers and topological properties","p-adic numbers"]}
{"upwards":["Rational number","Terminology"],"downwards":["Etymology"]}
{"upwards":["Rational number","Terminology","Etymology"],"downwards":[]}
{"upwards":["Rational number","Arithmetic"],"downwards":["Irreducible fraction","Embedding of integers","Equality","Ordering","Addition","Subtraction","Multiplication","Inverse","Division","Exponentiation to integer power"]}
{"upwards":["Rational number","Arithmetic","Irreducible fraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Embedding of integers"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Equality"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Ordering"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Addition"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Multiplication"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Inverse"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Division"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Exponentiation to integer power"],"downwards":[]}
{"upwards":["Rational number","Continued fraction representation"],"downwards":[]}
{"upwards":["Rational number","Other representations"],"downwards":[]}
{"upwards":["Rational number","Formal construction"],"downwards":[]}
{"upwards":["Rational number","Properties"],"downwards":["Countability"]}
{"upwards":["Rational number","Properties","Countability"],"downwards":[]}
{"upwards":["Rational number","Real numbers and topological properties"],"downwards":[]}
{"upwards":["Rational number","p-adic numbers"],"downwards":[]}
{"upwards":["Rational number"],"downwards":["Terminology","Arithmetic","Continued fraction representation","Other representations","Formal construction","Properties","Real numbers and topological properties","p-adic numbers"]}
{"upwards":["Rational number","Terminology"],"downwards":["Etymology"]}
{"upwards":["Rational number","Terminology","Etymology"],"downwards":[]}
{"upwards":["Rational number","Arithmetic"],"downwards":["Irreducible fraction","Embedding of integers","Equality","Ordering","Addition","Subtraction","Multiplication","Inverse","Division","Exponentiation to integer power"]}
{"upwards":["Rational number","Arithmetic","Irreducible fraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Embedding of integers"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Equality"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Ordering"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Addition"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Multiplication"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Inverse"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Division"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Exponentiation to integer power"],"downwards":[]}
{"upwards":["Rational number","Continued fraction representation"],"downwards":[]}
{"upwards":["Rational number","Other representations"],"downwards":[]}
{"upwards":["Rational number","Formal construction"],"downwards":[]}
{"upwards":["Rational number","Properties"],"downwards":["Countability"]}
{"upwards":["Rational number","Properties","Countability"],"downwards":[]}
{"upwards":["Rational number","Real numbers and topological properties"],"downwards":[]}
{"upwards":["Rational number","p-adic numbers"],"downwards":[]}
{"upwards":["Complex number"],"downwards":["Definition","Notation","Visualization","History","Relations and operations","Properties","Formal construction","Complex analysis","Applications","Generalizations and related notions"]}
{"upwards":["Complex number","Definition"],"downwards":[]}
{"upwards":["Complex number","Notation"],"downwards":[]}
{"upwards":["Complex number","Visualization"],"downwards":["Cartesian complex plane","Polar complex plane","Complex graphs"]}
{"upwards":["Complex number","Visualization","Cartesian complex plane"],"downwards":[]}
{"upwards":["Complex number","Visualization","Polar complex plane"],"downwards":["Modulus and argument"]}
{"upwards":["Complex number","Visualization","Polar complex plane","Modulus and argument"],"downwards":[]}
{"upwards":["Complex number","Visualization","Complex graphs"],"downwards":[]}
{"upwards":["Complex number","History"],"downwards":[]}
{"upwards":["Complex number","Relations and operations"],"downwards":["Equality","Ordering","Conjugate","Addition and subtraction","Multiplication and square","Reciprocal and division","Multiplication and division in polar form","Square root","Exponential function","Complex logarithm","Exponentiation"]}
{"upwards":["Complex number","Relations and operations","Equality"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Ordering"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Conjugate"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and square"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Reciprocal and division"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and division in polar form"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Square root"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function"],"downwards":["Functional equation","Euler's formula"]}
{"upwards":["Complex number","Relations and operations","Exponential function","Functional equation"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function","Euler's formula"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Complex logarithm"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponentiation"],"downwards":["Integer and fractional exponents"]}
{"upwards":["Complex number","Relations and operations","Exponentiation","Integer and fractional exponents"],"downwards":[]}
{"upwards":["Complex number","Properties"],"downwards":["Field structure","Solutions of polynomial equations","Algebraic characterization","Characterization as a topological field"]}
{"upwards":["Complex number","Properties","Field structure"],"downwards":[]}
{"upwards":["Complex number","Properties","Solutions of polynomial equations"],"downwards":[]}
{"upwards":["Complex number","Properties","Algebraic characterization"],"downwards":[]}
{"upwards":["Complex number","Properties","Characterization as a topological field"],"downwards":[]}
{"upwards":["Complex number","Formal construction"],"downwards":["Construction as ordered pairs","Construction as a quotient field","Matrix representation of complex numbers"]}
{"upwards":["Complex number","Formal construction","Construction as ordered pairs"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Construction as a quotient field"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Matrix representation of complex numbers"],"downwards":[]}
{"upwards":["Complex number","Complex analysis"],"downwards":["Complex exponential and related functions","Holomorphic functions"]}
{"upwards":["Complex number","Complex analysis","Complex exponential and related functions"],"downwards":[]}
{"upwards":["Complex number","Complex analysis","Holomorphic functions"],"downwards":[]}
{"upwards":["Complex number","Applications"],"downwards":["Geometry","Algebraic number theory","Analytic number theory","Improper integrals","Dynamic equations","Linear algebra","In applied mathematics","In physics"]}
{"upwards":["Complex number","Applications","Geometry"],"downwards":["Shapes","Fractal geometry","Triangles"]}
{"upwards":["Complex number","Applications","Geometry","Shapes"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Fractal geometry"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Triangles"],"downwards":[]}
{"upwards":["Complex number","Applications","Algebraic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Analytic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Improper integrals"],"downwards":[]}
{"upwards":["Complex number","Applications","Dynamic equations"],"downwards":[]}
{"upwards":["Complex number","Applications","Linear algebra"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics"],"downwards":["Control theory","Signal analysis"]}
{"upwards":["Complex number","Applications","In applied mathematics","Control theory"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics","Signal analysis"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics"],"downwards":["Electromagnetism and electrical engineering","Fluid dynamics","Quantum mechanics","Relativity"]}
{"upwards":["Complex number","Applications","In physics","Electromagnetism and electrical engineering"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Fluid dynamics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Quantum mechanics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Relativity"],"downwards":[]}
{"upwards":["Complex number","Generalizations and related notions"],"downwards":[]}
{"upwards":["Real number"],"downwards":["Characterizing properties","Arithmetic","Dedekind completeness","Decimal representation","Topological completeness","Cardinality","Other properties","History","Formal definitions","Applications and connections","Vocabulary and notation","Generalizations and extensions"]}
{"upwards":["Real number","Characterizing properties"],"downwards":[]}
{"upwards":["Real number","Arithmetic"],"downwards":["Auxiliary operations","Auxiliary order relations","Integers and fractions as real numbers"]}
{"upwards":["Real number","Arithmetic","Auxiliary operations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Auxiliary order relations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Integers and fractions as real numbers"],"downwards":[]}
{"upwards":["Real number","Dedekind completeness"],"downwards":[]}
{"upwards":["Real number","Decimal representation"],"downwards":[]}
{"upwards":["Real number","Topological completeness"],"downwards":["\"The complete ordered field\""]}
{"upwards":["Real number","Topological completeness","\"The complete ordered field\""],"downwards":[]}
{"upwards":["Real number","Cardinality"],"downwards":[]}
{"upwards":["Real number","Other properties"],"downwards":[]}
{"upwards":["Real number","History"],"downwards":[]}
{"upwards":["Real number","Formal definitions"],"downwards":["Axiomatic approach","Construction from the rational numbers"]}
{"upwards":["Real number","Formal definitions","Axiomatic approach"],"downwards":[]}
{"upwards":["Real number","Formal definitions","Construction from the rational numbers"],"downwards":[]}
{"upwards":["Real number","Applications and connections"],"downwards":["Physics","Logic","Computation","Set theory"]}
{"upwards":["Real number","Applications and connections","Physics"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Logic"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Computation"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Set theory"],"downwards":[]}
{"upwards":["Real number","Vocabulary and notation"],"downwards":[]}
{"upwards":["Real number","Generalizations and extensions"],"downwards":[]}
{"upwards":["Mathematical model"],"downwards":["Elements of a mathematical model","Classifications","Construction","Significance in the natural sciences","Some applications","Examples"]}
{"upwards":["Mathematical model","Elements of a mathematical model"],"downwards":[]}
{"upwards":["Mathematical model","Classifications"],"downwards":[]}
{"upwards":["Mathematical model","Construction"],"downwards":["A priori information","Complexity","Training, tuning, and fitting","Evaluation and assessment"]}
{"upwards":["Mathematical model","Construction","A priori information"],"downwards":["Subjective information"]}
{"upwards":["Mathematical model","Construction","A priori information","Subjective information"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Complexity"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Training, tuning, and fitting"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment"],"downwards":["Prediction of empirical data","Scope of the model","Philosophical considerations"]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Prediction of empirical data"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Scope of the model"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Philosophical considerations"],"downwards":[]}
{"upwards":["Mathematical model","Significance in the natural sciences"],"downwards":[]}
{"upwards":["Mathematical model","Some applications"],"downwards":[]}
{"upwards":["Mathematical model","Examples"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Plane (mathematics)"],"downwards":["Euclidean plane","Elliptic plane","Projective plane","Further generalizations","Topological and differential geometric notions"]}
{"upwards":["Plane (mathematics)","Euclidean plane"],"downwards":["Embedding in three-dimensional space"]}
{"upwards":["Plane (mathematics)","Euclidean plane","Embedding in three-dimensional space"],"downwards":[]}
{"upwards":["Plane (mathematics)","Elliptic plane"],"downwards":[]}
{"upwards":["Plane (mathematics)","Projective plane"],"downwards":[]}
{"upwards":["Plane (mathematics)","Further generalizations"],"downwards":[]}
{"upwards":["Plane (mathematics)","Topological and differential geometric notions"],"downwards":[]}
{"upwards":["Venn diagram"],"downwards":["Details","Example","History","Overview","Extensions to higher numbers of sets","Related concepts"]}
{"upwards":["Venn diagram","Details"],"downwards":[]}
{"upwards":["Venn diagram","Example"],"downwards":[]}
{"upwards":["Venn diagram","History"],"downwards":[]}
{"upwards":["Venn diagram","Overview"],"downwards":[]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets"],"downwards":["Edwards\u2013Venn diagrams","Other diagrams"]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets","Edwards\u2013Venn diagrams"],"downwards":[]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets","Other diagrams"],"downwards":[]}
{"upwards":["Venn diagram","Related concepts"],"downwards":[]}
{"upwards":["Line segment"],"downwards":["In real or complex vector spaces","Properties","In proofs","As a degenerate ellipse","In other geometric shapes","Directed line segment","Generalizations","Types of line segments"]}
{"upwards":["Line segment","In real or complex vector spaces"],"downwards":[]}
{"upwards":["Line segment","Properties"],"downwards":[]}
{"upwards":["Line segment","In proofs"],"downwards":[]}
{"upwards":["Line segment","As a degenerate ellipse"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes"],"downwards":["Triangles","Quadrilaterals","Circles and ellipses"]}
{"upwards":["Line segment","In other geometric shapes","Triangles"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes","Quadrilaterals"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes","Circles and ellipses"],"downwards":[]}
{"upwards":["Line segment","Directed line segment"],"downwards":[]}
{"upwards":["Line segment","Generalizations"],"downwards":[]}
{"upwards":["Line segment","Types of line segments"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
{"upwards":["Subset","Definition"],"downwards":[]}
{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["Ostensive definition"],"downwards":["Overview"]}
{"upwards":["Ostensive definition","Overview"],"downwards":[]}
{"upwards":["Independence (mathematical logic)"],"downwards":["Usage note","Independence results in set theory","Applications to physical theory"]}
{"upwards":["Independence (mathematical logic)","Usage note"],"downwards":[]}
{"upwards":["Independence (mathematical logic)","Independence results in set theory"],"downwards":[]}
{"upwards":["Independence (mathematical logic)","Applications to physical theory"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Power set"],"downwards":["Example","Properties","Representing subsets as functions","Relation to binomial theorem","Recursive definition","Subsets of limited cardinality","Power object","Functors and quantifiers"]}
{"upwards":["Power set","Example"],"downwards":[]}
{"upwards":["Power set","Properties"],"downwards":[]}
{"upwards":["Power set","Representing subsets as functions"],"downwards":[]}
{"upwards":["Power set","Relation to binomial theorem"],"downwards":[]}
{"upwards":["Power set","Recursive definition"],"downwards":[]}
{"upwards":["Power set","Subsets of limited cardinality"],"downwards":[]}
{"upwards":["Power set","Power object"],"downwards":[]}
{"upwards":["Power set","Functors and quantifiers"],"downwards":[]}
{"upwards":["Paul Cohen"],"downwards":["Early life and education","Career","Death","Selected publications"]}
{"upwards":["Paul Cohen","Early life and education"],"downwards":[]}
{"upwards":["Paul Cohen","Career"],"downwards":["Continuum hypothesis"]}
{"upwards":["Paul Cohen","Career","Continuum hypothesis"],"downwards":[]}
{"upwards":["Paul Cohen","Death"],"downwards":[]}
{"upwards":["Paul Cohen","Selected publications"],"downwards":[]}
{"upwards":["Union (set theory)"],"downwards":["Union of two sets","Algebraic properties","Finite unions","Arbitrary unions","Notation encoding"]}
{"upwards":["Union (set theory)","Union of two sets"],"downwards":[]}
{"upwards":["Union (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Union (set theory)","Finite unions"],"downwards":[]}
{"upwards":["Union (set theory)","Arbitrary unions"],"downwards":["Notations"]}
{"upwards":["Union (set theory)","Arbitrary unions","Notations"],"downwards":[]}
{"upwards":["Union (set theory)","Notation encoding"],"downwards":[]}
{"upwards":["Partition of a set"],"downwards":["Definition and notation","Examples","Partitions and equivalence relations","Refinement of partitions","Noncrossing partitions","Counting partitions"]}
{"upwards":["Partition of a set","Definition and notation"],"downwards":[]}
{"upwards":["Partition of a set","Examples"],"downwards":[]}
{"upwards":["Partition of a set","Partitions and equivalence relations"],"downwards":[]}
{"upwards":["Partition of a set","Refinement of partitions"],"downwards":[]}
{"upwards":["Partition of a set","Noncrossing partitions"],"downwards":[]}
{"upwards":["Partition of a set","Counting partitions"],"downwards":[]}
{"upwards":["Uncountable set"],"downwards":["Characterizations","Properties","Examples","Without the axiom of choice"]}
{"upwards":["Uncountable set","Characterizations"],"downwards":[]}
{"upwards":["Uncountable set","Properties"],"downwards":[]}
{"upwards":["Uncountable set","Examples"],"downwards":[]}
{"upwards":["Uncountable set","Without the axiom of choice"],"downwards":[]}
{"upwards":["Universe (mathematics)"],"downwards":["In a specific context","In ordinary mathematics","In set theory","In predicate calculus","In category theory","In type theory"]}
{"upwards":["Universe (mathematics)","In a specific context"],"downwards":[]}
{"upwards":["Universe (mathematics)","In ordinary mathematics"],"downwards":[]}
{"upwards":["Universe (mathematics)","In set theory"],"downwards":[]}
{"upwards":["Universe (mathematics)","In predicate calculus"],"downwards":[]}
{"upwards":["Universe (mathematics)","In category theory"],"downwards":[]}
{"upwards":["Universe (mathematics)","In type theory"],"downwards":[]}
{"upwards":["Algebra of sets"],"downwards":["Fundamentals","The fundamental properties of set algebra","The principle of duality","Some additional laws for unions and intersections","Some additional laws for complements","The algebra of inclusion","The algebra of relative complements"]}
{"upwards":["Algebra of sets","Fundamentals"],"downwards":[]}
{"upwards":["Algebra of sets","The fundamental properties of set algebra"],"downwards":[]}
{"upwards":["Algebra of sets","The principle of duality"],"downwards":[]}
{"upwards":["Algebra of sets","Some additional laws for unions and intersections"],"downwards":[]}
{"upwards":["Algebra of sets","Some additional laws for complements"],"downwards":[]}
{"upwards":["Algebra of sets","The algebra of inclusion"],"downwards":[]}
{"upwards":["Algebra of sets","The algebra of relative complements"],"downwards":[]}
{"upwards":["Bijection"],"downwards":["Definition","Examples","More mathematical examples","Inverses","Composition","Cardinality","Properties","Category theory","Generalization to partial functions","Gallery"]}
{"upwards":["Bijection","Definition"],"downwards":[]}
{"upwards":["Bijection","Examples"],"downwards":["Batting line-up of a baseball or cricket team","Seats and students of a classroom"]}
{"upwards":["Bijection","Examples","Batting line-up of a baseball or cricket team"],"downwards":[]}
{"upwards":["Bijection","Examples","Seats and students of a classroom"],"downwards":[]}
{"upwards":["Bijection","More mathematical examples"],"downwards":[]}
{"upwards":["Bijection","Inverses"],"downwards":[]}
{"upwards":["Bijection","Composition"],"downwards":[]}
{"upwards":["Bijection","Cardinality"],"downwards":[]}
{"upwards":["Bijection","Properties"],"downwards":[]}
{"upwards":["Bijection","Category theory"],"downwards":[]}
{"upwards":["Bijection","Generalization to partial functions"],"downwards":[]}
{"upwards":["Bijection","Gallery"],"downwards":[]}
{"upwards":["Cartesian product"],"downwards":["Set-theoretic definition","Examples","Most common implementation (set theory)","Cartesian products of several sets","Other forms","Definitions outside set theory"]}
{"upwards":["Cartesian product","Set-theoretic definition"],"downwards":[]}
{"upwards":["Cartesian product","Examples"],"downwards":["A deck of cards","A two-dimensional coordinate system"]}
{"upwards":["Cartesian product","Examples","A deck of cards"],"downwards":[]}
{"upwards":["Cartesian product","Examples","A two-dimensional coordinate system"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)"],"downwards":["Non-commutativity and non-associativity","Intersections, unions, and subsets","Cardinality"]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Non-commutativity and non-associativity"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Intersections, unions, and subsets"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Cardinality"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets"],"downwards":["n-ary Cartesian product","n-ary Cartesian power","Infinite Cartesian products"]}
{"upwards":["Cartesian product","Cartesian products of several sets","n-ary Cartesian product"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets","n-ary Cartesian power"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets","Infinite Cartesian products"],"downwards":[]}
{"upwards":["Cartesian product","Other forms"],"downwards":["Abbreviated form","Cartesian product of functions","Cylinder"]}
{"upwards":["Cartesian product","Other forms","Abbreviated form"],"downwards":[]}
{"upwards":["Cartesian product","Other forms","Cartesian product of functions"],"downwards":[]}
{"upwards":["Cartesian product","Other forms","Cylinder"],"downwards":[]}
{"upwards":["Cartesian product","Definitions outside set theory"],"downwards":["Category theory","Graph theory"]}
{"upwards":["Cartesian product","Definitions outside set theory","Category theory"],"downwards":[]}
{"upwards":["Cartesian product","Definitions outside set theory","Graph theory"],"downwards":[]}
{"upwards":["Disjoint sets"],"downwards":["Generalizations","Intersections","Disjoint unions and partitions"]}
{"upwards":["Disjoint sets","Generalizations"],"downwards":[]}
{"upwards":["Disjoint sets","Intersections"],"downwards":[]}
{"upwards":["Disjoint sets","Disjoint unions and partitions"],"downwards":[]}
{"upwards":["Ordered pair"],"downwards":["Generalities","Informal and formal definitions","Defining the ordered pair using set theory","Category theory"]}
{"upwards":["Ordered pair","Generalities"],"downwards":[]}
{"upwards":["Ordered pair","Informal and formal definitions"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory"],"downwards":["Wiener's definition","Hausdorff's definition","Kuratowski's definition","Quine\u2013Rosser definition","Cantor\u2013Frege definition","Morse definition","Axiomatic definition"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Wiener's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Hausdorff's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition"],"downwards":["Variants","Proving that definitions satisfy the characteristic property"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Variants"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Proving that definitions satisfy the characteristic property"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Quine\u2013Rosser definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Cantor\u2013Frege definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Morse definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Axiomatic definition"],"downwards":[]}
{"upwards":["Ordered pair","Category theory"],"downwards":[]}
{"upwards":["Symmetric difference"],"downwards":["Properties","n-ary symmetric difference","Symmetric difference on measure spaces","Hausdorff distance vs. symmetric difference"]}
{"upwards":["Symmetric difference","Properties"],"downwards":[]}
{"upwards":["Symmetric difference","n-ary symmetric difference"],"downwards":[]}
{"upwards":["Symmetric difference","Symmetric difference on measure spaces"],"downwards":[]}
{"upwards":["Symmetric difference","Hausdorff distance vs. symmetric difference"],"downwards":[]}
{"upwards":["Continuum hypothesis"],"downwards":["History","Cardinality of infinite sets","Independence from ZFC","Arguments for and against the continuum hypothesis","The generalized continuum hypothesis"]}
{"upwards":["Continuum hypothesis","History"],"downwards":[]}
{"upwards":["Continuum hypothesis","Cardinality of infinite sets"],"downwards":[]}
{"upwards":["Continuum hypothesis","Independence from ZFC"],"downwards":[]}
{"upwards":["Continuum hypothesis","Arguments for and against the continuum hypothesis"],"downwards":[]}
{"upwards":["Continuum hypothesis","The generalized continuum hypothesis"],"downwards":["Implications of GCH for cardinal exponentiation"]}
{"upwards":["Continuum hypothesis","The generalized continuum hypothesis","Implications of GCH for cardinal exponentiation"],"downwards":[]}
{"upwards":["Intersection (set theory)"],"downwards":["Notation and terminology","Definition","Algebraic properties","Arbitrary intersections","Nullary intersection"]}
{"upwards":["Intersection (set theory)","Notation and terminology"],"downwards":[]}
{"upwards":["Intersection (set theory)","Definition"],"downwards":["Intersecting and disjoint sets"]}
{"upwards":["Intersection (set theory)","Definition","Intersecting and disjoint sets"],"downwards":[]}
{"upwards":["Intersection (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Intersection (set theory)","Arbitrary intersections"],"downwards":[]}
{"upwards":["Intersection (set theory)","Nullary intersection"],"downwards":[]}
{"upwards":["De Morgan's laws"],"downwards":["Formal notation","History","Informal proof","Formal proof","Generalising De Morgan duality","Extension to predicate and modal logic","In intuitionistic logic","In computer engineering"]}
{"upwards":["De Morgan's laws","Formal notation"],"downwards":["Substitution form","Set theory and Boolean algebra","Engineering","Text searching"]}
{"upwards":["De Morgan's laws","Formal notation","Substitution form"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Set theory and Boolean algebra"],"downwards":["Unions and intersections of any number of sets"]}
{"upwards":["De Morgan's laws","Formal notation","Set theory and Boolean algebra","Unions and intersections of any number of sets"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Engineering"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Text searching"],"downwards":[]}
{"upwards":["De Morgan's laws","History"],"downwards":[]}
{"upwards":["De Morgan's laws","Informal proof"],"downwards":["Negation of a disjunction","Negation of a conjunction"]}
{"upwards":["De Morgan's laws","Informal proof","Negation of a disjunction"],"downwards":[]}
{"upwards":["De Morgan's laws","Informal proof","Negation of a conjunction"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof"],"downwards":["Part 1","Part 2","Conclusion"]}
{"upwards":["De Morgan's laws","Formal proof","Part 1"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof","Part 2"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof","Conclusion"],"downwards":[]}
{"upwards":["De Morgan's laws","Generalising De Morgan duality"],"downwards":[]}
{"upwards":["De Morgan's laws","Extension to predicate and modal logic"],"downwards":[]}
{"upwards":["De Morgan's laws","In intuitionistic logic"],"downwards":[]}
{"upwards":["De Morgan's laws","In computer engineering"],"downwards":[]}
{"upwards":["Euclidean space"],"downwards":["Definition","Prototypical examples","Affine structure","Metric structure","Isometries","Topology","Axiomatic definitions","Usage","Other geometric spaces"]}
{"upwards":["Euclidean space","Definition"],"downwards":["History of the definition","Motivation of the modern definition","Technical definition"]}
{"upwards":["Euclidean space","Definition","History of the definition"],"downwards":[]}
{"upwards":["Euclidean space","Definition","Motivation of the modern definition"],"downwards":[]}
{"upwards":["Euclidean space","Definition","Technical definition"],"downwards":[]}
{"upwards":["Euclidean space","Prototypical examples"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure"],"downwards":["Subspaces","Lines and segments","Parallelism"]}
{"upwards":["Euclidean space","Affine structure","Subspaces"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure","Lines and segments"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure","Parallelism"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure"],"downwards":["Distance and length","Orthogonality","Angle","Cartesian coordinates","Other coordinates"]}
{"upwards":["Euclidean space","Metric structure","Distance and length"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Orthogonality"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Angle"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Cartesian coordinates"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Other coordinates"],"downwards":[]}
{"upwards":["Euclidean space","Isometries"],"downwards":["Isometry with prototypical examples","Euclidean group"]}
{"upwards":["Euclidean space","Isometries","Isometry with prototypical examples"],"downwards":[]}
{"upwards":["Euclidean space","Isometries","Euclidean group"],"downwards":[]}
{"upwards":["Euclidean space","Topology"],"downwards":[]}
{"upwards":["Euclidean space","Axiomatic definitions"],"downwards":[]}
{"upwards":["Euclidean space","Usage"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces"],"downwards":["Affine space","Projective space","Non-Euclidean geometries","Curved spaces","Pseudo-Euclidean space"]}
{"upwards":["Euclidean space","Other geometric spaces","Affine space"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Projective space"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Non-Euclidean geometries"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Curved spaces"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Pseudo-Euclidean space"],"downwards":[]}
{"upwards":["Group (mathematics)"],"downwards":["Definition and illustration","History","Elementary consequences of the group axioms","Basic concepts","Examples and applications","Finite groups","Groups with additional structure","Generalizations"]}
{"upwards":["Group (mathematics)","Definition and illustration"],"downwards":["First example: the integers","Definition","Notation and terminology","Second example: a symmetry group"]}
{"upwards":["Group (mathematics)","Definition and illustration","First example: the integers"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Definition"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Notation and terminology"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Second example: a symmetry group"],"downwards":[]}
{"upwards":["Group (mathematics)","History"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms"],"downwards":["Uniqueness of identity element","Uniqueness of inverses","Division","Equivalent definition with relaxed axioms"]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Uniqueness of identity element"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Uniqueness of inverses"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Division"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Equivalent definition with relaxed axioms"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts"],"downwards":["Group homomorphisms","Subgroups","Cosets","Quotient groups","Presentations"]}
{"upwards":["Group (mathematics)","Basic concepts","Group homomorphisms"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Subgroups"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Cosets"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Quotient groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Presentations"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications"],"downwards":["Numbers","Modular arithmetic","Cyclic groups","Symmetry groups","General linear group and representation theory","Galois groups"]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers"],"downwards":["Integers","Rationals"]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers","Integers"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers","Rationals"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Modular arithmetic"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Cyclic groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Symmetry groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","General linear group and representation theory"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Galois groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups"],"downwards":["Finite abelian groups","Simple groups","Classification of finite simple groups"]}
{"upwards":["Group (mathematics)","Finite groups","Finite abelian groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups","Simple groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups","Classification of finite simple groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Groups with additional structure"],"downwards":["Topological groups","Lie groups"]}
{"upwards":["Group (mathematics)","Groups with additional structure","Topological groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Groups with additional structure","Lie groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Abstract algebra"],"downwards":["History","Basic concepts","Branches of abstract algebra","Applications"]}
{"upwards":["Abstract algebra","History"],"downwards":["Elementary algebra","Early group theory","Early ring theory","Early field theory","Other major areas","Modern algebra"]}
{"upwards":["Abstract algebra","History","Elementary algebra"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early group theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early ring theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early field theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Other major areas"],"downwards":[]}
{"upwards":["Abstract algebra","History","Modern algebra"],"downwards":[]}
{"upwards":["Abstract algebra","Basic concepts"],"downwards":[]}
{"upwards":["Abstract algebra","Branches of abstract algebra"],"downwards":["Group theory","Ring theory"]}
{"upwards":["Abstract algebra","Branches of abstract algebra","Group theory"],"downwards":[]}
{"upwards":["Abstract algebra","Branches of abstract algebra","Ring theory"],"downwards":[]}
{"upwards":["Abstract algebra","Applications"],"downwards":[]}
{"upwards":["Boolean ring"],"downwards":["Notations","Examples","Relation to Boolean algebras","Properties of Boolean rings","Unification"]}
{"upwards":["Boolean ring","Notations"],"downwards":[]}
{"upwards":["Boolean ring","Examples"],"downwards":[]}
{"upwards":["Boolean ring","Relation to Boolean algebras"],"downwards":[]}
{"upwards":["Boolean ring","Properties of Boolean rings"],"downwards":[]}
{"upwards":["Boolean ring","Unification"],"downwards":[]}
{"upwards":["Field (mathematics)"],"downwards":["Definition","Examples","Elementary notions","Finite fields","History","Constructing fields","Fields with additional structure","Galois theory","Invariants of fields","Applications","Related notions","Notes","References","External links"]}
{"upwards":["Field (mathematics)","Definition"],"downwards":["Classic definition","Alternative definition"]}
{"upwards":["Field (mathematics)","Definition","Classic definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Definition","Alternative definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples"],"downwards":["Rational numbers","Real and complex numbers","Constructible numbers","A field with four elements"]}
{"upwards":["Field (mathematics)","Examples","Rational numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","Real and complex numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","Constructible numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","A field with four elements"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions"],"downwards":["Consequences of the definition","The additive and the multiplicative group of a field","Characteristic","Subfields and prime fields"]}
{"upwards":["Field (mathematics)","Elementary notions","Consequences of the definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","The additive and the multiplicative group of a field"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","Characteristic"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","Subfields and prime fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Finite fields"],"downwards":[]}
{"upwards":["Field (mathematics)","History"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields"],"downwards":["Constructing fields from rings","Constructing fields within a bigger field","Field extensions","Closure operations"]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings"],"downwards":["Field of fractions","Residue fields"]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings","Field of fractions"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings","Residue fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields within a bigger field"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions"],"downwards":["Algebraic extensions","Transcendence bases"]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions","Algebraic extensions"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions","Transcendence bases"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Closure operations"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure"],"downwards":["Ordered fields","Topological fields","Differential fields"]}
{"upwards":["Field (mathematics)","Fields with additional structure","Ordered fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure","Topological fields"],"downwards":["Local fields"]}
{"upwards":["Field (mathematics)","Fields with additional structure","Topological fields","Local fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure","Differential fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Galois theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields"],"downwards":["Model theory of fields","The absolute Galois group","K-theory"]}
{"upwards":["Field (mathematics)","Invariants of fields","Model theory of fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields","The absolute Galois group"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields","K-theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications"],"downwards":["Linear algebra and commutative algebra","Finite fields: cryptography and coding theory","Geometry: field of functions","Number theory: global fields"]}
{"upwards":["Field (mathematics)","Applications","Linear algebra and commutative algebra"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Finite fields: cryptography and coding theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Geometry: field of functions"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Number theory: global fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Related notions"],"downwards":["Division rings"]}
{"upwards":["Field (mathematics)","Related notions","Division rings"],"downwards":[]}
{"upwards":["Field (mathematics)","Notes"],"downwards":[]}
{"upwards":["Field (mathematics)","References"],"downwards":[]}
{"upwards":["Field (mathematics)","External links"],"downwards":[]}
{"upwards":["Algebraic structure"],"downwards":["Introduction","Common axioms","Common algebraic structures","Hybrid structures","Universal algebra","Category theory","Different meanings of \"structure\""]}
{"upwards":["Algebraic structure","Introduction"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms"],"downwards":["Equational axioms","Existential axioms","Non-equational axioms"]}
{"upwards":["Algebraic structure","Common axioms","Equational axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms","Existential axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms","Non-equational axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common algebraic structures"],"downwards":["One set with operations","Two sets with operations"]}
{"upwards":["Algebraic structure","Common algebraic structures","One set with operations"],"downwards":[]}
{"upwards":["Algebraic structure","Common algebraic structures","Two sets with operations"],"downwards":[]}
{"upwards":["Algebraic structure","Hybrid structures"],"downwards":[]}
{"upwards":["Algebraic structure","Universal algebra"],"downwards":[]}
{"upwards":["Algebraic structure","Category theory"],"downwards":[]}
{"upwards":["Algebraic structure","Different meanings of \"structure\""],"downwards":[]}
{"upwards":["Ring (mathematics)"],"downwards":["Definition","Illustration","History","Basic examples","Basic concepts","Module","Constructions","Special kinds of rings","Rings with extra structure","Some examples of the ubiquity of rings","Category-theoretic description","Generalization","Other ring-like objects"]}
{"upwards":["Ring (mathematics)","Definition"],"downwards":["Notes on the definition"]}
{"upwards":["Ring (mathematics)","Definition","Notes on the definition"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration"],"downwards":["Some properties","Example: Integers modulo 4","Example: 2-by-2 matrices"]}
{"upwards":["Ring (mathematics)","Illustration","Some properties"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration","Example: Integers modulo 4"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration","Example: 2-by-2 matrices"],"downwards":[]}
{"upwards":["Ring (mathematics)","History"],"downwards":["Dedekind","Hilbert","Fraenkel and Noether","Multiplicative identity and the term \"ring\""]}
{"upwards":["Ring (mathematics)","History","Dedekind"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Hilbert"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Fraenkel and Noether"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Multiplicative identity and the term \"ring\""],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples"],"downwards":["Commutative rings","Noncommutative rings","Non-rings"]}
{"upwards":["Ring (mathematics)","Basic examples","Commutative rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples","Noncommutative rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples","Non-rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts"],"downwards":["Products and powers","Elements in a ring","Subring","Ideal","Homomorphism","Quotient ring"]}
{"upwards":["Ring (mathematics)","Basic concepts","Products and powers"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Elements in a ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Subring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Ideal"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Homomorphism"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Quotient ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Module"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions"],"downwards":["Direct product","Polynomial ring","Matrix ring and endomorphism ring","Limits and colimits of rings","Localization","Completion","Rings with generators and relations"]}
{"upwards":["Ring (mathematics)","Constructions","Direct product"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Polynomial ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Matrix ring and endomorphism ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Limits and colimits of rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Localization"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Completion"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Rings with generators and relations"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings"],"downwards":["Domains","Division ring","Semisimple rings","Central simple algebra and Brauer group","Valuation ring"]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Domains"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Division ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings"],"downwards":["Examples","Properties"]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings","Examples"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings","Properties"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Central simple algebra and Brauer group"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Valuation ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Rings with extra structure"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings"],"downwards":["Cohomology ring of a topological space","Burnside ring of a group","Representation ring of a group ring","Function field of an irreducible algebraic variety","Face ring of a simplicial complex"]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Cohomology ring of a topological space"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Burnside ring of a group"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Representation ring of a group ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Function field of an irreducible algebraic variety"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Face ring of a simplicial complex"],"downwards":[]}
{"upwards":["Ring (mathematics)","Category-theoretic description"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization"],"downwards":["Rng","Nonassociative ring","Semiring"]}
{"upwards":["Ring (mathematics)","Generalization","Rng"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization","Nonassociative ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization","Semiring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects"],"downwards":["Ring object in a category","Ring scheme","Ring spectrum"]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring object in a category"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring scheme"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring spectrum"],"downwards":[]}
{"upwards":["Closure (mathematics)"],"downwards":["Definitions","In topology","Binary relations","Other examples","Closure operator","Notes","References"]}
{"upwards":["Closure (mathematics)","Definitions"],"downwards":["In algebraic structures"]}
{"upwards":["Closure (mathematics)","Definitions","In algebraic structures"],"downwards":[]}
{"upwards":["Closure (mathematics)","In topology"],"downwards":[]}
{"upwards":["Closure (mathematics)","Binary relations"],"downwards":[]}
{"upwards":["Closure (mathematics)","Other examples"],"downwards":[]}
{"upwards":["Closure (mathematics)","Closure operator"],"downwards":["Closure operator vs. closed sets"]}
{"upwards":["Closure (mathematics)","Closure operator","Closure operator vs. closed sets"],"downwards":[]}
{"upwards":["Closure (mathematics)","Notes"],"downwards":[]}
{"upwards":["Closure (mathematics)","References"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle"],"downwards":["Formula","Examples","A special case","Formula generalization","In probability","Other formulas","Applications","Diluted inclusion\u2013exclusion principle","Proof of main statement"]}
{"upwards":["Inclusion\u2013exclusion principle","Formula"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Examples"],"downwards":["Counting integers","Counting derangements"]}
{"upwards":["Inclusion\u2013exclusion principle","Examples","Counting integers"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Examples","Counting derangements"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","A special case"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Formula generalization"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","In probability"],"downwards":["Special case"]}
{"upwards":["Inclusion\u2013exclusion principle","In probability","Special case"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Other formulas"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications"],"downwards":["Counting derangements","Counting intersections","Graph coloring","Bipartite graph perfect matchings","Number of onto functions","Permutations with forbidden positions","Stirling numbers of the second kind","Rook polynomials","Euler's phi function"]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Counting derangements"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Counting intersections"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Graph coloring"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Bipartite graph perfect matchings"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Number of onto functions"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Permutations with forbidden positions"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Stirling numbers of the second kind"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Rook polynomials"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Euler's phi function"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Diluted inclusion\u2013exclusion principle"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Proof of main statement"],"downwards":["Algebraic proof"]}
{"upwards":["Inclusion\u2013exclusion principle","Proof of main statement","Algebraic proof"],"downwards":[]}
{"upwards":["Codomain"],"downwards":["Examples"]}
{"upwards":["Codomain","Examples"],"downwards":[]}
{"upwards":["Naive set theory"],"downwards":["Method","Sets, membership and equality","Specifying sets","Subsets","Universal sets and absolute complements","Unions, intersections, and relative complements","Ordered pairs and Cartesian products","Some important sets","Paradoxes in early set theory"]}
{"upwards":["Naive set theory","Method"],"downwards":["Paradoxes","Cantor's theory","Axiomatic theories","Consistency","Utility"]}
{"upwards":["Naive set theory","Method","Paradoxes"],"downwards":[]}
{"upwards":["Naive set theory","Method","Cantor's theory"],"downwards":[]}
{"upwards":["Naive set theory","Method","Axiomatic theories"],"downwards":[]}
{"upwards":["Naive set theory","Method","Consistency"],"downwards":[]}
{"upwards":["Naive set theory","Method","Utility"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality"],"downwards":["Note on consistency","Membership","Equality","Empty set"]}
{"upwards":["Naive set theory","Sets, membership and equality","Note on consistency"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Membership"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Equality"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Empty set"],"downwards":[]}
{"upwards":["Naive set theory","Specifying sets"],"downwards":[]}
{"upwards":["Naive set theory","Subsets"],"downwards":[]}
{"upwards":["Naive set theory","Universal sets and absolute complements"],"downwards":[]}
{"upwards":["Naive set theory","Unions, intersections, and relative complements"],"downwards":[]}
{"upwards":["Naive set theory","Ordered pairs and Cartesian products"],"downwards":[]}
{"upwards":["Naive set theory","Some important sets"],"downwards":[]}
{"upwards":["Naive set theory","Paradoxes in early set theory"],"downwards":[]}
{"upwards":["Irrational number"],"downwards":["History","Examples","Types","Decimal expansions","Irrational powers","Open questions","In constructive mathematics","Set of all irrationals"]}
{"upwards":["Irrational number","History"],"downwards":["Ancient Greece","India","Middle Ages","Modern period"]}
{"upwards":["Irrational number","History","Ancient Greece"],"downwards":[]}
{"upwards":["Irrational number","History","India"],"downwards":[]}
{"upwards":["Irrational number","History","Middle Ages"],"downwards":[]}
{"upwards":["Irrational number","History","Modern period"],"downwards":[]}
{"upwards":["Irrational number","Examples"],"downwards":["Square roots","General roots","Logarithms"]}
{"upwards":["Irrational number","Examples","Square roots"],"downwards":[]}
{"upwards":["Irrational number","Examples","General roots"],"downwards":[]}
{"upwards":["Irrational number","Examples","Logarithms"],"downwards":[]}
{"upwards":["Irrational number","Types"],"downwards":["Transcendental\/algebraic"]}
{"upwards":["Irrational number","Types","Transcendental\/algebraic"],"downwards":[]}
{"upwards":["Irrational number","Decimal expansions"],"downwards":[]}
{"upwards":["Irrational number","Irrational powers"],"downwards":[]}
{"upwards":["Irrational number","Open questions"],"downwards":[]}
{"upwards":["Irrational number","In constructive mathematics"],"downwards":[]}
{"upwards":["Irrational number","Set of all irrationals"],"downwards":[]}
{"upwards":["Transcendental number"],"downwards":["History","Properties","Numbers proven to be transcendental","Possible transcendental numbers","Sketch of a proof that e is transcendental"]}
{"upwards":["Transcendental number","History"],"downwards":[]}
{"upwards":["Transcendental number","Properties"],"downwards":[]}
{"upwards":["Transcendental number","Numbers proven to be transcendental"],"downwards":[]}
{"upwards":["Transcendental number","Possible transcendental numbers"],"downwards":[]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental"],"downwards":["The transcendence of \u03c0"]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental","The transcendence of \u03c0"],"downwards":[]}
{"upwards":["Georg Cantor"],"downwards":["Biography","Mathematical work","Philosophy, religion, literature and Cantor's mathematics","Cantor's ancestry","Biographies"]}
{"upwards":["Georg Cantor","Biography"],"downwards":["Youth and studies","Teacher and researcher","Later years and death"]}
{"upwards":["Georg Cantor","Biography","Youth and studies"],"downwards":[]}
{"upwards":["Georg Cantor","Biography","Teacher and researcher"],"downwards":[]}
{"upwards":["Georg Cantor","Biography","Later years and death"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work"],"downwards":["Number theory, trigonometric series and ordinals","Set theory"]}
{"upwards":["Georg Cantor","Mathematical work","Number theory, trigonometric series and ordinals"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory"],"downwards":["One-to-one correspondence","Continuum hypothesis","Absolute infinite, well-ordering theorem, and paradoxes"]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","One-to-one correspondence"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","Continuum hypothesis"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","Absolute infinite, well-ordering theorem, and paradoxes"],"downwards":[]}
{"upwards":["Georg Cantor","Philosophy, religion, literature and Cantor's mathematics"],"downwards":[]}
{"upwards":["Georg Cantor","Cantor's ancestry"],"downwards":[]}
{"upwards":["Georg Cantor","Biographies"],"downwards":[]}
{"upwards":["Pi"],"downwards":["Fundamentals","History","Modern quest for more digits","Role and characterizations in mathematics","Outside mathematics"]}
{"upwards":["Pi","Fundamentals"],"downwards":["Name","Definition","Irrationality and normality","Transcendence","Continued fractions","Approximate value and digits","Complex numbers and Euler's identity"]}
{"upwards":["Pi","Fundamentals","Name"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Definition"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Irrationality and normality"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Transcendence"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Continued fractions"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Approximate value and digits"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Complex numbers and Euler's identity"],"downwards":[]}
{"upwards":["Pi","History"],"downwards":["Antiquity","Polygon approximation era","Infinite series","Irrationality and transcendence","Adoption of the symbol \u03c0"]}
{"upwards":["Pi","History","Antiquity"],"downwards":[]}
{"upwards":["Pi","History","Polygon approximation era"],"downwards":[]}
{"upwards":["Pi","History","Infinite series"],"downwards":["Rate of convergence"]}
{"upwards":["Pi","History","Infinite series","Rate of convergence"],"downwards":[]}
{"upwards":["Pi","History","Irrationality and transcendence"],"downwards":[]}
{"upwards":["Pi","History","Adoption of the symbol \u03c0"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits"],"downwards":["Computer era and iterative algorithms","Motives for computing \u03c0","Rapidly convergent series","Monte Carlo methods","Spigot algorithms"]}
{"upwards":["Pi","Modern quest for more digits","Computer era and iterative algorithms"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Motives for computing \u03c0"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Rapidly convergent series"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Monte Carlo methods"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Spigot algorithms"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics"],"downwards":["Geometry and trigonometry","Units of angle","Eigenvalues","Inequalities","Fourier transform and Heisenberg uncertainty principle","Gaussian integrals","Topology","Cauchy's integral formula","Total curvature","The gamma function and Stirling's approximation","Number theory and Riemann zeta function","Fourier series","Modular forms and theta functions","Cauchy distribution and potential theory","In the Mandelbrot set"]}
{"upwards":["Pi","Role and characterizations in mathematics","Geometry and trigonometry"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Units of angle"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Eigenvalues"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Inequalities"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Fourier transform and Heisenberg uncertainty principle"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Gaussian integrals"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Topology"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Cauchy's integral formula"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Total curvature"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","The gamma function and Stirling's approximation"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Number theory and Riemann zeta function"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Fourier series"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Modular forms and theta functions"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Cauchy distribution and potential theory"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","In the Mandelbrot set"],"downwards":[]}
{"upwards":["Pi","Outside mathematics"],"downwards":["Describing physical phenomena","Memorizing digits","In popular culture","In computer culture"]}
{"upwards":["Pi","Outside mathematics","Describing physical phenomena"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","Memorizing digits"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","In popular culture"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","In computer culture"],"downwards":[]}
{"upwards":["Empty set"],"downwards":["Notation","Properties","In other areas of mathematics","Questioned existence"]}
{"upwards":["Empty set","Notation"],"downwards":[]}
{"upwards":["Empty set","Properties"],"downwards":["Operations on the empty set"]}
{"upwards":["Empty set","Properties","Operations on the empty set"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics"],"downwards":["Extended real numbers","Topology","Category theory","Set theory"]}
{"upwards":["Empty set","In other areas of mathematics","Extended real numbers"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Topology"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Category theory"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Set theory"],"downwards":[]}
{"upwards":["Empty set","Questioned existence"],"downwards":["Axiomatic set theory","Philosophical issues"]}
{"upwards":["Empty set","Questioned existence","Axiomatic set theory"],"downwards":[]}
{"upwards":["Empty set","Questioned existence","Philosophical issues"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory"],"downwards":["History","Axioms","Motivation via the cumulative hierarchy","Metamathematics","Criticisms"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","History"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms"],"downwards":["1. Axiom of extensionality","2. Axiom of regularity (also called the axiom of foundation)","3. Axiom schema of specification (or of separation, or of restricted comprehension)","4. Axiom of pairing","5. Axiom of union","6. Axiom schema of replacement","7. Axiom of infinity","8. Axiom of power set","9. Well-ordering axiom","Zorn's lemma"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","1. Axiom of extensionality"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","2. Axiom of regularity (also called the axiom of foundation)"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","3. Axiom schema of specification (or of separation, or of restricted comprehension)"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","4. Axiom of pairing"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","5. Axiom of union"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","6. Axiom schema of replacement"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","7. Axiom of infinity"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","8. Axiom of power set"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","9. Well-ordering axiom"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","Zorn's lemma"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Motivation via the cumulative hierarchy"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics"],"downwards":["Virtual classes","Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Consistency","Independence","Proposed additions"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Virtual classes"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Von Neumann\u2013Bernays\u2013G\u00f6del set theory"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Consistency"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Independence"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Proposed additions"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Criticisms"],"downwards":[]}
{"upwards":["Domain of a function"],"downwards":["Natural domain","Other uses","Set theoretical notions"]}
{"upwards":["Domain of a function","Natural domain"],"downwards":["Examples"]}
{"upwards":["Domain of a function","Natural domain","Examples"],"downwards":[]}
{"upwards":["Domain of a function","Other uses"],"downwards":[]}
{"upwards":["Domain of a function","Set theoretical notions"],"downwards":[]}
{"upwards":["Complement (set theory)"],"downwards":["Absolute complement","Relative complement","Complementary relation","LaTeX notation","In programming languages"]}
{"upwards":["Complement (set theory)","Absolute complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Absolute complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Relative complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Complementary relation"],"downwards":[]}
{"upwards":["Complement (set theory)","LaTeX notation"],"downwards":[]}
{"upwards":["Complement (set theory)","In programming languages"],"downwards":[]}
{"upwards":["Axiom of choice"],"downwards":["Statement","Usage","Examples","Criticism and acceptance","In constructive mathematics","Independence","Stronger axioms","Equivalents","Weaker forms","Stronger forms of the negation of AC","Statements consistent with the negation of AC","Axiom of choice in type theory","Quotations","Notes","References","External links"]}
{"upwards":["Axiom of choice","Statement"],"downwards":["Nomenclature ZF, AC, and ZFC","Variants","Restriction to finite sets"]}
{"upwards":["Axiom of choice","Statement","Nomenclature ZF, AC, and ZFC"],"downwards":[]}
{"upwards":["Axiom of choice","Statement","Variants"],"downwards":[]}
{"upwards":["Axiom of choice","Statement","Restriction to finite sets"],"downwards":[]}
{"upwards":["Axiom of choice","Usage"],"downwards":[]}
{"upwards":["Axiom of choice","Examples"],"downwards":[]}
{"upwards":["Axiom of choice","Criticism and acceptance"],"downwards":[]}
{"upwards":["Axiom of choice","In constructive mathematics"],"downwards":[]}
{"upwards":["Axiom of choice","Independence"],"downwards":[]}
{"upwards":["Axiom of choice","Stronger axioms"],"downwards":[]}
{"upwards":["Axiom of choice","Equivalents"],"downwards":["Category theory"]}
{"upwards":["Axiom of choice","Equivalents","Category theory"],"downwards":[]}
{"upwards":["Axiom of choice","Weaker forms"],"downwards":["Results requiring AC (or weaker forms) but weaker than it","Possibly equivalent implications of AC"]}
{"upwards":["Axiom of choice","Weaker forms","Results requiring AC (or weaker forms) but weaker than it"],"downwards":[]}
{"upwards":["Axiom of choice","Weaker forms","Possibly equivalent implications of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Stronger forms of the negation of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Statements consistent with the negation of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Axiom of choice in type theory"],"downwards":[]}
{"upwards":["Axiom of choice","Quotations"],"downwards":[]}
{"upwards":["Axiom of choice","Notes"],"downwards":[]}
{"upwards":["Axiom of choice","References"],"downwards":[]}
{"upwards":["Axiom of choice","External links"],"downwards":[]}
{"upwards":["Function (mathematics)"],"downwards":["Definition","Notation","Other terms","Specifying a function","Representing a function","General properties","Multivariate function","In calculus","Function space","Multi-valued functions","In the foundations of mathematics and set theory","In computer science"]}
{"upwards":["Function (mathematics)","Definition"],"downwards":["Total, univalent relation","Set exponentiation"]}
{"upwards":["Function (mathematics)","Definition","Total, univalent relation"],"downwards":[]}
{"upwards":["Function (mathematics)","Definition","Set exponentiation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation"],"downwards":["Functional notation","Arrow notation","Index notation","Dot notation","Specialized notations"]}
{"upwards":["Function (mathematics)","Notation","Functional notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Arrow notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Index notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Dot notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Specialized notations"],"downwards":[]}
{"upwards":["Function (mathematics)","Other terms"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function"],"downwards":["By listing function values","By a formula","Inverse and implicit functions","Using differential calculus","By recurrence"]}
{"upwards":["Function (mathematics)","Specifying a function","By listing function values"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","By a formula"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","Inverse and implicit functions"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","Using differential calculus"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","By recurrence"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function"],"downwards":["Graphs and plots","Tables","Bar chart"]}
{"upwards":["Function (mathematics)","Representing a function","Graphs and plots"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function","Tables"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function","Bar chart"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties"],"downwards":["Standard functions","Function composition","Image and preimage","Injective, surjective and bijective functions","Restriction and extension"]}
{"upwards":["Function (mathematics)","General properties","Standard functions"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Function composition"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Image and preimage"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Injective, surjective and bijective functions"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Restriction and extension"],"downwards":[]}
{"upwards":["Function (mathematics)","Multivariate function"],"downwards":[]}
{"upwards":["Function (mathematics)","In calculus"],"downwards":["Real function","Vector-valued function"]}
{"upwards":["Function (mathematics)","In calculus","Real function"],"downwards":[]}
{"upwards":["Function (mathematics)","In calculus","Vector-valued function"],"downwards":[]}
{"upwards":["Function (mathematics)","Function space"],"downwards":[]}
{"upwards":["Function (mathematics)","Multi-valued functions"],"downwards":[]}
{"upwards":["Function (mathematics)","In the foundations of mathematics and set theory"],"downwards":[]}
{"upwards":["Function (mathematics)","In computer science"],"downwards":[]}
{"upwards":["Sequence"],"downwards":["Examples and notation","Formal definition and basic properties","Limits and convergence","Series","Use in other fields of mathematics"]}
{"upwards":["Sequence","Examples and notation"],"downwards":["Examples","Indexing","Defining a sequence by recursion"]}
{"upwards":["Sequence","Examples and notation","Examples"],"downwards":[]}
{"upwards":["Sequence","Examples and notation","Indexing"],"downwards":[]}
{"upwards":["Sequence","Examples and notation","Defining a sequence by recursion"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties"],"downwards":["Definition","Finite and infinite","Increasing and decreasing","Bounded","Subsequences","Other types of sequences"]}
{"upwards":["Sequence","Formal definition and basic properties","Definition"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Finite and infinite"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Increasing and decreasing"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Bounded"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Subsequences"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Other types of sequences"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence"],"downwards":["Formal definition of convergence","Applications and important results","Cauchy sequences","Infinite limits"]}
{"upwards":["Sequence","Limits and convergence","Formal definition of convergence"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Applications and important results"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Cauchy sequences"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Infinite limits"],"downwards":[]}
{"upwards":["Sequence","Series"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics"],"downwards":["Topology","Analysis","Linear algebra","Abstract algebra","Set theory","Computing","Streams"]}
{"upwards":["Sequence","Use in other fields of mathematics","Topology"],"downwards":["Product topology"]}
{"upwards":["Sequence","Use in other fields of mathematics","Topology","Product topology"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Analysis"],"downwards":["Sequence spaces"]}
{"upwards":["Sequence","Use in other fields of mathematics","Analysis","Sequence spaces"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Linear algebra"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra"],"downwards":["Free monoid","Exact sequences","Spectral sequences"]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Free monoid"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Exact sequences"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Spectral sequences"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Set theory"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Computing"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Streams"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems"],"downwards":["Formal systems: completeness, consistency, and effective axiomatization","First incompleteness theorem","Second incompleteness theorem","Examples of undecidable statements","Relationship with computability","Proof sketch for the first theorem","Proof sketch for the second theorem","Discussion and implications","History"]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization"],"downwards":["Effective axiomatization","Completeness","Consistency","Systems which contain arithmetic","Conflicting goals"]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Effective axiomatization"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Completeness"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Consistency"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Systems which contain arithmetic"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Conflicting goals"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem"],"downwards":["Syntactic form of the G\u00f6del sentence","Truth of the G\u00f6del sentence","Relationship with the liar paradox","Extensions of G\u00f6del's original result"]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Syntactic form of the G\u00f6del sentence"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Truth of the G\u00f6del sentence"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Relationship with the liar paradox"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Extensions of G\u00f6del's original result"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem"],"downwards":["Expressing consistency","The Hilbert\u2013Bernays conditions","Implications for consistency proofs"]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","Expressing consistency"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","The Hilbert\u2013Bernays conditions"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","Implications for consistency proofs"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Examples of undecidable statements"],"downwards":["Undecidable statements provable in larger systems"]}
{"upwards":["G\u00f6del's incompleteness theorems","Examples of undecidable statements","Undecidable statements provable in larger systems"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Relationship with computability"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem"],"downwards":["Arithmetization of syntax","Construction of a statement about \"provability\"","Diagonalization","Proof via Berry's paradox","Computer verified proofs"]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Arithmetization of syntax"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Construction of a statement about \"provability\""],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Diagonalization"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Proof via Berry's paradox"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Computer verified proofs"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the second theorem"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications"],"downwards":["Consequences for logicism and Hilbert's second problem","Minds and machines","Paraconsistent logic","Appeals to the incompleteness theorems in other fields"]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Consequences for logicism and Hilbert's second problem"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Minds and machines"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Paraconsistent logic"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Appeals to the incompleteness theorems in other fields"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History"],"downwards":["Announcement","Generalization and acceptance","Criticisms"]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Announcement"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Generalization and acceptance"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms"],"downwards":["Finsler","Zermelo","Wittgenstein"]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Finsler"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Zermelo"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Wittgenstein"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Category of sets"],"downwards":["Properties of the category of sets","Foundations for the category of sets"]}
{"upwards":["Category of sets","Properties of the category of sets"],"downwards":[]}
{"upwards":["Category of sets","Foundations for the category of sets"],"downwards":[]}
{"upwards":["Dense set"],"downwards":["Definition","Examples","Properties","Related notions"]}
{"upwards":["Dense set","Definition"],"downwards":["Density in metric spaces"]}
{"upwards":["Dense set","Definition","Density in metric spaces"],"downwards":[]}
{"upwards":["Dense set","Examples"],"downwards":[]}
{"upwards":["Dense set","Properties"],"downwards":[]}
{"upwards":["Dense set","Related notions"],"downwards":[]}
{"upwards":["Internal set"],"downwards":["Internal sets in the ultrapower construction","Internal subsets of the reals"]}
{"upwards":["Internal set","Internal sets in the ultrapower construction"],"downwards":[]}
{"upwards":["Internal set","Internal subsets of the reals"],"downwards":[]}
{"upwards":["Fuzzy set"],"downwards":["Definition","Fuzzy logic","Fuzzy number and only number","Fuzzy categories","Fuzzy relation equation","Entropy","Extensions"]}
{"upwards":["Fuzzy set","Definition"],"downwards":["Crisp sets related to a fuzzy set","Other definitions","Fuzzy set operations","Disjoint fuzzy sets","Scalar cardinality","Distance and similarity","L-fuzzy sets","Neutrosophic fuzzy sets","Pythagorean fuzzy sets"]}
{"upwards":["Fuzzy set","Definition","Crisp sets related to a fuzzy set"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Other definitions"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Fuzzy set operations"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Disjoint fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Scalar cardinality"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Distance and similarity"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","L-fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Neutrosophic fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Pythagorean fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy logic"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy number and only number"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy categories"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy relation equation"],"downwards":[]}
{"upwards":["Fuzzy set","Entropy"],"downwards":[]}
{"upwards":["Fuzzy set","Extensions"],"downwards":[]}
{"upwards":["Family of sets"],"downwards":["Examples","Properties","Related concepts","Special types of set families"]}
{"upwards":["Family of sets","Examples"],"downwards":[]}
{"upwards":["Family of sets","Properties"],"downwards":[]}
{"upwards":["Family of sets","Related concepts"],"downwards":["Covers and topologies"]}
{"upwards":["Family of sets","Related concepts","Covers and topologies"],"downwards":[]}
{"upwards":["Family of sets","Special types of set families"],"downwards":[]}
{"upwards":["Alternative set theory"],"downwards":["Vop\u011bnka's Alternative Set Theory","Other alternative set theories"]}
{"upwards":["Alternative set theory","Vop\u011bnka's Alternative Set Theory"],"downwards":[]}
{"upwards":["Alternative set theory","Other alternative set theories"],"downwards":[]}
{"upwards":["Mereology"],"downwards":["History","Axioms and primitive notions","Various systems","Set theory","Mathematics","Natural language","Metaphysics","Important surveys"]}
{"upwards":["Mereology","History"],"downwards":[]}
{"upwards":["Mereology","Axioms and primitive notions"],"downwards":[]}
{"upwards":["Mereology","Various systems"],"downwards":[]}
{"upwards":["Mereology","Set theory"],"downwards":[]}
{"upwards":["Mereology","Mathematics"],"downwards":[]}
{"upwards":["Mereology","Natural language"],"downwards":[]}
{"upwards":["Mereology","Metaphysics"],"downwards":["Mereological constitution","Mereological composition"]}
{"upwards":["Mereology","Metaphysics","Mereological constitution"],"downwards":[]}
{"upwards":["Mereology","Metaphysics","Mereological composition"],"downwards":["Fundamentality","Special composition question (SCQ)"]}
{"upwards":["Mereology","Metaphysics","Mereological composition","Fundamentality"],"downwards":[]}
{"upwards":["Mereology","Metaphysics","Mereological composition","Special composition question (SCQ)"],"downwards":[]}
{"upwards":["Mereology","Important surveys"],"downwards":[]}
{"upwards":["Multiset"],"downwards":["History","Examples","Definition","Basic properties and operations","Counting multisets","Applications","Generalizations"]}
{"upwards":["Multiset","History"],"downwards":[]}
{"upwards":["Multiset","Examples"],"downwards":[]}
{"upwards":["Multiset","Definition"],"downwards":[]}
{"upwards":["Multiset","Basic properties and operations"],"downwards":[]}
{"upwards":["Multiset","Counting multisets"],"downwards":["Recurrence relation","Generating series","Generalization and connection to the negative binomial series"]}
{"upwards":["Multiset","Counting multisets","Recurrence relation"],"downwards":[]}
{"upwards":["Multiset","Counting multisets","Generating series"],"downwards":[]}
{"upwards":["Multiset","Counting multisets","Generalization and connection to the negative binomial series"],"downwards":[]}
{"upwards":["Multiset","Applications"],"downwards":[]}
{"upwards":["Multiset","Generalizations"],"downwards":[]}
{"upwards":["Bertrand Russell"],"downwards":["Biography","Views","Selected works"]}
{"upwards":["Bertrand Russell","Biography"],"downwards":["Early life and background","Childhood and adolescence","University and first marriage","Early career","First World War","G. H. Hardy on the Trinity controversy","Between the wars","Second World War","Later life","Political causes","Final years, death and legacy","Titles and honours from birth"]}
{"upwards":["Bertrand Russell","Biography","Early life and background"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Childhood and adolescence"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","University and first marriage"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Early career"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","First World War"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","G. H. Hardy on the Trinity controversy"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Between the wars"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Second World War"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Later life"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Political causes"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Final years, death and legacy"],"downwards":["Marriages and issue"]}
{"upwards":["Bertrand Russell","Biography","Final years, death and legacy","Marriages and issue"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Titles and honours from birth"],"downwards":[]}
{"upwards":["Bertrand Russell","Views"],"downwards":["Philosophy","Religion","Society","Freedom of opinion and expression","Education"]}
{"upwards":["Bertrand Russell","Views","Philosophy"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Religion"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Society"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Freedom of opinion and expression"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Education"],"downwards":[]}
{"upwards":["Bertrand Russell","Selected works"],"downwards":[]}
{"upwards":["Crelle's Journal"],"downwards":["History","Past editors","References","External links"]}
{"upwards":["Crelle's Journal","History"],"downwards":[]}
{"upwards":["Crelle's Journal","Past editors"],"downwards":[]}
{"upwards":["Crelle's Journal","References"],"downwards":[]}
{"upwards":["Crelle's Journal","External links"],"downwards":[]}
{"upwards":["Digital object identifier"],"downwards":["Nomenclature and syntax","Content","Features and benefits","Comparison with other identifier schemes","Resolution","IDF organizational structure","Standardization"]}
{"upwards":["Digital object identifier","Nomenclature and syntax"],"downwards":["Display"]}
{"upwards":["Digital object identifier","Nomenclature and syntax","Display"],"downwards":[]}
{"upwards":["Digital object identifier","Content"],"downwards":[]}
{"upwards":["Digital object identifier","Features and benefits"],"downwards":[]}
{"upwards":["Digital object identifier","Comparison with other identifier schemes"],"downwards":[]}
{"upwards":["Digital object identifier","Resolution"],"downwards":[]}
{"upwards":["Digital object identifier","IDF organizational structure"],"downwards":[]}
{"upwards":["Digital object identifier","Standardization"],"downwards":[]}
{"upwards":["PubMed Central"],"downwards":["Adoption","Technology","Reception","PMCID"]}
{"upwards":["PubMed Central","Adoption"],"downwards":[]}
{"upwards":["PubMed Central","Technology"],"downwards":[]}
{"upwards":["PubMed Central","Reception"],"downwards":[]}
{"upwards":["PubMed Central","PMCID"],"downwards":[]}
{"upwards":["PubMed"],"downwards":["Content","Characteristics"]}
{"upwards":["PubMed","Content"],"downwards":[]}
{"upwards":["PubMed","Characteristics"],"downwards":["Website design","Search","Journal article parameters"]}
{"upwards":["PubMed","Characteristics","Website design"],"downwards":["PubMed for handhelds\/mobiles"]}
{"upwards":["PubMed","Characteristics","Website design","PubMed for handhelds\/mobiles"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Search"],"downwards":["Standard search","Comprehensive search"]}
{"upwards":["PubMed","Characteristics","Search","Standard search"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Search","Comprehensive search"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Journal article parameters"],"downwards":["Publication Type: Clinical queries\/systematic reviews","Secondary ID"]}
{"upwards":["PubMed","Characteristics","Journal article parameters","Publication Type: Clinical queries\/systematic reviews"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Journal article parameters","Secondary ID"],"downwards":[]}
{"upwards":["Joseph Dauben"],"downwards":["Positions","Publications","Articles, reviews, and essays"]}
{"upwards":["Joseph Dauben","Positions"],"downwards":[]}
{"upwards":["Joseph Dauben","Publications"],"downwards":[]}
{"upwards":["Joseph Dauben","Articles, reviews, and essays"],"downwards":[]}
{"upwards":["Harvard University Press"],"downwards":["Related publishers, imprints, and series","Awards","Publications"]}
{"upwards":["Harvard University Press","Related publishers, imprints, and series"],"downwards":[]}
{"upwards":["Harvard University Press","Awards"],"downwards":[]}
{"upwards":["Harvard University Press","Publications"],"downwards":[]}
{"upwards":["Paul Halmos"],"downwards":["Early life and education","Career","Accomplishments","Books by Halmos"]}
{"upwards":["Paul Halmos","Early life and education"],"downwards":[]}
{"upwards":["Paul Halmos","Career"],"downwards":[]}
{"upwards":["Paul Halmos","Accomplishments"],"downwards":[]}
{"upwards":["Paul Halmos","Books by Halmos"],"downwards":[]}
{"upwards":["Dover Publications"],"downwards":["Classic reprints","History","Publications","References","External links"]}
{"upwards":["Dover Publications","Classic reprints"],"downwards":[]}
{"upwards":["Dover Publications","History"],"downwards":[]}
{"upwards":["Dover Publications","Publications"],"downwards":[]}
{"upwards":["Dover Publications","References"],"downwards":[]}
{"upwards":["Dover Publications","External links"],"downwards":[]}
{"upwards":["Cambridge University Press"],"downwards":["History","Print and typographic heritage","Current publications","Open access","Nobel prize winners published by Cambridge University Press","Organisational governance and operational structure","Cambridge University Press partnerships and acquisitions","Digital developments","Controversies","Corporate social responsibility","References","External links"]}
{"upwards":["Cambridge University Press","History"],"downwards":[]}
{"upwards":["Cambridge University Press","Print and typographic heritage"],"downwards":["People","Publications"]}
{"upwards":["Cambridge University Press","Print and typographic heritage","People"],"downwards":[]}
{"upwards":["Cambridge University Press","Print and typographic heritage","Publications"],"downwards":[]}
{"upwards":["Cambridge University Press","Current publications"],"downwards":[]}
{"upwards":["Cambridge University Press","Open access"],"downwards":[]}
{"upwards":["Cambridge University Press","Nobel prize winners published by Cambridge University Press"],"downwards":[]}
{"upwards":["Cambridge University Press","Organisational governance and operational structure"],"downwards":["Relationship with the University of Cambridge","Operational structure"]}
{"upwards":["Cambridge University Press","Organisational governance and operational structure","Relationship with the University of Cambridge"],"downwards":[]}
{"upwards":["Cambridge University Press","Organisational governance and operational structure","Operational structure"],"downwards":[]}
{"upwards":["Cambridge University Press","Cambridge University Press partnerships and acquisitions"],"downwards":[]}
{"upwards":["Cambridge University Press","Digital developments"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies"],"downwards":["Tax exemption controversy","Alms for Jihad","Cambridge University Press v. Patton","The China Quarterly","The Cambridge Handbook of Privatization"]}
{"upwards":["Cambridge University Press","Controversies","Tax exemption controversy"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","Alms for Jihad"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","Cambridge University Press v. Patton"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","The China Quarterly"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","The Cambridge Handbook of Privatization"],"downwards":[]}
{"upwards":["Cambridge University Press","Corporate social responsibility"],"downwards":["Community","Environment"]}
{"upwards":["Cambridge University Press","Corporate social responsibility","Community"],"downwards":[]}
{"upwards":["Cambridge University Press","Corporate social responsibility","Environment"],"downwards":[]}
{"upwards":["Cambridge University Press","References"],"downwards":["Citations","Sources"]}
{"upwards":["Cambridge University Press","References","Citations"],"downwards":[]}
{"upwards":["Cambridge University Press","References","Sources"],"downwards":[]}
{"upwards":["Cambridge University Press","External links"],"downwards":[]}
{"upwards":["Axiom"],"downwards":["Etymology","Historical development","Mathematical logic"]}
{"upwards":["Axiom","Etymology"],"downwards":[]}
{"upwards":["Axiom","Historical development"],"downwards":["Early Greeks","Modern development","Other sciences"]}
{"upwards":["Axiom","Historical development","Early Greeks"],"downwards":[]}
{"upwards":["Axiom","Historical development","Modern development"],"downwards":[]}
{"upwards":["Axiom","Historical development","Other sciences"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic"],"downwards":["Logical axioms","Non-logical axioms","Role in mathematical logic","Further discussion"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic"],"downwards":["Deductive systems and completeness"]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic","Deductive systems and completeness"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Further discussion"],"downwards":[]}
{"upwards":["List of axioms"],"downwards":["ZF (the Zermelo\u2013Fraenkel axioms without the axiom of choice)","Axiom of choice","Other axioms of mathematical logic","Geometry","Other axioms"]}
{"upwards":["List of axioms","ZF (the Zermelo\u2013Fraenkel axioms without the axiom of choice)"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice"],"downwards":["Equivalents of AC","Stronger than AC","Weaker than AC","Alternates incompatible with AC"]}
{"upwards":["List of axioms","Axiom of choice","Equivalents of AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Stronger than AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Weaker than AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Alternates incompatible with AC"],"downwards":[]}
{"upwards":["List of axioms","Other axioms of mathematical logic"],"downwards":[]}
{"upwards":["List of axioms","Geometry"],"downwards":[]}
{"upwards":["List of axioms","Other axioms"],"downwards":[]}
{"upwards":["Formal proof"],"downwards":["Background"]}
{"upwards":["Formal proof","Background"],"downwards":["Formal language","Formal grammar","Formal systems","Interpretations"]}
{"upwards":["Formal proof","Background","Formal language"],"downwards":[]}
{"upwards":["Formal proof","Background","Formal grammar"],"downwards":[]}
{"upwards":["Formal proof","Background","Formal systems"],"downwards":[]}
{"upwards":["Formal proof","Background","Interpretations"],"downwards":[]}
{"upwards":["Semantics of logic"],"downwards":["Overview"]}
{"upwards":["Semantics of logic","Overview"],"downwards":[]}
{"upwards":["Information theory"],"downwards":["Overview","Historical background","Quantities of information","Coding theory","Applications to other fields"]}
{"upwards":["Information theory","Overview"],"downwards":[]}
{"upwards":["Information theory","Historical background"],"downwards":[]}
{"upwards":["Information theory","Quantities of information"],"downwards":["Entropy of an information source","Joint entropy","Conditional entropy (equivocation)","Mutual information (transinformation)","Kullback\u2013Leibler divergence (information gain)","Directed Information","Other quantities"]}
{"upwards":["Information theory","Quantities of information","Entropy of an information source"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Joint entropy"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Conditional entropy (equivocation)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Mutual information (transinformation)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Kullback\u2013Leibler divergence (information gain)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Directed Information"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Other quantities"],"downwards":[]}
{"upwards":["Information theory","Coding theory"],"downwards":["Source theory","Channel capacity"]}
{"upwards":["Information theory","Coding theory","Source theory"],"downwards":["Rate"]}
{"upwards":["Information theory","Coding theory","Source theory","Rate"],"downwards":[]}
{"upwards":["Information theory","Coding theory","Channel capacity"],"downwards":["Capacity of particular channel models","Channels with memory and directed information"]}
{"upwards":["Information theory","Coding theory","Channel capacity","Capacity of particular channel models"],"downwards":[]}
{"upwards":["Information theory","Coding theory","Channel capacity","Channels with memory and directed information"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields"],"downwards":["Intelligence uses and secrecy applications","Pseudorandom number generation","Seismic exploration","Semiotics","Integrated process organization of neural information","Miscellaneous applications"]}
{"upwards":["Information theory","Applications to other fields","Intelligence uses and secrecy applications"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Pseudorandom number generation"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Seismic exploration"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Semiotics"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Integrated process organization of neural information"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Miscellaneous applications"],"downwards":[]}
{"upwards":["Philosophy"],"downwards":["Definitions","Historical overview","Branches of philosophy","Methods of philosophy","Applied and professional philosophy"]}
{"upwards":["Philosophy","Definitions"],"downwards":[]}
{"upwards":["Philosophy","Historical overview"],"downwards":["Western philosophy","Middle Eastern philosophy","Eastern philosophy","African philosophy","Indigenous American philosophy"]}
{"upwards":["Philosophy","Historical overview","Western philosophy"],"downwards":["Ancient era","Medieval era","Modern era"]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Ancient era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Medieval era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Modern era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy"],"downwards":["Pre-Islamic philosophy","Islamic philosophy"]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy","Pre-Islamic philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy","Islamic philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy"],"downwards":["Indian philosophy","Buddhist philosophy","East Asian philosophy"]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","Indian philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","Buddhist philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","East Asian philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","African philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Indigenous American philosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy"],"downwards":["Aesthetics","Ethics","Epistemology","Metaphysics","Logic","Mind and language","Philosophy of science","Political philosophy","Philosophy of religion","Metaphilosophy","Other subdivisions"]}
{"upwards":["Philosophy","Branches of philosophy","Aesthetics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Ethics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Epistemology"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Metaphysics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Logic"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Mind and language"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Philosophy of science"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Political philosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Philosophy of religion"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Metaphilosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Other subdivisions"],"downwards":[]}
{"upwards":["Philosophy","Methods of philosophy"],"downwards":["Philosophical progress"]}
{"upwards":["Philosophy","Methods of philosophy","Philosophical progress"],"downwards":[]}
{"upwards":["Philosophy","Applied and professional philosophy"],"downwards":["Women in philosophy"]}
{"upwards":["Philosophy","Applied and professional philosophy","Women in philosophy"],"downwards":[]}
{"upwards":["Lemma (mathematics)"],"downwards":["Etymology","Comparison with theorem","Well-known lemmas"]}
{"upwards":["Lemma (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Lemma (mathematics)","Comparison with theorem"],"downwards":[]}
{"upwards":["Lemma (mathematics)","Well-known lemmas"],"downwards":[]}
{"upwards":["Mathematical logic"],"downwards":["Subfields and scope","History","Formal logical systems","Set theory","Model theory","Recursion theory","Proof theory and constructive mathematics","Applications","Connections with computer science","Foundations of mathematics"]}
{"upwards":["Mathematical logic","Subfields and scope"],"downwards":[]}
{"upwards":["Mathematical logic","History"],"downwards":["Early history","19th century","20th century"]}
{"upwards":["Mathematical logic","History","Early history"],"downwards":[]}
{"upwards":["Mathematical logic","History","19th century"],"downwards":["Foundational theories"]}
{"upwards":["Mathematical logic","History","19th century","Foundational theories"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century"],"downwards":["Set theory and paradoxes","Symbolic logic","Beginnings of the other branches"]}
{"upwards":["Mathematical logic","History","20th century","Set theory and paradoxes"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Symbolic logic"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Beginnings of the other branches"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems"],"downwards":["First-order logic","Other classical logics","Nonclassical and modal logic","Algebraic logic"]}
{"upwards":["Mathematical logic","Formal logical systems","First-order logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Other classical logics"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Nonclassical and modal logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Algebraic logic"],"downwards":[]}
{"upwards":["Mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Mathematical logic","Model theory"],"downwards":[]}
{"upwards":["Mathematical logic","Recursion theory"],"downwards":["Algorithmically unsolvable problems"]}
{"upwards":["Mathematical logic","Recursion theory","Algorithmically unsolvable problems"],"downwards":[]}
{"upwards":["Mathematical logic","Proof theory and constructive mathematics"],"downwards":[]}
{"upwards":["Mathematical logic","Applications"],"downwards":[]}
{"upwards":["Mathematical logic","Connections with computer science"],"downwards":[]}
{"upwards":["Mathematical logic","Foundations of mathematics"],"downwards":[]}
{"upwards":["Logical consequence"],"downwards":["Formal accounts","A priori property of logical consequence","Proofs and models","Modal accounts"]}
{"upwards":["Logical consequence","Formal accounts"],"downwards":[]}
{"upwards":["Logical consequence","A priori property of logical consequence"],"downwards":[]}
{"upwards":["Logical consequence","Proofs and models"],"downwards":["Syntactic consequence","Semantic consequence"]}
{"upwards":["Logical consequence","Proofs and models","Syntactic consequence"],"downwards":[]}
{"upwards":["Logical consequence","Proofs and models","Semantic consequence"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts"],"downwards":["Modal-formal accounts","Warrant-based accounts","Non-monotonic logical consequence"]}
{"upwards":["Logical consequence","Modal accounts","Modal-formal accounts"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts","Warrant-based accounts"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts","Non-monotonic logical consequence"],"downwards":[]}
{"upwards":["Structure (mathematical logic)"],"downwards":["Definition","Induced substructures and closed subsets","Homomorphisms and embeddings","Structures and first-order logic","Many-sorted structures","Other generalizations"]}
{"upwards":["Structure (mathematical logic)","Definition"],"downwards":["Domain","Signature","Interpretation function","Examples"]}
{"upwards":["Structure (mathematical logic)","Definition","Domain"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Signature"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Interpretation function"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets"],"downwards":["Examples"]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings"],"downwards":["Homomorphisms","Embeddings","Example","Homomorphism problem"]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphisms"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Embeddings"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Example"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphism problem"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic"],"downwards":["Satisfaction relation","Definable relations"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Satisfaction relation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations"],"downwards":["Definability with parameters","Implicit definability"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Definability with parameters"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Implicit definability"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Many-sorted structures"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations"],"downwards":["Partial algebras","Structures for typed languages","Higher-order languages","Structures that are proper classes"]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Partial algebras"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures for typed languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Higher-order languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures that are proper classes"],"downwards":[]}
{"upwards":["Theorem"],"downwards":["Theoremhood and truth","Epistemological considerations","Informal account of theorems","Relation with scientific theories","Terminology","Layout","Lore","Theorems in logic"]}
{"upwards":["Theorem","Theoremhood and truth"],"downwards":[]}
{"upwards":["Theorem","Epistemological considerations"],"downwards":[]}
{"upwards":["Theorem","Informal account of theorems"],"downwards":[]}
{"upwards":["Theorem","Relation with scientific theories"],"downwards":[]}
{"upwards":["Theorem","Terminology"],"downwards":[]}
{"upwards":["Theorem","Layout"],"downwards":[]}
{"upwards":["Theorem","Lore"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic"],"downwards":["Syntax and semantics","Interpretation of a formal theorem","Theorems and theories"]}
{"upwards":["Theorem","Theorems in logic","Syntax and semantics"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Interpretation of a formal theorem"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Theorems and theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)"],"downwards":["General theories (as expressed in formal language)","First-order theories","Examples"]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)"],"downwards":["Subtheories and extensions","Deductive theories","Consistency and completeness","Interpretation of a theory","Theories associated with a structure"]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Subtheories and extensions"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Deductive theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Consistency and completeness"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Interpretation of a theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Theories associated with a structure"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories"],"downwards":["Derivation in a first-order theory","Syntactic consequence in a first-order theory","Interpretation of a first-order theory","First-order theories with identity","Topics related to first-order theories"]}
{"upwards":["Theory (mathematical logic)","First-order theories","Derivation in a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Syntactic consequence in a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Interpretation of a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","First-order theories with identity"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Topics related to first-order theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","Examples"],"downwards":[]}
{"upwards":["Type theory"],"downwards":["History","Introduction","Differences from set theory","Technical details","Interpretations","List of type theories","Applications"]}
{"upwards":["Type theory","History"],"downwards":[]}
{"upwards":["Type theory","Introduction"],"downwards":["Basics","Functions","Common \"by rule\" types and terms"]}
{"upwards":["Type theory","Introduction","Basics"],"downwards":["Terms and types","Computation"]}
{"upwards":["Type theory","Introduction","Basics","Terms and types"],"downwards":[]}
{"upwards":["Type theory","Introduction","Basics","Computation"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions"],"downwards":["Lambda terms","Dependent typing","Universes"]}
{"upwards":["Type theory","Introduction","Functions","Lambda terms"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions","Dependent typing"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions","Universes"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms"],"downwards":["Empty type","Unit type","Boolean type","Product type","Sum type","Natural numbers","Identity type","Inductive types"]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Empty type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Unit type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Boolean type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Product type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Sum type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Natural numbers"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Identity type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Inductive types"],"downwards":[]}
{"upwards":["Type theory","Differences from set theory"],"downwards":[]}
{"upwards":["Type theory","Technical details"],"downwards":["Terms","Judgements","Rules","Properties of type theories","Axioms","Decision problems","Research areas"]}
{"upwards":["Type theory","Technical details","Terms"],"downwards":[]}
{"upwards":["Type theory","Technical details","Judgements"],"downwards":[]}
{"upwards":["Type theory","Technical details","Rules"],"downwards":[]}
{"upwards":["Type theory","Technical details","Properties of type theories"],"downwards":[]}
{"upwards":["Type theory","Technical details","Axioms"],"downwards":[]}
{"upwards":["Type theory","Technical details","Decision problems"],"downwards":["Type inhabitation","Type inference"]}
{"upwards":["Type theory","Technical details","Decision problems","Type inhabitation"],"downwards":[]}
{"upwards":["Type theory","Technical details","Decision problems","Type inference"],"downwards":[]}
{"upwards":["Type theory","Technical details","Research areas"],"downwards":[]}
{"upwards":["Type theory","Interpretations"],"downwards":["Types are propositions; terms are proofs","Curry-Howard correspondence","Intuitionistic logic","Constructive mathematics","Category theory"]}
{"upwards":["Type theory","Interpretations","Types are propositions; terms are proofs"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Curry-Howard correspondence"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Intuitionistic logic"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Constructive mathematics"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Category theory"],"downwards":[]}
{"upwards":["Type theory","List of type theories"],"downwards":["Major","Minor","Active research"]}
{"upwards":["Type theory","List of type theories","Major"],"downwards":[]}
{"upwards":["Type theory","List of type theories","Minor"],"downwards":[]}
{"upwards":["Type theory","List of type theories","Active research"],"downwards":[]}
{"upwards":["Type theory","Applications"],"downwards":["Mathematical foundations","Proof assistants","Programming languages","Linguistics","Social sciences"]}
{"upwards":["Type theory","Applications","Mathematical foundations"],"downwards":[]}
{"upwards":["Type theory","Applications","Proof assistants"],"downwards":[]}
{"upwards":["Type theory","Applications","Programming languages"],"downwards":[]}
{"upwards":["Type theory","Applications","Linguistics"],"downwards":[]}
{"upwards":["Type theory","Applications","Social sciences"],"downwards":[]}
{"upwards":["Paradoxes of set theory"],"downwards":["Basics","Paradoxes of the infinite sets","Paradoxes of the Supertask","Paradoxes of proof and definability"]}
{"upwards":["Paradoxes of set theory","Basics"],"downwards":["Cardinal numbers","Ordinal numbers","Power sets"]}
{"upwards":["Paradoxes of set theory","Basics","Cardinal numbers"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Basics","Ordinal numbers"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Basics","Power sets"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets"],"downwards":["Paradoxes of enumeration","Je le vois, mais je ne crois pas","Paradoxes of well-ordering"]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Paradoxes of enumeration"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Je le vois, mais je ne crois pas"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Paradoxes of well-ordering"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask"],"downwards":["The diary of Tristram Shandy","The Ross-Littlewood paradox"]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask","The diary of Tristram Shandy"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask","The Ross-Littlewood paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability"],"downwards":["Early paradoxes: the set of all sets","Paradoxes by change of language","Paradox of L\u00f6wenheim and Skolem"]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Early paradoxes: the set of all sets"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language"],"downwards":["K\u00f6nig's paradox","Richard's paradox"]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language","K\u00f6nig's paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language","Richard's paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradox of L\u00f6wenheim and Skolem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem"],"downwards":["Preliminaries","Statement","Consequences","Relationship to the incompleteness theorems","Relationship to the compactness theorem","Completeness in other logics","Proofs"]}
{"upwards":["G\u00f6del's completeness theorem","Preliminaries"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement"],"downwards":["G\u00f6del's original formulation","More general form","Model existence theorem","As a theorem of arithmetic"]}
{"upwards":["G\u00f6del's completeness theorem","Statement","G\u00f6del's original formulation"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","More general form"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","Model existence theorem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","As a theorem of arithmetic"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Consequences"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Relationship to the incompleteness theorems"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Relationship to the compactness theorem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Completeness in other logics"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Proofs"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem"],"downwards":["History","Statement","General form","Discussion"]}
{"upwards":["Tarski's undefinability theorem","History"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","Statement"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","General form"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","Discussion"],"downwards":[]}
{"upwards":["JSTOR"],"downwards":["History","Content","Access","Usage"]}
{"upwards":["JSTOR","History"],"downwards":[]}
{"upwards":["JSTOR","Content"],"downwards":[]}
{"upwards":["JSTOR","Access"],"downwards":["Aaron Swartz incident","Limitations","Increasing public access"]}
{"upwards":["JSTOR","Access","Aaron Swartz incident"],"downwards":[]}
{"upwards":["JSTOR","Access","Limitations"],"downwards":[]}
{"upwards":["JSTOR","Access","Increasing public access"],"downwards":[]}
{"upwards":["JSTOR","Usage"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox"],"downwards":["Banach and Tarski publication","Formal treatment","Connection with earlier work and the role of the axiom of choice","A sketch of the proof","Obtaining infinitely many balls from one","Von Neumann paradox in the Euclidean plane"]}
{"upwards":["Banach\u2013Tarski paradox","Banach and Tarski publication"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Formal treatment"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Connection with earlier work and the role of the axiom of choice"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof"],"downwards":["Step 1","Step 2","Step 3","Step 4","Some details, fleshed out"]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 1"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 2"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 3"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 4"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Some details, fleshed out"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Obtaining infinitely many balls from one"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Von Neumann paradox in the Euclidean plane"],"downwards":["Recent progress"]}
{"upwards":["Banach\u2013Tarski paradox","Von Neumann paradox in the Euclidean plane","Recent progress"],"downwards":[]}
{"upwards":["Bibcode"],"downwards":["Adoption","Format","Examples"]}
{"upwards":["Bibcode","Adoption"],"downwards":[]}
{"upwards":["Bibcode","Format"],"downwards":[]}
{"upwards":["Bibcode","Examples"],"downwards":[]}
{"upwards":["Cantor's theorem"],"downwards":["Proof","When A is countably infinite","Related paradoxes","History","Generalizations"]}
{"upwards":["Cantor's theorem","Proof"],"downwards":[]}
{"upwards":["Cantor's theorem","When A is countably infinite"],"downwards":[]}
{"upwards":["Cantor's theorem","Related paradoxes"],"downwards":[]}
{"upwards":["Cantor's theorem","History"],"downwards":[]}
{"upwards":["Cantor's theorem","Generalizations"],"downwards":[]}
{"upwards":["Cantor's diagonal argument"],"downwards":["Uncountable set","Consequences","Version for Quine's New Foundations"]}
{"upwards":["Cantor's diagonal argument","Uncountable set"],"downwards":["Real numbers","General sets"]}
{"upwards":["Cantor's diagonal argument","Uncountable set","Real numbers"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Uncountable set","General sets"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences"],"downwards":["Ordering of cardinals","Open questions","Diagonalization in broader context"]}
{"upwards":["Cantor's diagonal argument","Consequences","Ordering of cardinals"],"downwards":["In the absence of excluded middle"]}
{"upwards":["Cantor's diagonal argument","Consequences","Ordering of cardinals","In the absence of excluded middle"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences","Open questions"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences","Diagonalization in broader context"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Version for Quine's New Foundations"],"downwards":[]}
{"upwards":["Compactness theorem"],"downwards":["History","Applications","Proofs"]}
{"upwards":["Compactness theorem","History"],"downwards":[]}
{"upwards":["Compactness theorem","Applications"],"downwards":["Robinson's principle","Upward L\u00f6wenheim\u2013Skolem theorem","Non-standard analysis"]}
{"upwards":["Compactness theorem","Applications","Robinson's principle"],"downwards":[]}
{"upwards":["Compactness theorem","Applications","Upward L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["Compactness theorem","Applications","Non-standard analysis"],"downwards":[]}
{"upwards":["Compactness theorem","Proofs"],"downwards":[]}
{"upwards":["Halting problem"],"downwards":["Background","History","Formalization","Computability theory","Generalization"]}
{"upwards":["Halting problem","Background"],"downwards":["Programming consequences","Common pitfalls"]}
{"upwards":["Halting problem","Background","Programming consequences"],"downwards":[]}
{"upwards":["Halting problem","Background","Common pitfalls"],"downwards":[]}
{"upwards":["Halting problem","History"],"downwards":["Timeline","Origin of the halting problem"]}
{"upwards":["Halting problem","History","Timeline"],"downwards":[]}
{"upwards":["Halting problem","History","Origin of the halting problem"],"downwards":[]}
{"upwards":["Halting problem","Formalization"],"downwards":["Representation as a set","Proof concept","Sketch of rigorous proof"]}
{"upwards":["Halting problem","Formalization","Representation as a set"],"downwards":[]}
{"upwards":["Halting problem","Formalization","Proof concept"],"downwards":[]}
{"upwards":["Halting problem","Formalization","Sketch of rigorous proof"],"downwards":[]}
{"upwards":["Halting problem","Computability theory"],"downwards":["Approximations","G\u00f6del's incompleteness theorems"]}
{"upwards":["Halting problem","Computability theory","Approximations"],"downwards":[]}
{"upwards":["Halting problem","Computability theory","G\u00f6del's incompleteness theorems"],"downwards":[]}
{"upwards":["Halting problem","Generalization"],"downwards":["Halting on all inputs","Recognizing partial solutions","Lossy computation","Oracle machines"]}
{"upwards":["Halting problem","Generalization","Halting on all inputs"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Recognizing partial solutions"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Lossy computation"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Oracle machines"],"downwards":[]}
{"upwards":["The Principles of Mathematics"],"downwards":["Contents","Early reviews","Second edition","Later reviews"]}
{"upwards":["The Principles of Mathematics","Contents"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Early reviews"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Second edition"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Later reviews"],"downwards":[]}
{"upwards":["ISBN"],"downwards":["History","Overview","Check digits","EAN format used in barcodes, and upgrading"]}
{"upwards":["ISBN","History"],"downwards":[]}
{"upwards":["ISBN","Overview"],"downwards":["Issuing process","Registration group element","Registrant element"]}
{"upwards":["ISBN","Overview","Issuing process"],"downwards":[]}
{"upwards":["ISBN","Overview","Registration group element"],"downwards":[]}
{"upwards":["ISBN","Overview","Registrant element"],"downwards":["English language pattern"]}
{"upwards":["ISBN","Overview","Registrant element","English language pattern"],"downwards":[]}
{"upwards":["ISBN","Check digits"],"downwards":["ISBN-10 check digits","ISBN-10 check digit calculation","ISBN-13 check digit calculation","ISBN-10 to ISBN-13 conversion","Errors in usage","eISBN"]}
{"upwards":["ISBN","Check digits","ISBN-10 check digits"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-10 check digit calculation"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-13 check digit calculation"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-10 to ISBN-13 conversion"],"downwards":[]}
{"upwards":["ISBN","Check digits","Errors in usage"],"downwards":[]}
{"upwards":["ISBN","Check digits","eISBN"],"downwards":[]}
{"upwards":["ISBN","EAN format used in barcodes, and upgrading"],"downwards":[]}
{"upwards":["Lindstr\u00f6m's theorem"],"downwards":["Notes","References"]}
{"upwards":["Lindstr\u00f6m's theorem","Notes"],"downwards":[]}
{"upwards":["Lindstr\u00f6m's theorem","References"],"downwards":[]}
{"upwards":["Principia Mathematica"],"downwards":["Scope of foundations laid","Theoretical basis","Ramified types and the axiom of reducibility","Notation","Consistency and criticisms","Contents","Comparison with set theory","Differences between editions","Editions"]}
{"upwards":["Principia Mathematica","Scope of foundations laid"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis"],"downwards":["Contemporary construction of a formal theory","Construction","Primitive ideas","Primitive propositions"]}
{"upwards":["Principia Mathematica","Theoretical basis","Contemporary construction of a formal theory"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Construction"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Primitive ideas"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Primitive propositions"],"downwards":[]}
{"upwards":["Principia Mathematica","Ramified types and the axiom of reducibility"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation"],"downwards":["An introduction to the notation of \"Section A Mathematical Logic\" (formulas \u27381\u2013\u27385.71)","An introduction to the notation of \"Section B Theory of Apparent Variables\" (formulas \u27388\u2013\u273814.34)","Introduction to the notation of the theory of classes and relations"]}
{"upwards":["Principia Mathematica","Notation","An introduction to the notation of \"Section A Mathematical Logic\" (formulas \u27381\u2013\u27385.71)"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation","An introduction to the notation of \"Section B Theory of Apparent Variables\" (formulas \u27388\u2013\u273814.34)"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation","Introduction to the notation of the theory of classes and relations"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms"],"downwards":["G\u00f6del 1930, 1931","Wittgenstein 1919, 1939","G\u00f6del 1944"]}
{"upwards":["Principia Mathematica","Consistency and criticisms","G\u00f6del 1930, 1931"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms","Wittgenstein 1919, 1939"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms","G\u00f6del 1944"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents"],"downwards":["Part I Mathematical logic. Volume I \u27381 to \u273843","Part II Prolegomena to cardinal arithmetic. Volume I \u273850 to \u273897","Part III Cardinal arithmetic. Volume II \u2738100 to \u2738126","Part IV Relation-arithmetic. Volume II \u2738150 to \u2738186","Part V Series. Volume II \u2738200 to \u2738234 and volume III \u2738250 to \u2738276","Part VI Quantity. Volume III \u2738300 to \u2738375"]}
{"upwards":["Principia Mathematica","Contents","Part I Mathematical logic. Volume I \u27381 to \u273843"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part II Prolegomena to cardinal arithmetic. Volume I \u273850 to \u273897"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part III Cardinal arithmetic. Volume II \u2738100 to \u2738126"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part IV Relation-arithmetic. Volume II \u2738150 to \u2738186"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part V Series. Volume II \u2738200 to \u2738234 and volume III \u2738250 to \u2738276"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part VI Quantity. Volume III \u2738300 to \u2738375"],"downwards":[]}
{"upwards":["Principia Mathematica","Comparison with set theory"],"downwards":[]}
{"upwards":["Principia Mathematica","Differences between editions"],"downwards":[]}
{"upwards":["Principia Mathematica","Editions"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem"],"downwards":["Theorem","Discussion","Proof sketch","In other logics","Historical notes","References","Sources","External links"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Theorem"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion"],"downwards":["Concepts","Consequences"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts"],"downwards":["Signatures","Structures \/ Models"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts","Signatures"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts","Structures \/ Models"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Consequences"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch"],"downwards":["Downward part","Upward part"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch","Downward part"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch","Upward part"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","In other logics"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Historical notes"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","References"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources"],"downwards":["Historical publications","Secondary sources"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources","Historical publications"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources","Secondary sources"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","External links"],"downwards":[]}
{"upwards":["Classical logic"],"downwards":["Characteristics","History","Generalized semantics","References","Further reading"]}
{"upwards":["Classical logic","Characteristics"],"downwards":[]}
{"upwards":["Classical logic","History"],"downwards":[]}
{"upwards":["Classical logic","Generalized semantics"],"downwards":[]}
{"upwards":["Classical logic","References"],"downwards":[]}
{"upwards":["Classical logic","Further reading"],"downwards":[]}
{"upwards":["Term logic"],"downwards":["Aristotle's system","Basics","Term","Proposition","Singular terms","Syllogism in the first figure","Syllogism in the second figure","Syllogism in the third figure","Table of syllogisms","Decline of term logic","Revival","Boole\u2019s acceptance of Aristotle"]}
{"upwards":["Term logic","Aristotle's system"],"downwards":[]}
{"upwards":["Term logic","Basics"],"downwards":[]}
{"upwards":["Term logic","Term"],"downwards":[]}
{"upwards":["Term logic","Proposition"],"downwards":[]}
{"upwards":["Term logic","Singular terms"],"downwards":["The three figures","The fourth figure"]}
{"upwards":["Term logic","Singular terms","The three figures"],"downwards":[]}
{"upwards":["Term logic","Singular terms","The fourth figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the first figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the second figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the third figure"],"downwards":[]}
{"upwards":["Term logic","Table of syllogisms"],"downwards":[]}
{"upwards":["Term logic","Decline of term logic"],"downwards":[]}
{"upwards":["Term logic","Revival"],"downwards":[]}
{"upwards":["Term logic","Boole\u2019s acceptance of Aristotle"],"downwards":[]}
{"upwards":["Logical truth"],"downwards":["Logical truths and analytic truths","Truth values and tautologies","Logical truth and logical constants","Logical truth and rules of inference","Logical truth and logical positivism","Non-classical logics"]}
{"upwards":["Logical truth","Logical truths and analytic truths"],"downwards":[]}
{"upwards":["Logical truth","Truth values and tautologies"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and logical constants"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and rules of inference"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and logical positivism"],"downwards":[]}
{"upwards":["Logical truth","Non-classical logics"],"downwards":[]}
{"upwards":["Equiconsistency"],"downwards":["Consistency","Consistency strength"]}
{"upwards":["Equiconsistency","Consistency"],"downwards":[]}
{"upwards":["Equiconsistency","Consistency strength"],"downwards":[]}
{"upwards":["Logical equivalence"],"downwards":["Logical equivalences","Examples","Relation to material equivalence"]}
{"upwards":["Logical equivalence","Logical equivalences"],"downwards":["General logical equivalences","Logical equivalences involving conditional statements","Logical equivalences involving biconditionals"]}
{"upwards":["Logical equivalence","Logical equivalences","General logical equivalences"],"downwards":[]}
{"upwards":["Logical equivalence","Logical equivalences","Logical equivalences involving conditional statements"],"downwards":[]}
{"upwards":["Logical equivalence","Logical equivalences","Logical equivalences involving biconditionals"],"downwards":[]}
{"upwards":["Logical equivalence","Examples"],"downwards":["In logic"]}
{"upwards":["Logical equivalence","Examples","In logic"],"downwards":[]}
{"upwards":["Logical equivalence","Relation to material equivalence"],"downwards":[]}
{"upwards":["Inference"],"downwards":["Definition","Examples","Incorrect inference","Applications"]}
{"upwards":["Inference","Definition"],"downwards":[]}
{"upwards":["Inference","Examples"],"downwards":["Example for definition #1","Example for definition #2"]}
{"upwards":["Inference","Examples","Example for definition #1"],"downwards":[]}
{"upwards":["Inference","Examples","Example for definition #2"],"downwards":[]}
{"upwards":["Inference","Incorrect inference"],"downwards":[]}
{"upwards":["Inference","Applications"],"downwards":["Inference engines","Semantic web","Bayesian statistics and probability logic","Fuzzy logic","Non-monotonic logic"]}
{"upwards":["Inference","Applications","Inference engines"],"downwards":["Prolog engine"]}
{"upwards":["Inference","Applications","Inference engines","Prolog engine"],"downwards":[]}
{"upwards":["Inference","Applications","Semantic web"],"downwards":[]}
{"upwards":["Inference","Applications","Bayesian statistics and probability logic"],"downwards":[]}
{"upwards":["Inference","Applications","Fuzzy logic"],"downwards":[]}
{"upwards":["Inference","Applications","Non-monotonic logic"],"downwards":[]}
{"upwards":["Consistency"],"downwards":["Consistency and completeness in arithmetic and set theory","First-order logic","Model theory"]}
{"upwards":["Consistency","Consistency and completeness in arithmetic and set theory"],"downwards":[]}
{"upwards":["Consistency","First-order logic"],"downwards":["Notation","Definition","Basic results","Henkin's theorem","Sketch of proof"]}
{"upwards":["Consistency","First-order logic","Notation"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Definition"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Basic results"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Henkin's theorem"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Sketch of proof"],"downwards":[]}
{"upwards":["Consistency","Model theory"],"downwards":[]}
{"upwards":["Proposition"],"downwards":["Historical usage","Relation to the mind","Treatment in logic","Objections to propositions"]}
{"upwards":["Proposition","Historical usage"],"downwards":["By Aristotle","By the logical positivists","By Russell"]}
{"upwards":["Proposition","Historical usage","By Aristotle"],"downwards":[]}
{"upwards":["Proposition","Historical usage","By the logical positivists"],"downwards":[]}
{"upwards":["Proposition","Historical usage","By Russell"],"downwards":[]}
{"upwards":["Proposition","Relation to the mind"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic"],"downwards":["Aristotelian logic","Syntactic characterization","Semantic characterization"]}
{"upwards":["Proposition","Treatment in logic","Aristotelian logic"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic","Syntactic characterization"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic","Semantic characterization"],"downwards":[]}
{"upwards":["Proposition","Objections to propositions"],"downwards":[]}
{"upwards":["Tautology (logic)"],"downwards":["History","Background","Definition and examples","Verifying tautologies","Tautological implication","Substitution","Semantic completeness and soundness","Efficient verification and the Boolean satisfiability problem","Tautologies versus validities in first-order logic"]}
{"upwards":["Tautology (logic)","History"],"downwards":[]}
{"upwards":["Tautology (logic)","Background"],"downwards":[]}
{"upwards":["Tautology (logic)","Definition and examples"],"downwards":[]}
{"upwards":["Tautology (logic)","Verifying tautologies"],"downwards":[]}
{"upwards":["Tautology (logic)","Tautological implication"],"downwards":[]}
{"upwards":["Tautology (logic)","Substitution"],"downwards":[]}
{"upwards":["Tautology (logic)","Semantic completeness and soundness"],"downwards":[]}
{"upwards":["Tautology (logic)","Efficient verification and the Boolean satisfiability problem"],"downwards":[]}
{"upwards":["Tautology (logic)","Tautologies versus validities in first-order logic"],"downwards":[]}
{"upwards":["Isomorphism"],"downwards":["Examples","Applications","Category theoretic view","Relation with equality"]}
{"upwards":["Isomorphism","Examples"],"downwards":["Logarithm and exponential","Integers modulo 6","Relation-preserving isomorphism"]}
{"upwards":["Isomorphism","Examples","Logarithm and exponential"],"downwards":[]}
{"upwards":["Isomorphism","Examples","Integers modulo 6"],"downwards":[]}
{"upwards":["Isomorphism","Examples","Relation-preserving isomorphism"],"downwards":[]}
{"upwards":["Isomorphism","Applications"],"downwards":[]}
{"upwards":["Isomorphism","Category theoretic view"],"downwards":["Isomorphism vs. bijective morphism"]}
{"upwards":["Isomorphism","Category theoretic view","Isomorphism vs. bijective morphism"],"downwards":[]}
{"upwards":["Isomorphism","Relation with equality"],"downwards":[]}
{"upwards":["Aleph number"],"downwards":["Aleph-nought","Aleph-one","Continuum hypothesis","Aleph-omega","Aleph-\u03b1 for general \u03b1","Fixed points of omega","Role of axiom of choice"]}
{"upwards":["Aleph number","Aleph-nought"],"downwards":[]}
{"upwards":["Aleph number","Aleph-one"],"downwards":[]}
{"upwards":["Aleph number","Continuum hypothesis"],"downwards":[]}
{"upwards":["Aleph number","Aleph-omega"],"downwards":[]}
{"upwards":["Aleph number","Aleph-\u03b1 for general \u03b1"],"downwards":[]}
{"upwards":["Aleph number","Fixed points of omega"],"downwards":[]}
{"upwards":["Aleph number","Role of axiom of choice"],"downwards":[]}
{"upwards":["Enumeration"],"downwards":["Combinatorics","Set theory","Computability and complexity theory"]}
{"upwards":["Enumeration","Combinatorics"],"downwards":[]}
{"upwards":["Enumeration","Set theory"],"downwards":["Listing","Countable vs. uncountable","Ordinals","Comparison of cardinalities"]}
{"upwards":["Enumeration","Set theory","Listing"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Countable vs. uncountable"],"downwards":["Examples","Properties"]}
{"upwards":["Enumeration","Set theory","Countable vs. uncountable","Examples"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Countable vs. uncountable","Properties"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Ordinals"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Comparison of cardinalities"],"downwards":[]}
{"upwards":["Enumeration","Computability and complexity theory"],"downwards":[]}
{"upwards":["Large cardinal"],"downwards":["Partial definition","Hierarchy of consistency strength","Motivations and epistemic status"]}
{"upwards":["Large cardinal","Partial definition"],"downwards":[]}
{"upwards":["Large cardinal","Hierarchy of consistency strength"],"downwards":[]}
{"upwards":["Large cardinal","Motivations and epistemic status"],"downwards":[]}
{"upwards":["Operation (mathematics)"],"downwards":["Types of operation","Definition"]}
{"upwards":["Operation (mathematics)","Types of operation"],"downwards":[]}
{"upwards":["Operation (mathematics)","Definition"],"downwards":[]}
{"upwards":["Surjective function"],"downwards":["Definition","Examples","Properties","Space of surjections","Gallery"]}
{"upwards":["Surjective function","Definition"],"downwards":[]}
{"upwards":["Surjective function","Examples"],"downwards":[]}
{"upwards":["Surjective function","Properties"],"downwards":["Surjections as right invertible functions","Surjections as epimorphisms","Surjections as binary relations","Cardinality of the domain of a surjection","Composition and decomposition","Induced surjection and induced bijection"]}
{"upwards":["Surjective function","Properties","Surjections as right invertible functions"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as epimorphisms"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as binary relations"],"downwards":[]}
{"upwards":["Surjective function","Properties","Cardinality of the domain of a surjection"],"downwards":[]}
{"upwards":["Surjective function","Properties","Composition and decomposition"],"downwards":[]}
{"upwards":["Surjective function","Properties","Induced surjection and induced bijection"],"downwards":[]}
{"upwards":["Surjective function","Space of surjections"],"downwards":[]}
{"upwards":["Surjective function","Gallery"],"downwards":[]}
{"upwards":["Inaccessible cardinal"],"downwards":["Models and consistency","Existence of a proper class of inaccessibles","\u03b1-inaccessible cardinals and hyper-inaccessible cardinals","Two model-theoretic characterisations of inaccessibility"]}
{"upwards":["Inaccessible cardinal","Models and consistency"],"downwards":[]}
{"upwards":["Inaccessible cardinal","Existence of a proper class of inaccessibles"],"downwards":[]}
{"upwards":["Inaccessible cardinal","\u03b1-inaccessible cardinals and hyper-inaccessible cardinals"],"downwards":[]}
{"upwards":["Inaccessible cardinal","Two model-theoretic characterisations of inaccessibility"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem"],"downwards":["Proof","History","Prerequisites"]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Proof"],"downwards":["Another proof","Examples"]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Proof","Another proof"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Proof","Examples"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","History"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Prerequisites"],"downwards":[]}
{"upwards":["G\u00f6del numbering"],"downwards":["Simplified overview","G\u00f6del's encoding","Lack of uniqueness","Application to formal arithmetic","Generalizations","G\u00f6del  sets"]}
{"upwards":["G\u00f6del numbering","Simplified overview"],"downwards":[]}
{"upwards":["G\u00f6del numbering","G\u00f6del's encoding"],"downwards":["Example"]}
{"upwards":["G\u00f6del numbering","G\u00f6del's encoding","Example"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Lack of uniqueness"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Application to formal arithmetic"],"downwards":["Recursion","Expressing statements and proofs by numbers"]}
{"upwards":["G\u00f6del numbering","Application to formal arithmetic","Recursion"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Application to formal arithmetic","Expressing statements and proofs by numbers"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Generalizations"],"downwards":[]}
{"upwards":["G\u00f6del numbering","G\u00f6del  sets"],"downwards":[]}
{"upwards":["Binary operation"],"downwards":["Terminology","Properties and examples","Notation","Binary operations as ternary relations","External binary operations"]}
{"upwards":["Binary operation","Terminology"],"downwards":[]}
{"upwards":["Binary operation","Properties and examples"],"downwards":[]}
{"upwards":["Binary operation","Notation"],"downwards":[]}
{"upwards":["Binary operation","Binary operations as ternary relations"],"downwards":[]}
{"upwards":["Binary operation","External binary operations"],"downwards":[]}
{"upwards":["General set theory"],"downwards":["Ontology","Axioms","Discussion","History","Footnotes","References","External links"]}
{"upwards":["General set theory","Ontology"],"downwards":[]}
{"upwards":["General set theory","Axioms"],"downwards":[]}
{"upwards":["General set theory","Discussion"],"downwards":["Metamathematics","Peano arithmetic","Infinite sets"]}
{"upwards":["General set theory","Discussion","Metamathematics"],"downwards":[]}
{"upwards":["General set theory","Discussion","Peano arithmetic"],"downwards":[]}
{"upwards":["General set theory","Discussion","Infinite sets"],"downwards":[]}
{"upwards":["General set theory","History"],"downwards":[]}
{"upwards":["General set theory","Footnotes"],"downwards":[]}
{"upwards":["General set theory","References"],"downwards":[]}
{"upwards":["General set theory","External links"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory"],"downwards":["MK axioms and ontology","Discussion","The axioms in Kelley's General Topology","Notes","References","External links"]}
{"upwards":["Morse\u2013Kelley set theory","MK axioms and ontology"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Discussion"],"downwards":["Model theory","History"]}
{"upwards":["Morse\u2013Kelley set theory","Discussion","Model theory"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Discussion","History"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","The axioms in Kelley's General Topology"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Notes"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","References"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","External links"],"downwards":[]}
{"upwards":["Constructive set theory"],"downwards":["Introduction","Subtheories of ZF","Sorted theories"]}
{"upwards":["Constructive set theory","Introduction"],"downwards":["Constructive outlook","History and overview"]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook"],"downwards":["Preliminary on the use of intuitionistic logic","Imposed restrictions on a set theory","Metalogic","Anti-classical principles"]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Preliminary on the use of intuitionistic logic"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Imposed restrictions on a set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Anti-classical principles"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","History and overview"],"downwards":["Models"]}
{"upwards":["Constructive set theory","Introduction","History and overview","Models"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF"],"downwards":["Notation","Equality","Merging sets","Set existence","BCST","ECST","Functions","Choice functions","Arithmetic","Exponentiation","Analysis","Non-constructive principles","Induction","Strong Collection","Constructive Zermelo\u2013Fraenkel","Metalogic of CZF","Intuitionistic Zermelo\u2013Fraenkel","Intuitionistic Z","Intuitionistic KP"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation"],"downwards":["Language","Variables","Classes"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Language"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Variables"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Classes"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Equality"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Merging sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Set existence"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST"],"downwards":["Separation","No universal set","Predicativity","Replacement"]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Separation"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","No universal set"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Predicativity"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Replacement"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST"],"downwards":["Weaker formulations of infinity","Number bounds"]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST","Weaker formulations of infinity"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST","Number bounds"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions"],"downwards":["Finiteness","General note on programs and functions","Total functional relations","Characteristic functions","Computable sets","Boundedness criteria"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Finiteness"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","General note on programs and functions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Total functional relations"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Characteristic functions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Computable sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Boundedness criteria"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions"],"downwards":["Diaconescu's theorem","Analysis of Diaconescu's theorem","Regularity implies PEM"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Diaconescu's theorem"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Analysis of Diaconescu's theorem"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Regularity implies PEM"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic"],"downwards":["Recursion","Moderate induction in ECST","Induction without infinite sets"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Recursion"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Moderate induction in ECST"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Induction without infinite sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation"],"downwards":["On countable sets","The class of subsets of a set","Metalogic","Category and type theoretic notions"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","On countable sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","The class of subsets of a set"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","Category and type theoretic notions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis"],"downwards":["Cauchy sequences","Cauchy reals","Towards the Dedekind reals","Constructive schools"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Cauchy sequences"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Cauchy reals"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Towards the Dedekind reals"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Constructive schools"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Non-constructive principles"],"downwards":["Infinite trees"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Non-constructive principles","Infinite trees"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction"],"downwards":["Mathematical induction","Set Induction","Metalogic","Relation to ZF"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Mathematical induction"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Set Induction"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Relation to ZF"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Strong Collection"],"downwards":["Metalogic"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Strong Collection","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel"],"downwards":["Binary refinement","Subset Collection","Fullness"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Binary refinement"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Subset Collection"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Fullness"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF"],"downwards":["Unprovable claims","Ordinal analysis","Models","Indecomposability","Breaking with ZF"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Unprovable claims"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Ordinal analysis"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Models"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Indecomposability"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Breaking with ZF"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Zermelo\u2013Fraenkel"],"downwards":["Metalogic"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Zermelo\u2013Fraenkel","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Z"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic KP"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories"],"downwards":["Constructive set theory","Bishop style set theory","Category theories"]}
{"upwards":["Constructive set theory","Sorted theories","Constructive set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories","Bishop style set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories","Category theories"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory"],"downwards":["Axioms","Related definitions","Theorems","Metalogic"]}
{"upwards":["Kripke\u2013Platek set theory","Axioms"],"downwards":["Empty set","Comparison with Zermelo-Fraenkel set theory"]}
{"upwards":["Kripke\u2013Platek set theory","Axioms","Empty set"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Axioms","Comparison with Zermelo-Fraenkel set theory"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Related definitions"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Theorems"],"downwards":["Admissible sets","Cartesian products exist"]}
{"upwards":["Kripke\u2013Platek set theory","Theorems","Admissible sets"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Theorems","Cartesian products exist"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Metalogic"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory"],"downwards":["Classes in set theory","Axiomatization of NBG","History","NBG, ZFC, and MK","Category theory","Notes","References","Bibliography","External links"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory"],"downwards":["The uses of classes","Axiom schema versus class existence theorem"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory","The uses of classes"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory","Axiom schema versus class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG"],"downwards":["Classes and sets","Definitions and axioms of extensionality and pairing","Class existence axioms and axiom of regularity","Class existence theorem","Extending the class existence theorem","Set axioms","Axiom of global choice"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Classes and sets"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Definitions and axioms of extensionality and pairing"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Class existence axioms and axiom of regularity"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Extending the class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Set axioms"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Axiom of global choice"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History"],"downwards":["Von Neumann's 1925 axiom system","Von Neumann's 1929 axiom system","Bernays' axiom system","G\u00f6del's axiom system (NBG)"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Von Neumann's 1925 axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Von Neumann's 1929 axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Bernays' axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","G\u00f6del's axiom system (NBG)"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","NBG, ZFC, and MK"],"downwards":["Models"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","NBG, ZFC, and MK","Models"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Category theory"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Notes"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","References"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Bibliography"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","External links"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory"],"downwards":["Axioms","Implementation in the Mizar system","Implementation in Metamath"]}
{"upwards":["Tarski\u2013Grothendieck set theory","Axioms"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory","Implementation in the Mizar system"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory","Implementation in Metamath"],"downwards":[]}
{"upwards":["Formal system"],"downwards":["Background","History","Examples","Variants"]}
{"upwards":["Formal system","Background"],"downwards":["Recursive","Inference and entailment"]}
{"upwards":["Formal system","Background","Recursive"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment"],"downwards":["Formal language","Deductive system","Logical system"]}
{"upwards":["Formal system","Background","Inference and entailment","Formal language"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Deductive system"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Logical system"],"downwards":[]}
{"upwards":["Formal system","History"],"downwards":["Formalism","QED manifesto"]}
{"upwards":["Formal system","History","Formalism"],"downwards":["Hilbert's program"]}
{"upwards":["Formal system","History","Formalism","Hilbert's program"],"downwards":[]}
{"upwards":["Formal system","History","QED manifesto"],"downwards":[]}
{"upwards":["Formal system","Examples"],"downwards":[]}
{"upwards":["Formal system","Variants"],"downwards":["Proof system"]}
{"upwards":["Formal system","Variants","Proof system"],"downwards":[]}
{"upwards":["Alphabet (formal languages)"],"downwards":["Notation","Applications"]}
{"upwards":["Alphabet (formal languages)","Notation"],"downwards":[]}
{"upwards":["Alphabet (formal languages)","Applications"],"downwards":[]}
{"upwards":["List of formal systems"],"downwards":["Mathematical","Other formal systems"]}
{"upwards":["List of formal systems","Mathematical"],"downwards":[]}
{"upwards":["List of formal systems","Other formal systems"],"downwards":[]}
{"upwards":["Expression (mathematics)"],"downwards":["Examples","Syntax versus semantics","Variables"]}
{"upwards":["Expression (mathematics)","Examples"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics"],"downwards":["Syntax","Semantics","Formal languages and lambda calculus"]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Syntax"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Semantics"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Formal languages and lambda calculus"],"downwards":[]}
{"upwards":["Expression (mathematics)","Variables"],"downwards":[]}
{"upwards":["Ackermann set theory"],"downwards":["The language","The axioms","Relation to Zermelo\u2013Fraenkel set theory","AST and category theory"]}
{"upwards":["Ackermann set theory","The language"],"downwards":[]}
{"upwards":["Ackermann set theory","The axioms"],"downwards":[]}
{"upwards":["Ackermann set theory","Relation to Zermelo\u2013Fraenkel set theory"],"downwards":[]}
{"upwards":["Ackermann set theory","AST and category theory"],"downwards":[]}
{"upwards":["Arity"],"downwards":["Examples","Terminology"]}
{"upwards":["Arity","Examples"],"downwards":["Nullary","Unary","Binary","Ternary","n-ary","Varying arity"]}
{"upwards":["Arity","Examples","Nullary"],"downwards":[]}
{"upwards":["Arity","Examples","Unary"],"downwards":[]}
{"upwards":["Arity","Examples","Binary"],"downwards":[]}
{"upwards":["Arity","Examples","Ternary"],"downwards":[]}
{"upwards":["Arity","Examples","n-ary"],"downwards":[]}
{"upwards":["Arity","Examples","Varying arity"],"downwards":[]}
{"upwards":["Arity","Terminology"],"downwards":[]}
{"upwards":["Formal language"],"downwards":["History","Words over an alphabet","Definition","Examples","Language-specification formalisms","Operations on languages","Applications"]}
{"upwards":["Formal language","History"],"downwards":[]}
{"upwards":["Formal language","Words over an alphabet"],"downwards":[]}
{"upwards":["Formal language","Definition"],"downwards":[]}
{"upwards":["Formal language","Examples"],"downwards":["Constructions"]}
{"upwards":["Formal language","Examples","Constructions"],"downwards":[]}
{"upwards":["Formal language","Language-specification formalisms"],"downwards":[]}
{"upwards":["Formal language","Operations on languages"],"downwards":[]}
{"upwards":["Formal language","Applications"],"downwards":["Programming languages","Formal theories, systems, and proofs"]}
{"upwards":["Formal language","Applications","Programming languages"],"downwards":[]}
{"upwards":["Formal language","Applications","Formal theories, systems, and proofs"],"downwards":["Interpretations and models"]}
{"upwards":["Formal language","Applications","Formal theories, systems, and proofs","Interpretations and models"],"downwards":[]}
{"upwards":["Syntax (logic)"],"downwards":["Syntactic entities"]}
{"upwards":["Syntax (logic)","Syntactic entities"],"downwards":["Symbols","Formal language","Formation rules","Propositions","Formal theories","Formal systems","Interpretations"]}
{"upwards":["Syntax (logic)","Syntactic entities","Symbols"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal language"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formation rules"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Propositions"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal theories"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems"],"downwards":["Syntactic consequence within a formal system","Syntactic completeness of a formal system"]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems","Syntactic consequence within a formal system"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems","Syntactic completeness of a formal system"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Interpretations"],"downwards":[]}
{"upwards":["Axiom schema"],"downwards":["Formal definition","Finite axiomatization","Examples","Finitely axiomatized theories","In higher-order logic"]}
{"upwards":["Axiom schema","Formal definition"],"downwards":[]}
{"upwards":["Axiom schema","Finite axiomatization"],"downwards":[]}
{"upwards":["Axiom schema","Examples"],"downwards":[]}
{"upwards":["Axiom schema","Finitely axiomatized theories"],"downwards":[]}
{"upwards":["Axiom schema","In higher-order logic"],"downwards":[]}
{"upwards":["New Foundations"],"downwards":["The Type Theory TST","Quinean set theory","Consequences","The consistency problem and related partial results","Models of NFU","Strong axioms of infinity","The system ML (Mathematical Logic)"]}
{"upwards":["New Foundations","The Type Theory TST"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory"],"downwards":["Axioms and stratification","Finite axiomatization","Ordered pairs","Natural numbers and the axiom of infinity"]}
{"upwards":["New Foundations","Quinean set theory","Axioms and stratification"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Finite axiomatization"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Ordered pairs"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Natural numbers and the axiom of infinity"],"downwards":[]}
{"upwards":["New Foundations","Consequences"],"downwards":["Admissibility of useful large sets","Resolution of set-theoretic paradoxes","Cartesian closure"]}
{"upwards":["New Foundations","Consequences","Admissibility of useful large sets"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes"],"downwards":["Cantor's paradox and Cantorian sets","Burali-Forti paradox and the T operation"]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes","Cantor's paradox and Cantorian sets"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes","Burali-Forti paradox and the T operation"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Cartesian closure"],"downwards":[]}
{"upwards":["New Foundations","The consistency problem and related partial results"],"downwards":[]}
{"upwards":["New Foundations","Models of NFU"],"downwards":["Self-sufficiency of mathematical foundations in NFU","Facts about the automorphism j"]}
{"upwards":["New Foundations","Models of NFU","Self-sufficiency of mathematical foundations in NFU"],"downwards":[]}
{"upwards":["New Foundations","Models of NFU","Facts about the automorphism j"],"downwards":[]}
{"upwards":["New Foundations","Strong axioms of infinity"],"downwards":[]}
{"upwards":["New Foundations","The system ML (Mathematical Logic)"],"downwards":[]}
{"upwards":["Automata theory"],"downwards":["History","Automata","Variant definitions of automata","Types of automata","Hierarchy in terms of powers","Applications","Automata simulators","Connection to category theory"]}
{"upwards":["Automata theory","History"],"downwards":[]}
{"upwards":["Automata theory","Automata"],"downwards":["Informal description","Formal definition"]}
{"upwards":["Automata theory","Automata","Informal description"],"downwards":[]}
{"upwards":["Automata theory","Automata","Formal definition"],"downwards":[]}
{"upwards":["Automata theory","Variant definitions of automata"],"downwards":[]}
{"upwards":["Automata theory","Types of automata"],"downwards":["Discrete, continuous, and hybrid automata"]}
{"upwards":["Automata theory","Types of automata","Discrete, continuous, and hybrid automata"],"downwards":[]}
{"upwards":["Automata theory","Hierarchy in terms of powers"],"downwards":[]}
{"upwards":["Automata theory","Applications"],"downwards":[]}
{"upwards":["Automata theory","Automata simulators"],"downwards":[]}
{"upwards":["Automata theory","Connection to category theory"],"downwards":[]}
{"upwards":["Ground expression"],"downwards":["Examples","Formal definitions"]}
{"upwards":["Ground expression","Examples"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions"],"downwards":["Ground term","Ground atom","Ground formula"]}
{"upwards":["Ground expression","Formal definitions","Ground term"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground atom"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground formula"],"downwards":[]}
{"upwards":["Free variables and bound variables"],"downwards":["Examples","Formal explanation","Natural language"]}
{"upwards":["Free variables and bound variables","Examples"],"downwards":["Variable-binding operators"]}
{"upwards":["Free variables and bound variables","Examples","Variable-binding operators"],"downwards":[]}
{"upwards":["Free variables and bound variables","Formal explanation"],"downwards":["Function expressions"]}
{"upwards":["Free variables and bound variables","Formal explanation","Function expressions"],"downwards":[]}
{"upwards":["Free variables and bound variables","Natural language"],"downwards":[]}
{"upwards":["Extension by new constant and function names"],"downwards":[]}
{"upwards":["Extension by definitions"],"downwards":["Definition of relation symbols","Definition of function symbols","Extensions by definitions","Examples"]}
{"upwards":["Extension by definitions","Definition of relation symbols"],"downwards":[]}
{"upwards":["Extension by definitions","Definition of function symbols"],"downwards":[]}
{"upwards":["Extension by definitions","Extensions by definitions"],"downwards":[]}
{"upwards":["Extension by definitions","Examples"],"downwards":[]}
{"upwards":["Well-formed formula"],"downwards":["Introduction","Propositional calculus","Predicate logic","Atomic and open formulas","Closed formulas","Properties applicable to formulas","Usage of the terminology"]}
{"upwards":["Well-formed formula","Introduction"],"downwards":[]}
{"upwards":["Well-formed formula","Propositional calculus"],"downwards":[]}
{"upwards":["Well-formed formula","Predicate logic"],"downwards":[]}
{"upwards":["Well-formed formula","Atomic and open formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Closed formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Properties applicable to formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Usage of the terminology"],"downwards":[]}
{"upwards":["Finitary relation"],"downwards":["Example","Definitions","History"]}
{"upwards":["Finitary relation","Example"],"downwards":[]}
{"upwards":["Finitary relation","Definitions"],"downwards":[]}
{"upwards":["Finitary relation","History"],"downwards":[]}
{"upwards":["Logical biconditional"],"downwards":["Definition","Properties","Rules of inference","Colloquial usage"]}
{"upwards":["Logical biconditional","Definition"],"downwards":["Truth table","Venn diagrams"]}
{"upwards":["Logical biconditional","Definition","Truth table"],"downwards":[]}
{"upwards":["Logical biconditional","Definition","Venn diagrams"],"downwards":[]}
{"upwards":["Logical biconditional","Properties"],"downwards":[]}
{"upwards":["Logical biconditional","Rules of inference"],"downwards":["Biconditional introduction","Biconditional elimination"]}
{"upwards":["Logical biconditional","Rules of inference","Biconditional introduction"],"downwards":[]}
{"upwards":["Logical biconditional","Rules of inference","Biconditional elimination"],"downwards":[]}
{"upwards":["Logical biconditional","Colloquial usage"],"downwards":[]}
{"upwards":["Logical conjunction"],"downwards":["Notation","Definition","Introduction and elimination rules","Negation","Properties","Applications in computer engineering","Set-theoretic correspondence","Natural language"]}
{"upwards":["Logical conjunction","Notation"],"downwards":[]}
{"upwards":["Logical conjunction","Definition"],"downwards":["Truth table","Defined by other operators"]}
{"upwards":["Logical conjunction","Definition","Truth table"],"downwards":[]}
{"upwards":["Logical conjunction","Definition","Defined by other operators"],"downwards":[]}
{"upwards":["Logical conjunction","Introduction and elimination rules"],"downwards":[]}
{"upwards":["Logical conjunction","Negation"],"downwards":["Definition","Other proof strategies"]}
{"upwards":["Logical conjunction","Negation","Definition"],"downwards":[]}
{"upwards":["Logical conjunction","Negation","Other proof strategies"],"downwards":[]}
{"upwards":["Logical conjunction","Properties"],"downwards":[]}
{"upwards":["Logical conjunction","Applications in computer engineering"],"downwards":[]}
{"upwards":["Logical conjunction","Set-theoretic correspondence"],"downwards":[]}
{"upwards":["Logical conjunction","Natural language"],"downwards":[]}
{"upwards":["Formal grammar"],"downwards":["History","Introductory example","Formal definition","The Chomsky hierarchy","Analytic grammars"]}
{"upwards":["Formal grammar","History"],"downwards":[]}
{"upwards":["Formal grammar","Introductory example"],"downwards":["Example 1","Examples 2 and 3"]}
{"upwards":["Formal grammar","Introductory example","Example 1"],"downwards":[]}
{"upwards":["Formal grammar","Introductory example","Examples 2 and 3"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition"],"downwards":["The syntax of grammars","Some mathematical constructs regarding formal grammars","Example"]}
{"upwards":["Formal grammar","Formal definition","The syntax of grammars"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition","Some mathematical constructs regarding formal grammars"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition","Example"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy"],"downwards":["Context-free grammars","Regular grammars","Other forms of generative grammars","Recursive grammars"]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Context-free grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Regular grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Other forms of generative grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Recursive grammars"],"downwards":[]}
{"upwards":["Formal grammar","Analytic grammars"],"downwards":[]}
{"upwards":["Atomic formula"],"downwards":["Atomic formula in first-order logic"]}
{"upwards":["Atomic formula","Atomic formula in first-order logic"],"downwards":[]}
{"upwards":["Conservative extension"],"downwards":["Examples","Model-theoretic conservative extension"]}
{"upwards":["Conservative extension","Examples"],"downwards":[]}
{"upwards":["Conservative extension","Model-theoretic conservative extension"],"downwards":[]}
{"upwards":["Metalanguage"],"downwards":["Types of metalanguage","In natural language","Types of expressions","Role in metaphor","Metaprogramming"]}
{"upwards":["Metalanguage","Types of metalanguage"],"downwards":["Embedded","Ordered","Nested"]}
{"upwards":["Metalanguage","Types of metalanguage","Embedded"],"downwards":[]}
{"upwards":["Metalanguage","Types of metalanguage","Ordered"],"downwards":[]}
{"upwards":["Metalanguage","Types of metalanguage","Nested"],"downwards":[]}
{"upwards":["Metalanguage","In natural language"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions"],"downwards":["Deductive systems","Metavariables","Metatheories and metatheorems","Interpretations"]}
{"upwards":["Metalanguage","Types of expressions","Deductive systems"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Metavariables"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Metatheories and metatheorems"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Interpretations"],"downwards":[]}
{"upwards":["Metalanguage","Role in metaphor"],"downwards":[]}
{"upwards":["Metalanguage","Metaprogramming"],"downwards":[]}
{"upwards":["Sentence (mathematical logic)"],"downwards":["Example"]}
{"upwards":["Sentence (mathematical logic)","Example"],"downwards":[]}
{"upwards":["Open formula"],"downwards":[]}
{"upwards":["Material conditional"],"downwards":["Notation","Definitions","Formal properties","Discrepancies with natural language"]}
{"upwards":["Material conditional","Notation"],"downwards":["History"]}
{"upwards":["Material conditional","Notation","History"],"downwards":[]}
{"upwards":["Material conditional","Definitions"],"downwards":["Semantics","Truth table","Deductive definition"]}
{"upwards":["Material conditional","Definitions","Semantics"],"downwards":[]}
{"upwards":["Material conditional","Definitions","Truth table"],"downwards":[]}
{"upwards":["Material conditional","Definitions","Deductive definition"],"downwards":[]}
{"upwards":["Material conditional","Formal properties"],"downwards":[]}
{"upwards":["Material conditional","Discrepancies with natural language"],"downwards":[]}
{"upwards":["Formation rule"],"downwards":["Formal language","Formal systems","Propositional and predicate logic"]}
{"upwards":["Formation rule","Formal language"],"downwards":[]}
{"upwards":["Formation rule","Formal systems"],"downwards":[]}
{"upwards":["Formation rule","Propositional and predicate logic"],"downwards":[]}
{"upwards":["Negation"],"downwards":["Definition","Notation","Properties","Rules of inference","Programming language and ordinary language","Kripke semantics"]}
{"upwards":["Negation","Definition"],"downwards":[]}
{"upwards":["Negation","Notation"],"downwards":[]}
{"upwards":["Negation","Properties"],"downwards":["Double negation","Distributivity","Linearity","Self dual","Negations of quantifiers"]}
{"upwards":["Negation","Properties","Double negation"],"downwards":[]}
{"upwards":["Negation","Properties","Distributivity"],"downwards":[]}
{"upwards":["Negation","Properties","Linearity"],"downwards":[]}
{"upwards":["Negation","Properties","Self dual"],"downwards":[]}
{"upwards":["Negation","Properties","Negations of quantifiers"],"downwards":[]}
{"upwards":["Negation","Rules of inference"],"downwards":[]}
{"upwards":["Negation","Programming language and ordinary language"],"downwards":[]}
{"upwards":["Negation","Kripke semantics"],"downwards":[]}
{"upwards":["Logical disjunction"],"downwards":["Inclusive and exclusive disjunction","Notation","Classical disjunction","Applications in computer science","Set theory","Natural language"]}
{"upwards":["Logical disjunction","Inclusive and exclusive disjunction"],"downwards":[]}
{"upwards":["Logical disjunction","Notation"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction"],"downwards":["Semantics","Defined by other operators","Properties"]}
{"upwards":["Logical disjunction","Classical disjunction","Semantics"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction","Defined by other operators"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction","Properties"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science"],"downwards":["Bitwise operation","Logical operation","Constructive disjunction"]}
{"upwards":["Logical disjunction","Applications in computer science","Bitwise operation"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science","Logical operation"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science","Constructive disjunction"],"downwards":[]}
{"upwards":["Logical disjunction","Set theory"],"downwards":[]}
{"upwards":["Logical disjunction","Natural language"],"downwards":[]}
{"upwards":["Ground expression"],"downwards":["Examples","Formal definitions"]}
{"upwards":["Ground expression","Examples"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions"],"downwards":["Ground term","Ground atom","Ground formula"]}
{"upwards":["Ground expression","Formal definitions","Ground term"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground atom"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground formula"],"downwards":[]}
{"upwards":["Logic"],"downwards":["Definition","Basic concepts","Systems of logic","Areas of research","History"]}
{"upwards":["Logic","Definition"],"downwards":["Formal logic","Informal logic"]}
{"upwards":["Logic","Definition","Formal logic"],"downwards":[]}
{"upwards":["Logic","Definition","Informal logic"],"downwards":[]}
{"upwards":["Logic","Basic concepts"],"downwards":["Premises, conclusions, and truth","Arguments and inferences","Fallacies","Definitory and strategic rules","Formal systems"]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth"],"downwards":["Premises and conclusions","Internal structure","Logical truth","Truth tables"]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Premises and conclusions"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Internal structure"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Logical truth"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Truth tables"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Arguments and inferences"],"downwards":["Deductive","Ampliative"]}
{"upwards":["Logic","Basic concepts","Arguments and inferences","Deductive"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Arguments and inferences","Ampliative"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Fallacies"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Definitory and strategic rules"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Formal systems"],"downwards":[]}
{"upwards":["Logic","Systems of logic"],"downwards":["Aristotelian","Classical","Extended","Deviant","Informal"]}
{"upwards":["Logic","Systems of logic","Aristotelian"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Classical"],"downwards":["Propositional logic","First-order logic"]}
{"upwards":["Logic","Systems of logic","Classical","Propositional logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Classical","First-order logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Extended"],"downwards":["Modal logic","Higher order logic"]}
{"upwards":["Logic","Systems of logic","Extended","Modal logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Extended","Higher order logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Deviant"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Informal"],"downwards":[]}
{"upwards":["Logic","Areas of research"],"downwards":["Philosophy of logic and philosophical logic","Metalogic","Mathematical logic","Computational logic","Formal semantics of natural language","Epistemology of logic"]}
{"upwards":["Logic","Areas of research","Philosophy of logic and philosophical logic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Metalogic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Mathematical logic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Computational logic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Formal semantics of natural language"],"downwards":[]}
{"upwards":["Logic","Areas of research","Epistemology of logic"],"downwards":[]}
{"upwards":["Logic","History"],"downwards":[]}
{"upwards":["Universal quantification"],"downwards":["Basics","Properties","Universal closure","As adjoint"]}
{"upwards":["Universal quantification","Basics"],"downwards":["Notation"]}
{"upwards":["Universal quantification","Basics","Notation"],"downwards":[]}
{"upwards":["Universal quantification","Properties"],"downwards":["Negation","Other connectives","Rules of inference","The empty set"]}
{"upwards":["Universal quantification","Properties","Negation"],"downwards":[]}
{"upwards":["Universal quantification","Properties","Other connectives"],"downwards":[]}
{"upwards":["Universal quantification","Properties","Rules of inference"],"downwards":[]}
{"upwards":["Universal quantification","Properties","The empty set"],"downwards":[]}
{"upwards":["Universal quantification","Universal closure"],"downwards":[]}
{"upwards":["Universal quantification","As adjoint"],"downwards":[]}
{"upwards":["Atomic sentence"],"downwards":["Examples","Interpretations","Translating sentences from a natural language into an artificial language","Philosophical significance"]}
{"upwards":["Atomic sentence","Examples"],"downwards":["Assumptions","Atomic sentences","Atomic formulae","Compound sentences","Compound formulae"]}
{"upwards":["Atomic sentence","Examples","Assumptions"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Atomic sentences"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Atomic formulae"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Compound sentences"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Compound formulae"],"downwards":[]}
{"upwards":["Atomic sentence","Interpretations"],"downwards":[]}
{"upwards":["Atomic sentence","Translating sentences from a natural language into an artificial language"],"downwards":[]}
{"upwards":["Atomic sentence","Philosophical significance"],"downwards":[]}
{"upwards":["Uniqueness quantification"],"downwards":["Proving uniqueness","Reduction to ordinary existential and universal quantification","Generalizations"]}
{"upwards":["Uniqueness quantification","Proving uniqueness"],"downwards":[]}
{"upwards":["Uniqueness quantification","Reduction to ordinary existential and universal quantification"],"downwards":[]}
{"upwards":["Uniqueness quantification","Generalizations"],"downwards":[]}
{"upwards":["Predicate variable"],"downwards":["Notation","Usage"]}
{"upwards":["Predicate variable","Notation"],"downwards":[]}
{"upwards":["Predicate variable","Usage"],"downwards":[]}
{"upwards":["Propositional variable"],"downwards":["Uses","Predicate logic"]}
{"upwards":["Propositional variable","Uses"],"downwards":[]}
{"upwards":["Propositional variable","Predicate logic"],"downwards":[]}
{"upwards":["Quantifier rank"],"downwards":["Definition","Examples"]}
{"upwards":["Quantifier rank","Definition"],"downwards":[]}
{"upwards":["Quantifier rank","Examples"],"downwards":[]}
{"upwards":["Existential quantification"],"downwards":["Basics","Properties","As adjoint","Encoding","Origin"]}
{"upwards":["Existential quantification","Basics"],"downwards":[]}
{"upwards":["Existential quantification","Properties"],"downwards":["Negation","Rules of inference","The empty set"]}
{"upwards":["Existential quantification","Properties","Negation"],"downwards":[]}
{"upwards":["Existential quantification","Properties","Rules of inference"],"downwards":[]}
{"upwards":["Existential quantification","Properties","The empty set"],"downwards":[]}
{"upwards":["Existential quantification","As adjoint"],"downwards":[]}
{"upwards":["Existential quantification","Encoding"],"downwards":[]}
{"upwards":["Existential quantification","Origin"],"downwards":[]}
{"upwards":["Functional predicate"],"downwards":["Introducing new function symbols","Doing without functional predicates"]}
{"upwards":["Functional predicate","Introducing new function symbols"],"downwards":[]}
{"upwards":["Functional predicate","Doing without functional predicates"],"downwards":[]}
{"upwards":["Spectrum of a sentence"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Spectrum of a sentence","Definition"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Examples"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Properties"],"downwards":["Equivalence to Turing machines","Other properties"]}
{"upwards":["Spectrum of a sentence","Properties","Equivalence to Turing machines"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Properties","Other properties"],"downwards":[]}
{"upwards":["String (computer science)"],"downwards":["Purpose","History","String datatypes","Literal strings","Non-text strings","String processing algorithms","Character string-oriented languages and utilities","Character string functions","Formal theory"]}
{"upwards":["String (computer science)","Purpose"],"downwards":[]}
{"upwards":["String (computer science)","History"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes"],"downwards":["String length","Character encoding","Implementations","Representations","Security concerns"]}
{"upwards":["String (computer science)","String datatypes","String length"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Character encoding"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Implementations"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations"],"downwards":["Null-terminated","Byte- and bit-terminated","Length-prefixed","Strings as records","Other representations"]}
{"upwards":["String (computer science)","String datatypes","Representations","Null-terminated"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Byte- and bit-terminated"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Length-prefixed"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Strings as records"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Other representations"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Security concerns"],"downwards":[]}
{"upwards":["String (computer science)","Literal strings"],"downwards":[]}
{"upwards":["String (computer science)","Non-text strings"],"downwards":[]}
{"upwards":["String (computer science)","String processing algorithms"],"downwards":[]}
{"upwards":["String (computer science)","Character string-oriented languages and utilities"],"downwards":[]}
{"upwards":["String (computer science)","Character string functions"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory"],"downwards":["Concatenation and substrings","Prefixes and suffixes","Reversal","Rotations","Lexicographical ordering","String operations","Topology"]}
{"upwards":["String (computer science)","Formal theory","Concatenation and substrings"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Prefixes and suffixes"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Reversal"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Rotations"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Lexicographical ordering"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","String operations"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Topology"],"downwards":[]}
{"upwards":["Signature (logic)"],"downwards":["Definition","Other conventions","Use of signatures in logic and algebra","Many-sorted signatures"]}
{"upwards":["Signature (logic)","Definition"],"downwards":[]}
{"upwards":["Signature (logic)","Other conventions"],"downwards":[]}
{"upwards":["Signature (logic)","Use of signatures in logic and algebra"],"downwards":[]}
{"upwards":["Signature (logic)","Many-sorted signatures"],"downwards":["Symbol types","Signature"]}
{"upwards":["Signature (logic)","Many-sorted signatures","Symbol types"],"downwards":[]}
{"upwards":["Signature (logic)","Many-sorted signatures","Signature"],"downwards":[]}
{"upwards":["List of mathematical theories"],"downwards":[]}
{"upwards":["Non-logical symbol"],"downwards":["Signatures","Models","Informal semantics","Descriptive signs"]}
{"upwards":["Non-logical symbol","Signatures"],"downwards":[]}
{"upwards":["Non-logical symbol","Models"],"downwards":[]}
{"upwards":["Non-logical symbol","Informal semantics"],"downwards":[]}
{"upwards":["Non-logical symbol","Descriptive signs"],"downwards":[]}
{"upwards":["Logical constant"],"downwards":[]}
{"upwards":["Term (logic)"],"downwards":["Formal definition","Operations with terms","Related concepts"]}
{"upwards":["Term (logic)","Formal definition"],"downwards":["Term structure vs. representation","Structural equality","Ground and linear terms","Building formulas from terms"]}
{"upwards":["Term (logic)","Formal definition","Term structure vs. representation"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Structural equality"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Ground and linear terms"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Building formulas from terms"],"downwards":[]}
{"upwards":["Term (logic)","Operations with terms"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts"],"downwards":["Sorted terms","Lambda terms"]}
{"upwards":["Term (logic)","Related concepts","Sorted terms"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts","Lambda terms"],"downwards":["Motivation","Definition"]}
{"upwards":["Term (logic)","Related concepts","Lambda terms","Motivation"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts","Lambda terms","Definition"],"downwards":[]}
{"upwards":["Uninterpreted function"],"downwards":["Example","Discussion"]}
{"upwards":["Uninterpreted function","Example"],"downwards":[]}
{"upwards":["Uninterpreted function","Discussion"],"downwards":[]}
{"upwards":["Symbol (formal)"],"downwards":["Overview","Can words be modeled as formal symbols?","References"]}
{"upwards":["Symbol (formal)","Overview"],"downwards":[]}
{"upwards":["Symbol (formal)","Can words be modeled as formal symbols?"],"downwards":[]}
{"upwards":["Symbol (formal)","References"],"downwards":[]}
{"upwards":["Variable (mathematics)"],"downwards":["History","Notation","Specific kinds of variables","Moduli spaces","Conventional variable names"]}
{"upwards":["Variable (mathematics)","History"],"downwards":[]}
{"upwards":["Variable (mathematics)","Notation"],"downwards":[]}
{"upwards":["Variable (mathematics)","Specific kinds of variables"],"downwards":["Dependent and independent variables","Examples"]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Dependent and independent variables"],"downwards":[]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Examples"],"downwards":["Example: the ideal gas law"]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Examples","Example: the ideal gas law"],"downwards":[]}
{"upwards":["Variable (mathematics)","Moduli spaces"],"downwards":[]}
{"upwards":["Variable (mathematics)","Conventional variable names"],"downwards":[]}
{"upwards":["Axiomatic system"],"downwards":["Properties","Relative consistency","Models","Axiomatic method"]}
{"upwards":["Axiomatic system","Properties"],"downwards":[]}
{"upwards":["Axiomatic system","Relative consistency"],"downwards":[]}
{"upwards":["Axiomatic system","Models"],"downwards":["Example"]}
{"upwards":["Axiomatic system","Models","Example"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method"],"downwards":["History","Issues","Example: The Peano axiomatization of natural numbers","Axiomatization"]}
{"upwards":["Axiomatic system","Axiomatic method","History"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Issues"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Example: The Peano axiomatization of natural numbers"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Axiomatization"],"downwards":[]}
{"upwards":["Substitution (logic)"],"downwards":["Propositional logic","First-order logic","Algebra"]}
{"upwards":["Substitution (logic)","Propositional logic"],"downwards":["Definition","Tautologies"]}
{"upwards":["Substitution (logic)","Propositional logic","Definition"],"downwards":[]}
{"upwards":["Substitution (logic)","Propositional logic","Tautologies"],"downwards":[]}
{"upwards":["Substitution (logic)","First-order logic"],"downwards":[]}
{"upwards":["Substitution (logic)","Algebra"],"downwards":[]}
{"upwards":["Minimal axioms for Boolean algebra"],"downwards":["References"]}
{"upwards":["Minimal axioms for Boolean algebra","References"],"downwards":[]}
{"upwards":["Euclid's Elements"],"downwards":["History","Influence","Contents","Euclid's method and style of presentation","Criticism","Apocrypha","Editions","References","External links"]}
{"upwards":["Euclid's Elements","History"],"downwards":["Basis in earlier work","Transmission of the text"]}
{"upwards":["Euclid's Elements","History","Basis in earlier work"],"downwards":[]}
{"upwards":["Euclid's Elements","History","Transmission of the text"],"downwards":[]}
{"upwards":["Euclid's Elements","Influence"],"downwards":["In modern mathematics"]}
{"upwards":["Euclid's Elements","Influence","In modern mathematics"],"downwards":[]}
{"upwards":["Euclid's Elements","Contents"],"downwards":[]}
{"upwards":["Euclid's Elements","Euclid's method and style of presentation"],"downwards":[]}
{"upwards":["Euclid's Elements","Criticism"],"downwards":[]}
{"upwards":["Euclid's Elements","Apocrypha"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions"],"downwards":["Translations","Currently in print","Free versions"]}
{"upwards":["Euclid's Elements","Editions","Translations"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions","Currently in print"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions","Free versions"],"downwards":[]}
{"upwards":["Euclid's Elements","References"],"downwards":["Notes","Citations","Sources"]}
{"upwards":["Euclid's Elements","References","Notes"],"downwards":[]}
{"upwards":["Euclid's Elements","References","Citations"],"downwards":[]}
{"upwards":["Euclid's Elements","References","Sources"],"downwards":[]}
{"upwards":["Euclid's Elements","External links"],"downwards":[]}
{"upwards":["Construction of the real numbers"],"downwards":["Axiomatic definitions","Explicit constructions of models"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions"],"downwards":["Axioms","Tarski's axiomatization of the reals"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms"],"downwards":["On the least upper bound property","On models"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms","On the least upper bound property"],"downwards":[]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms","On models"],"downwards":[]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Tarski's axiomatization of the reals"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models"],"downwards":["Construction from Cauchy sequences","Construction by Dedekind cuts","Construction using hyperreal numbers","Construction from surreal numbers","Construction from integers (Eudoxus reals)","Other constructions"]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from Cauchy sequences"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction by Dedekind cuts"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction using hyperreal numbers"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from surreal numbers"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from integers (Eudoxus reals)"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Other constructions"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined"],"downwards":["Definition","Basis","Truth tables","Examples","Boolean algebras of Boolean operations","Axiomatizing Boolean algebras","Underlying lattice structure","Boolean homomorphisms","Infinitary extensions","Other definitions of Boolean algebra"]}
{"upwards":["Boolean algebras canonically defined","Definition"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Basis"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Truth tables"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples"],"downwards":["Bit vectors","Power set algebra","Representation theorems","Other examples"]}
{"upwards":["Boolean algebras canonically defined","Examples","Bit vectors"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Power set algebra"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Representation theorems"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Other examples"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Boolean algebras of Boolean operations"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Axiomatizing Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Underlying lattice structure"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Boolean homomorphisms"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Infinitary extensions"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Other definitions of Boolean algebra"],"downwards":[]}
{"upwards":["Euclidean geometry"],"downwards":["The Elements","Notation and terminology","Some important or well known results","System of measurement and arithmetic","Applications","Later history","As a description of the structure of space","Treatment of infinity","Logical basis"]}
{"upwards":["Euclidean geometry","The Elements"],"downwards":["Axioms","Parallel postulate","Methods of proof"]}
{"upwards":["Euclidean geometry","The Elements","Axioms"],"downwards":[]}
{"upwards":["Euclidean geometry","The Elements","Parallel postulate"],"downwards":[]}
{"upwards":["Euclidean geometry","The Elements","Methods of proof"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology"],"downwards":["Naming of points and figures","Complementary and supplementary angles","Modern versions of Euclid's notation"]}
{"upwards":["Euclidean geometry","Notation and terminology","Naming of points and figures"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology","Complementary and supplementary angles"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology","Modern versions of Euclid's notation"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results"],"downwards":["Pons asinorum","Congruence of triangles","Triangle angle sum","Pythagorean theorem","Thales' theorem","Scaling of area and volume"]}
{"upwards":["Euclidean geometry","Some important or well known results","Pons asinorum"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Congruence of triangles"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Triangle angle sum"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Pythagorean theorem"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Thales' theorem"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Scaling of area and volume"],"downwards":[]}
{"upwards":["Euclidean geometry","System of measurement and arithmetic"],"downwards":[]}
{"upwards":["Euclidean geometry","Applications"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history"],"downwards":["Archimedes and Apollonius","17th century: Descartes","18th century","19th century","20th century and relativity"]}
{"upwards":["Euclidean geometry","Later history","Archimedes and Apollonius"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","17th century: Descartes"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","18th century"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","19th century"],"downwards":["Higher dimensions","Non-Euclidean geometry"]}
{"upwards":["Euclidean geometry","Later history","19th century","Higher dimensions"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","19th century","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","20th century and relativity"],"downwards":[]}
{"upwards":["Euclidean geometry","As a description of the structure of space"],"downwards":[]}
{"upwards":["Euclidean geometry","Treatment of infinity"],"downwards":["Infinite objects","Infinite processes"]}
{"upwards":["Euclidean geometry","Treatment of infinity","Infinite objects"],"downwards":[]}
{"upwards":["Euclidean geometry","Treatment of infinity","Infinite processes"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis"],"downwards":["Classical logic","Modern standards of rigor","Axiomatic formulations"]}
{"upwards":["Euclidean geometry","Logical basis","Classical logic"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis","Modern standards of rigor"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis","Axiomatic formulations"],"downwards":[]}
{"upwards":["Skolem arithmetic"],"downwards":["Expressive power","Idea of decidability","Complexity","Decidable extensions","Undecidable extensions"]}
{"upwards":["Skolem arithmetic","Expressive power"],"downwards":[]}
{"upwards":["Skolem arithmetic","Idea of decidability"],"downwards":[]}
{"upwards":["Skolem arithmetic","Complexity"],"downwards":[]}
{"upwards":["Skolem arithmetic","Decidable extensions"],"downwards":[]}
{"upwards":["Skolem arithmetic","Undecidable extensions"],"downwards":[]}
{"upwards":["True arithmetic"],"downwards":["Definition","Arithmetic undefinability","Computability properties","Model-theoretic properties","True theory of second-order arithmetic","Notes","References","External links"]}
{"upwards":["True arithmetic","Definition"],"downwards":[]}
{"upwards":["True arithmetic","Arithmetic undefinability"],"downwards":[]}
{"upwards":["True arithmetic","Computability properties"],"downwards":[]}
{"upwards":["True arithmetic","Model-theoretic properties"],"downwards":[]}
{"upwards":["True arithmetic","True theory of second-order arithmetic"],"downwards":[]}
{"upwards":["True arithmetic","Notes"],"downwards":[]}
{"upwards":["True arithmetic","References"],"downwards":[]}
{"upwards":["True arithmetic","External links"],"downwards":[]}
{"upwards":["Boolean algebra (structure)"],"downwards":["History","Definition","Examples","Homomorphisms and isomorphisms","Boolean rings","Ideals and filters","Representations","Axiomatics","Generalizations"]}
{"upwards":["Boolean algebra (structure)","History"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Definition"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Examples"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Homomorphisms and isomorphisms"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Boolean rings"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Ideals and filters"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Representations"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Axiomatics"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Generalizations"],"downwards":[]}
{"upwards":["Peano axioms"],"downwards":["Historical second-order formulation","Peano arithmetic as first-order theory"]}
{"upwards":["Peano axioms","Historical second-order formulation"],"downwards":["Defining arithmetic operations and relations","Models","Consistency"]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations"],"downwards":["Addition","Multiplication","Inequalities"]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Addition"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Multiplication"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Inequalities"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Models"],"downwards":["Set-theoretic models","Interpretation in category theory"]}
{"upwards":["Peano axioms","Historical second-order formulation","Models","Set-theoretic models"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Models","Interpretation in category theory"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Consistency"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory"],"downwards":["Equivalent axiomatizations","Undecidability and incompleteness","Nonstandard models"]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Equivalent axiomatizations"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Undecidability and incompleteness"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Nonstandard models"],"downwards":["Overspill"]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Nonstandard models","Overspill"],"downwards":[]}
{"upwards":["Hilbert's axioms"],"downwards":["The axioms","Hilbert's discarded axiom","Editions and translations of Grundlagen der Geometrie","Application"]}
{"upwards":["Hilbert's axioms","The axioms"],"downwards":["I. Incidence","II. Order","III. Congruence","IV. Parallels","V. Continuity"]}
{"upwards":["Hilbert's axioms","The axioms","I. Incidence"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","II. Order"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","III. Congruence"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","IV. Parallels"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","V. Continuity"],"downwards":[]}
{"upwards":["Hilbert's axioms","Hilbert's discarded axiom"],"downwards":[]}
{"upwards":["Hilbert's axioms","Editions and translations of Grundlagen der Geometrie"],"downwards":[]}
{"upwards":["Hilbert's axioms","Application"],"downwards":[]}
{"upwards":["Foundations of geometry"],"downwards":["Axiomatic systems","Euclidean geometry","Non-Euclidean geometry","Other geometries"]}
{"upwards":["Foundations of geometry","Axiomatic systems"],"downwards":["Properties of axiomatic systems"]}
{"upwards":["Foundations of geometry","Axiomatic systems","Properties of axiomatic systems"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry"],"downwards":["Euclid's Elements","A critique of Euclid","Pasch and Peano","Pieri and the Italian school of geometers","Hilbert's axioms","Birkhoff's axioms","School geometry","Other systems"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Euclid's Elements"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","A critique of Euclid"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Pasch and Peano"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Pieri and the Italian school of geometers"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms"],"downwards":["Changes in Hilbert's axioms","Consistency and independence"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms","Changes in Hilbert's axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms","Consistency and independence"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Birkhoff's axioms"],"downwards":["Postulates"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Birkhoff's axioms","Postulates"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry"],"downwards":["Mac Lane's axioms","SMSG (School Mathematics Study Group) axioms","UCSMP (University of Chicago School Mathematics Project) axioms"]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","Mac Lane's axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","SMSG (School Mathematics Study Group) axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","UCSMP (University of Chicago School Mathematics Project) axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Other systems"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry"],"downwards":["Euclid's parallel postulate","Neutral (or absolute) geometry","Hyperbolic geometry","Elliptic geometry"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Euclid's parallel postulate"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry"],"downwards":["Relation to other geometries","Incompleteness"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry","Relation to other geometries"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry","Incompleteness"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Hyperbolic geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Elliptic geometry"],"downwards":["Spherical geometry"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Elliptic geometry","Spherical geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries"],"downwards":["Projective geometry","Affine geometry","Ordered geometry","Finite geometry"]}
{"upwards":["Foundations of geometry","Other geometries","Projective geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Affine geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Ordered geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Finite geometry"],"downwards":[]}
{"upwards":["Second-order arithmetic"],"downwards":["Definition","Models","Subsystems","Projective determinacy","Coding mathematics"]}
{"upwards":["Second-order arithmetic","Definition"],"downwards":["Syntax","Semantics","Axioms","The full system"]}
{"upwards":["Second-order arithmetic","Definition","Syntax"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Semantics"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Axioms"],"downwards":["Basic","Induction and comprehension schema"]}
{"upwards":["Second-order arithmetic","Definition","Axioms","Basic"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Axioms","Induction and comprehension schema"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","The full system"],"downwards":[]}
{"upwards":["Second-order arithmetic","Models"],"downwards":["Definable functions","More types of models"]}
{"upwards":["Second-order arithmetic","Models","Definable functions"],"downwards":[]}
{"upwards":["Second-order arithmetic","Models","More types of models"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems"],"downwards":["Arithmetical comprehension","The arithmetical hierarchy for formulas","Recursive comprehension","Weaker systems","Stronger systems"]}
{"upwards":["Second-order arithmetic","Subsystems","Arithmetical comprehension"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","The arithmetical hierarchy for formulas"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Recursive comprehension"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Weaker systems"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Stronger systems"],"downwards":[]}
{"upwards":["Second-order arithmetic","Projective determinacy"],"downwards":[]}
{"upwards":["Second-order arithmetic","Coding mathematics"],"downwards":[]}
{"upwards":["Elementary function arithmetic"],"downwards":["Definition","Friedman's grand conjecture","Related systems"]}
{"upwards":["Elementary function arithmetic","Definition"],"downwards":[]}
{"upwards":["Elementary function arithmetic","Friedman's grand conjecture"],"downwards":[]}
{"upwards":["Elementary function arithmetic","Related systems"],"downwards":[]}
{"upwards":["Tarski's axiomatization of the reals"],"downwards":["The axioms","How these axioms imply a field"]}
{"upwards":["Tarski's axiomatization of the reals","The axioms"],"downwards":[]}
{"upwards":["Tarski's axiomatization of the reals","How these axioms imply a field"],"downwards":[]}
{"upwards":["Robinson arithmetic"],"downwards":["Axioms","Metamathematics"]}
{"upwards":["Robinson arithmetic","Axioms"],"downwards":["Variant axiomatizations"]}
{"upwards":["Robinson arithmetic","Axioms","Variant axiomatizations"],"downwards":[]}
{"upwards":["Robinson arithmetic","Metamathematics"],"downwards":[]}
{"upwards":["Tarski's axioms"],"downwards":["Overview","The axioms","Discussion","Comparison with Hilbert"]}
{"upwards":["Tarski's axioms","Overview"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms"],"downwards":["Fundamental relations","Congruence axioms","Betweenness axioms","Congruence and betweenness"]}
{"upwards":["Tarski's axioms","The axioms","Fundamental relations"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Congruence axioms"],"downwards":["Commentary"]}
{"upwards":["Tarski's axioms","The axioms","Congruence axioms","Commentary"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Betweenness axioms"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Congruence and betweenness"],"downwards":[]}
{"upwards":["Tarski's axioms","Discussion"],"downwards":[]}
{"upwards":["Tarski's axioms","Comparison with Hilbert"],"downwards":[]}
{"upwards":["Proof theory"],"downwards":["History","Structural proof theory","Ordinal analysis","Provability logic","Reverse mathematics","Functional interpretations","Formal and informal proof","Proof-theoretic semantics"]}
{"upwards":["Proof theory","History"],"downwards":[]}
{"upwards":["Proof theory","Structural proof theory"],"downwards":[]}
{"upwards":["Proof theory","Ordinal analysis"],"downwards":[]}
{"upwards":["Proof theory","Provability logic"],"downwards":[]}
{"upwards":["Proof theory","Reverse mathematics"],"downwards":[]}
{"upwards":["Proof theory","Functional interpretations"],"downwards":[]}
{"upwards":["Proof theory","Formal and informal proof"],"downwards":[]}
{"upwards":["Proof theory","Proof-theoretic semantics"],"downwards":[]}
{"upwards":["Non-Euclidean geometry"],"downwards":["Principles","History","Axiomatic basis of non-Euclidean geometry","Models","Uncommon properties","Importance","Planar algebras","Kinematic geometries","Fiction"]}
{"upwards":["Non-Euclidean geometry","Principles"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History"],"downwards":["Background","Discovery of non-Euclidean geometry","Terminology"]}
{"upwards":["Non-Euclidean geometry","History","Background"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History","Discovery of non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History","Terminology"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Axiomatic basis of non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models"],"downwards":["Elliptic geometry","Hyperbolic geometry","Three-dimensional non-Euclidean geometry"]}
{"upwards":["Non-Euclidean geometry","Models","Elliptic geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models","Hyperbolic geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models","Three-dimensional non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Uncommon properties"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Importance"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Planar algebras"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Kinematic geometries"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Fiction"],"downwards":[]}
{"upwards":["Proof of impossibility"],"downwards":["Types of proof","Types of disproof","Pythagorean proof of the existence of irrational numbers","Impossible constructions sought by the ancient Greeks","Euclid's parallel postulate","Fermat's Last Theorem","Richard's paradox","Incompleteness of axiomatic systems: G\u00f6del's proof","Halting problem: Turing's first proof","String compressibility: Chaitin's proof","Integer solutions of Diophantine equations: Hilbert's tenth problem","In social science","In natural science"]}
{"upwards":["Proof of impossibility","Types of proof"],"downwards":["By contradiction"]}
{"upwards":["Proof of impossibility","Types of proof","By contradiction"],"downwards":["By descent"]}
{"upwards":["Proof of impossibility","Types of proof","By contradiction","By descent"],"downwards":[]}
{"upwards":["Proof of impossibility","Types of disproof"],"downwards":[]}
{"upwards":["Proof of impossibility","Pythagorean proof of the existence of irrational numbers"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks"],"downwards":["Angle trisection and doubling the cube","Squaring the circle","Constructing an equilateral n-gon"]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Angle trisection and doubling the cube"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Squaring the circle"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Constructing an equilateral n-gon"],"downwards":[]}
{"upwards":["Proof of impossibility","Euclid's parallel postulate"],"downwards":[]}
{"upwards":["Proof of impossibility","Fermat's Last Theorem"],"downwards":[]}
{"upwards":["Proof of impossibility","Richard's paradox"],"downwards":[]}
{"upwards":["Proof of impossibility","Incompleteness of axiomatic systems: G\u00f6del's proof"],"downwards":[]}
{"upwards":["Proof of impossibility","Halting problem: Turing's first proof"],"downwards":[]}
{"upwards":["Proof of impossibility","String compressibility: Chaitin's proof"],"downwards":[]}
{"upwards":["Proof of impossibility","Integer solutions of Diophantine equations: Hilbert's tenth problem"],"downwards":[]}
{"upwards":["Proof of impossibility","In social science"],"downwards":[]}
{"upwards":["Proof of impossibility","In natural science"],"downwards":[]}
{"upwards":["Ordinal analysis"],"downwards":["History","Definition","Upper bound","Examples","Table of ordinal analyses"]}
{"upwards":["Ordinal analysis","History"],"downwards":[]}
{"upwards":["Ordinal analysis","Definition"],"downwards":["Ordinal notations"]}
{"upwards":["Ordinal analysis","Definition","Ordinal notations"],"downwards":[]}
{"upwards":["Ordinal analysis","Upper bound"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples"],"downwards":["Theories with proof-theoretic ordinal \u03c9","Theories with proof-theoretic ordinal \u03c92","Theories with proof-theoretic ordinal \u03c93","Theories with proof-theoretic ordinal \u03c9n (for n = 2, 3, ... \u03c9)","Theories with proof-theoretic ordinal \u03c9\u03c9","Theories with proof-theoretic ordinal \u03b50","Theories with proof-theoretic ordinal the Feferman\u2013Sch\u00fctte ordinal \u03930","Theories with proof-theoretic ordinal the Bachmann\u2013Howard ordinal","Theories with larger proof-theoretic ordinals"]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c92"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c93"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9n (for n = 2, 3, ... \u03c9)"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9\u03c9"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03b50"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal the Feferman\u2013Sch\u00fctte ordinal \u03930"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal the Bachmann\u2013Howard ordinal"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with larger proof-theoretic ordinals"],"downwards":[]}
{"upwards":["Ordinal analysis","Table of ordinal analyses"],"downwards":["Key"]}
{"upwards":["Ordinal analysis","Table of ordinal analyses","Key"],"downwards":[]}
{"upwards":["Reverse mathematics"],"downwards":["General principles","The big five subsystems of second-order arithmetic","Additional systems","\u03c9-models and \u03b2-models"]}
{"upwards":["Reverse mathematics","General principles"],"downwards":["Use of second-order arithmetic","Use of higher-order arithmetic"]}
{"upwards":["Reverse mathematics","General principles","Use of second-order arithmetic"],"downwards":[]}
{"upwards":["Reverse mathematics","General principles","Use of higher-order arithmetic"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic"],"downwards":["The base system RCA0","Weak K\u0151nig's lemma WKL0","Arithmetical comprehension ACA0","Arithmetical transfinite recursion ATR0","\u03a011 comprehension \u03a011-CA0"]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","The base system RCA0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Weak K\u0151nig's lemma WKL0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Arithmetical comprehension ACA0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Arithmetical transfinite recursion ATR0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","\u03a011 comprehension \u03a011-CA0"],"downwards":[]}
{"upwards":["Reverse mathematics","Additional systems"],"downwards":["Stronger Systems"]}
{"upwards":["Reverse mathematics","Additional systems","Stronger Systems"],"downwards":[]}
{"upwards":["Reverse mathematics","\u03c9-models and \u03b2-models"],"downwards":[]}
{"upwards":["List of statements independent of ZFC"],"downwards":["Axiomatic set theory","Set theory of the real line","Order theory","Abstract algebra","Number theory","Measure theory","Topology","Functional analysis","Model theory","Computability theory","References","External links"]}
{"upwards":["List of statements independent of ZFC","Axiomatic set theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Set theory of the real line"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Order theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Abstract algebra"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Number theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Measure theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Topology"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Functional analysis"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Model theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Computability theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","References"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","External links"],"downwards":[]}
{"upwards":["Complete theory"],"downwards":["Examples"]}
{"upwards":["Complete theory","Examples"],"downwards":[]}
{"upwards":["List of Hilbert systems"],"downwards":["Classical propositional calculus systems","Implicational propositional calculus","Intuitionistic and intermediate logics","Positive implicational calculus","Positive propositional calculus","Equivalential calculus"]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems"],"downwards":["Implication and negation","Implication and falsum","Negation and disjunction","Sheffer's stroke"]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Implication and negation"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Implication and falsum"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Negation and disjunction"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Sheffer's stroke"],"downwards":[]}
{"upwards":["List of Hilbert systems","Implicational propositional calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Intuitionistic and intermediate logics"],"downwards":[]}
{"upwards":["List of Hilbert systems","Positive implicational calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Positive propositional calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Equivalential calculus"],"downwards":[]}
{"upwards":["Model theory"],"downwards":["Overview","Fundamental notions of first-order model theory","Definability","Types","Constructing models","Categoricity","Stability theory","Non-elementary model theory","Selected applications","History","Connections to related branches of mathematical logic"]}
{"upwards":["Model theory","Overview"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory"],"downwards":["First-order logic","Basic model-theoretic concepts","Compactness and the L\u00f6wenheim-Skolem theorem"]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","First-order logic"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","Basic model-theoretic concepts"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","Compactness and the L\u00f6wenheim-Skolem theorem"],"downwards":[]}
{"upwards":["Model theory","Definability"],"downwards":["Definable sets","Eliminating quantifiers","Minimality","Definable and interpretable structures"]}
{"upwards":["Model theory","Definability","Definable sets"],"downwards":[]}
{"upwards":["Model theory","Definability","Eliminating quantifiers"],"downwards":[]}
{"upwards":["Model theory","Definability","Minimality"],"downwards":[]}
{"upwards":["Model theory","Definability","Definable and interpretable structures"],"downwards":[]}
{"upwards":["Model theory","Types"],"downwards":["Basic notions","Structures and types","Stone spaces"]}
{"upwards":["Model theory","Types","Basic notions"],"downwards":[]}
{"upwards":["Model theory","Types","Structures and types"],"downwards":[]}
{"upwards":["Model theory","Types","Stone spaces"],"downwards":[]}
{"upwards":["Model theory","Constructing models"],"downwards":["Realising and omitting types","Ultraproducts"]}
{"upwards":["Model theory","Constructing models","Realising and omitting types"],"downwards":[]}
{"upwards":["Model theory","Constructing models","Ultraproducts"],"downwards":[]}
{"upwards":["Model theory","Categoricity"],"downwards":["\u03c9-categoricity","Uncountable categoricity"]}
{"upwards":["Model theory","Categoricity","\u03c9-categoricity"],"downwards":[]}
{"upwards":["Model theory","Categoricity","Uncountable categoricity"],"downwards":[]}
{"upwards":["Model theory","Stability theory"],"downwards":["The stability hierarchy","Geometric stability theory"]}
{"upwards":["Model theory","Stability theory","The stability hierarchy"],"downwards":[]}
{"upwards":["Model theory","Stability theory","Geometric stability theory"],"downwards":[]}
{"upwards":["Model theory","Non-elementary model theory"],"downwards":[]}
{"upwards":["Model theory","Selected applications"],"downwards":[]}
{"upwards":["Model theory","History"],"downwards":[]}
{"upwards":["Model theory","Connections to related branches of mathematical logic"],"downwards":["Finite model theory","Set theory"]}
{"upwards":["Model theory","Connections to related branches of mathematical logic","Finite model theory"],"downwards":[]}
{"upwards":["Model theory","Connections to related branches of mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Self-verifying theories"],"downwards":["References","External links"]}
{"upwards":["Self-verifying theories","References"],"downwards":[]}
{"upwards":["Self-verifying theories","External links"],"downwards":[]}
{"upwards":["Interpretation (logic)"],"downwards":["Formal languages","General properties of truth-functional interpretations","Interpretation of a theory","Interpretations for propositional logic","First-order logic","Higher-order predicate logics","Non-classical interpretations","Intended interpretations","Other concepts of interpretation"]}
{"upwards":["Interpretation (logic)","Formal languages"],"downwards":["Example","Logical constants"]}
{"upwards":["Interpretation (logic)","Formal languages","Example"],"downwards":[]}
{"upwards":["Interpretation (logic)","Formal languages","Logical constants"],"downwards":[]}
{"upwards":["Interpretation (logic)","General properties of truth-functional interpretations"],"downwards":["Logical connectives"]}
{"upwards":["Interpretation (logic)","General properties of truth-functional interpretations","Logical connectives"],"downwards":[]}
{"upwards":["Interpretation (logic)","Interpretation of a theory"],"downwards":[]}
{"upwards":["Interpretation (logic)","Interpretations for propositional logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic"],"downwards":["Formal languages for first-order logic","Interpretations of a first-order language","Example of a first-order interpretation","Non-empty domain requirement","Interpreting equality","Many-sorted first-order logic"]}
{"upwards":["Interpretation (logic)","First-order logic","Formal languages for first-order logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Interpretations of a first-order language"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Example of a first-order interpretation"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Non-empty domain requirement"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Interpreting equality"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Many-sorted first-order logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","Higher-order predicate logics"],"downwards":[]}
{"upwards":["Interpretation (logic)","Non-classical interpretations"],"downwards":[]}
{"upwards":["Interpretation (logic)","Intended interpretations"],"downwards":["Example"]}
{"upwards":["Interpretation (logic)","Intended interpretations","Example"],"downwards":[]}
{"upwards":["Interpretation (logic)","Other concepts of interpretation"],"downwards":[]}
{"upwards":["Hilbert system"],"downwards":["Formal deductions","Conservative extensions","Metatheorems","Some useful theorems and their proofs","Alternative axiomatizations","Further connections"]}
{"upwards":["Hilbert system","Formal deductions"],"downwards":["Logical axioms"]}
{"upwards":["Hilbert system","Formal deductions","Logical axioms"],"downwards":[]}
{"upwards":["Hilbert system","Conservative extensions"],"downwards":["Existential quantification","Conjunction and disjunction"]}
{"upwards":["Hilbert system","Conservative extensions","Existential quantification"],"downwards":[]}
{"upwards":["Hilbert system","Conservative extensions","Conjunction and disjunction"],"downwards":[]}
{"upwards":["Hilbert system","Metatheorems"],"downwards":[]}
{"upwards":["Hilbert system","Some useful theorems and their proofs"],"downwards":[]}
{"upwards":["Hilbert system","Alternative axiomatizations"],"downwards":[]}
{"upwards":["Hilbert system","Further connections"],"downwards":[]}
{"upwards":["Formal system"],"downwards":["Background","History","Examples","Variants"]}
{"upwards":["Formal system","Background"],"downwards":["Recursive","Inference and entailment"]}
{"upwards":["Formal system","Background","Recursive"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment"],"downwards":["Formal language","Deductive system","Logical system"]}
{"upwards":["Formal system","Background","Inference and entailment","Formal language"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Deductive system"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Logical system"],"downwards":[]}
{"upwards":["Formal system","History"],"downwards":["Formalism","QED manifesto"]}
{"upwards":["Formal system","History","Formalism"],"downwards":["Hilbert's program"]}
{"upwards":["Formal system","History","Formalism","Hilbert's program"],"downwards":[]}
{"upwards":["Formal system","History","QED manifesto"],"downwards":[]}
{"upwards":["Formal system","Examples"],"downwards":[]}
{"upwards":["Formal system","Variants"],"downwards":["Proof system"]}
{"upwards":["Formal system","Variants","Proof system"],"downwards":[]}
{"upwards":["Sequent calculus"],"downwards":["Overview","Proving logical formulas","The system LK","Variants"]}
{"upwards":["Sequent calculus","Overview"],"downwards":["Hilbert-style deduction systems","Natural deduction systems","Sequent calculus systems","Distinction between natural deduction and sequent calculus","Origin of word \"sequent\""]}
{"upwards":["Sequent calculus","Overview","Hilbert-style deduction systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Natural deduction systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Sequent calculus systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Distinction between natural deduction and sequent calculus"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Origin of word \"sequent\""],"downwards":[]}
{"upwards":["Sequent calculus","Proving logical formulas"],"downwards":["Reduction trees","Relation to standard axiomatizations"]}
{"upwards":["Sequent calculus","Proving logical formulas","Reduction trees"],"downwards":[]}
{"upwards":["Sequent calculus","Proving logical formulas","Relation to standard axiomatizations"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK"],"downwards":["Inference rules","An intuitive explanation","Example derivations","Relation to analytic tableaux","Structural rules","Properties of the system LK"]}
{"upwards":["Sequent calculus","The system LK","Inference rules"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","An intuitive explanation"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Example derivations"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Relation to analytic tableaux"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Structural rules"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Properties of the system LK"],"downwards":[]}
{"upwards":["Sequent calculus","Variants"],"downwards":["Minor structural alternatives","Absurdity","Substructural logics","Intuitionistic sequent calculus: System LJ"]}
{"upwards":["Sequent calculus","Variants","Minor structural alternatives"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Absurdity"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Substructural logics"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Intuitionistic sequent calculus: System LJ"],"downwards":[]}
{"upwards":["Natural deduction"],"downwards":["Motivation","Judgments and propositions","Introduction and elimination","Hypothetical derivations","Consistency, completeness, and normal forms","First and higher-order extensions","Different presentations of natural deduction","Proofs and type theory","Classical and modal logics","Comparison with other foundational approaches"]}
{"upwards":["Natural deduction","Motivation"],"downwards":[]}
{"upwards":["Natural deduction","Judgments and propositions"],"downwards":[]}
{"upwards":["Natural deduction","Introduction and elimination"],"downwards":[]}
{"upwards":["Natural deduction","Hypothetical derivations"],"downwards":[]}
{"upwards":["Natural deduction","Consistency, completeness, and normal forms"],"downwards":[]}
{"upwards":["Natural deduction","First and higher-order extensions"],"downwards":[]}
{"upwards":["Natural deduction","Different presentations of natural deduction"],"downwards":["Tree-like presentations","Sequential presentations"]}
{"upwards":["Natural deduction","Different presentations of natural deduction","Tree-like presentations"],"downwards":[]}
{"upwards":["Natural deduction","Different presentations of natural deduction","Sequential presentations"],"downwards":[]}
{"upwards":["Natural deduction","Proofs and type theory"],"downwards":["Example: Dependent Type Theory"]}
{"upwards":["Natural deduction","Proofs and type theory","Example: Dependent Type Theory"],"downwards":[]}
{"upwards":["Natural deduction","Classical and modal logics"],"downwards":[]}
{"upwards":["Natural deduction","Comparison with other foundational approaches"],"downwards":["Sequent calculus"]}
{"upwards":["Natural deduction","Comparison with other foundational approaches","Sequent calculus"],"downwards":[]}
{"upwards":["Rule of inference"],"downwards":["Standard form","Example: Hilbert systems for two propositional logics","Admissibility and derivability"]}
{"upwards":["Rule of inference","Standard form"],"downwards":[]}
{"upwards":["Rule of inference","Example: Hilbert systems for two propositional logics"],"downwards":[]}
{"upwards":["Rule of inference","Admissibility and derivability"],"downwards":[]}
{"upwards":["Primitive recursive arithmetic"],"downwards":["Language and axioms","Logic-free calculus"]}
{"upwards":["Primitive recursive arithmetic","Language and axioms"],"downwards":[]}
{"upwards":["Primitive recursive arithmetic","Logic-free calculus"],"downwards":[]}
{"upwards":["Finite model theory"],"downwards":["Axiomatisability","Zero-one laws","Descriptive complexity theory","Applications","History","Citations","References","Further reading","External links"]}
{"upwards":["Finite model theory","Axiomatisability"],"downwards":["Characterisation of a single structure","Characterisation of a class of structures"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure"],"downwards":["Problem","Approach","Extension to a fixed number of structures","Extension to an infinite structure"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Problem"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Approach"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Extension to a fixed number of structures"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Extension to an infinite structure"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures"],"downwards":["Problem","Approach","Example"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Problem"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Approach"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Example"],"downwards":[]}
{"upwards":["Finite model theory","Zero-one laws"],"downwards":[]}
{"upwards":["Finite model theory","Descriptive complexity theory"],"downwards":[]}
{"upwards":["Finite model theory","Applications"],"downwards":["Database theory","Querying and search"]}
{"upwards":["Finite model theory","Applications","Database theory"],"downwards":[]}
{"upwards":["Finite model theory","Applications","Querying and search"],"downwards":[]}
{"upwards":["Finite model theory","History"],"downwards":[]}
{"upwards":["Finite model theory","Citations"],"downwards":[]}
{"upwards":["Finite model theory","References"],"downwards":[]}
{"upwards":["Finite model theory","Further reading"],"downwards":[]}
{"upwards":["Finite model theory","External links"],"downwards":[]}
{"upwards":["Non-standard model"],"downwards":["Existence","Importance"]}
{"upwards":["Non-standard model","Existence"],"downwards":[]}
{"upwards":["Non-standard model","Importance"],"downwards":[]}
{"upwards":["Spectrum of a theory"],"downwards":["Early results","List of possible spectra of a countable theory"]}
{"upwards":["Spectrum of a theory","Early results"],"downwards":[]}
{"upwards":["Spectrum of a theory","List of possible spectra of a countable theory"],"downwards":[]}
{"upwards":["Diagram (mathematical logic)"],"downwards":["Definition"]}
{"upwards":["Diagram (mathematical logic)","Definition"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic"],"downwards":["Existence","Structure of countable non-standard models","References"]}
{"upwards":["Non-standard model of arithmetic","Existence"],"downwards":["From the compactness theorem","From the incompleteness theorems","From an ultraproduct"]}
{"upwards":["Non-standard model of arithmetic","Existence","From the compactness theorem"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Existence","From the incompleteness theorems"],"downwards":["Arithmetic unsoundness for models with ~G true"]}
{"upwards":["Non-standard model of arithmetic","Existence","From the incompleteness theorems","Arithmetic unsoundness for models with ~G true"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Existence","From an ultraproduct"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Structure of countable non-standard models"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","References"],"downwards":["Citations","Sources"]}
{"upwards":["Non-standard model of arithmetic","References","Citations"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","References","Sources"],"downwards":[]}
{"upwards":["Substructure (mathematics)"],"downwards":["Definition","Example","As subobjects","Submodel"]}
{"upwards":["Substructure (mathematics)","Definition"],"downwards":[]}
{"upwards":["Substructure (mathematics)","Example"],"downwards":[]}
{"upwards":["Substructure (mathematics)","As subobjects"],"downwards":[]}
{"upwards":["Substructure (mathematics)","Submodel"],"downwards":[]}
{"upwards":["Saturated model"],"downwards":["Definition","Motivation","Examples","Relationship to prime models","Notes","References"]}
{"upwards":["Saturated model","Definition"],"downwards":[]}
{"upwards":["Saturated model","Motivation"],"downwards":[]}
{"upwards":["Saturated model","Examples"],"downwards":[]}
{"upwards":["Saturated model","Relationship to prime models"],"downwards":[]}
{"upwards":["Saturated model","Notes"],"downwards":[]}
{"upwards":["Saturated model","References"],"downwards":[]}
{"upwards":["Categorical theory"],"downwards":["History and motivation","Examples","Properties"]}
{"upwards":["Categorical theory","History and motivation"],"downwards":[]}
{"upwards":["Categorical theory","Examples"],"downwards":[]}
{"upwards":["Categorical theory","Properties"],"downwards":[]}
{"upwards":["Elementary diagram"],"downwards":["Definition"]}
{"upwards":["Elementary diagram","Definition"],"downwards":[]}
{"upwards":["Model complete theory"],"downwards":["Model companion and model completion","Examples","Non-examples","Sufficient condition for completeness of model-complete theories","Notes","References"]}
{"upwards":["Model complete theory","Model companion and model completion"],"downwards":[]}
{"upwards":["Model complete theory","Examples"],"downwards":[]}
{"upwards":["Model complete theory","Non-examples"],"downwards":[]}
{"upwards":["Model complete theory","Sufficient condition for completeness of model-complete theories"],"downwards":[]}
{"upwards":["Model complete theory","Notes"],"downwards":[]}
{"upwards":["Model complete theory","References"],"downwards":[]}
{"upwards":["Strength (mathematical logic)"],"downwards":[]}
{"upwards":["Truth"],"downwards":["Definition and etymology","Major theories","Formal theories","Folk beliefs","Notable views"]}
{"upwards":["Truth","Definition and etymology"],"downwards":[]}
{"upwards":["Truth","Major theories"],"downwards":["Substantive","Minimalist","Pluralist"]}
{"upwards":["Truth","Major theories","Substantive"],"downwards":["Correspondence","Coherence","Pragmatic","Constructivist","Consensus"]}
{"upwards":["Truth","Major theories","Substantive","Correspondence"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Coherence"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Pragmatic"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Constructivist"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Consensus"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist"],"downwards":["Deflationary","Performative","Redundancy and related","Philosophical skepticism"]}
{"upwards":["Truth","Major theories","Minimalist","Deflationary"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Performative"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Redundancy and related"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Philosophical skepticism"],"downwards":[]}
{"upwards":["Truth","Major theories","Pluralist"],"downwards":[]}
{"upwards":["Truth","Formal theories"],"downwards":["Logic","Mathematics","Tarski's semantics","Kripke's semantics"]}
{"upwards":["Truth","Formal theories","Logic"],"downwards":[]}
{"upwards":["Truth","Formal theories","Mathematics"],"downwards":[]}
{"upwards":["Truth","Formal theories","Tarski's semantics"],"downwards":[]}
{"upwards":["Truth","Formal theories","Kripke's semantics"],"downwards":[]}
{"upwards":["Truth","Folk beliefs"],"downwards":[]}
{"upwards":["Truth","Notable views"],"downwards":["Ancient Greek philosophy","Medieval philosophy","Modern philosophy","Theological views"]}
{"upwards":["Truth","Notable views","Ancient Greek philosophy"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy"],"downwards":["Avicenna (980\u20131037)","Aquinas (1225\u20131274)","Changing concepts of truth in the Middle Ages"]}
{"upwards":["Truth","Notable views","Medieval philosophy","Avicenna (980\u20131037)"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy","Aquinas (1225\u20131274)"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy","Changing concepts of truth in the Middle Ages"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy"],"downwards":["Kant (1724\u20131804)","Hegel (1770\u20131831)","Schopenhauer (1788\u20131860)","Kierkegaard (1813\u20131855)","Nietzsche (1844\u20131900)","Heidegger (1889\u20131976)","Sartre (1905\u20131980)","Camus (1913\u20131960)","Whitehead (1861\u20131947)","Peirce (1839\u20131914)","Nishida (1870\u20131945)","Fromm (1900\u20131980)","Foucault (1926\u20131984)","Baudrillard (1929\u20132007)","Other contemporary positions"]}
{"upwards":["Truth","Notable views","Modern philosophy","Kant (1724\u20131804)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Hegel (1770\u20131831)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Schopenhauer (1788\u20131860)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Kierkegaard (1813\u20131855)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Nietzsche (1844\u20131900)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Heidegger (1889\u20131976)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Sartre (1905\u20131980)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Camus (1913\u20131960)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Whitehead (1861\u20131947)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Peirce (1839\u20131914)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Nishida (1870\u20131945)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Fromm (1900\u20131980)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Foucault (1926\u20131984)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Baudrillard (1929\u20132007)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Other contemporary positions"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views"],"downwards":["Hinduism","Buddhism","Christianity"]}
{"upwards":["Truth","Notable views","Theological views","Hinduism"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views","Buddhism"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views","Christianity"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Satisfiability"],"downwards":["Reduction of validity to satisfiability","Propositional satisfiability for classical logic","Satisfiability in first-order logic","Satisfiability in model theory","Finite satisfiability","Numerical constraints"]}
{"upwards":["Satisfiability","Reduction of validity to satisfiability"],"downwards":[]}
{"upwards":["Satisfiability","Propositional satisfiability for classical logic"],"downwards":[]}
{"upwards":["Satisfiability","Satisfiability in first-order logic"],"downwards":[]}
{"upwards":["Satisfiability","Satisfiability in model theory"],"downwards":[]}
{"upwards":["Satisfiability","Finite satisfiability"],"downwards":[]}
{"upwards":["Satisfiability","Numerical constraints"],"downwards":[]}
{"upwards":["Truth value"],"downwards":["Computing","Classical logic","Intuitionistic and constructive logic","Multi-valued logic","Algebraic semantics","In other theories"]}
{"upwards":["Truth value","Computing"],"downwards":[]}
{"upwards":["Truth value","Classical logic"],"downwards":[]}
{"upwards":["Truth value","Intuitionistic and constructive logic"],"downwards":[]}
{"upwards":["Truth value","Multi-valued logic"],"downwards":[]}
{"upwards":["Truth value","Algebraic semantics"],"downwards":[]}
{"upwards":["Truth value","In other theories"],"downwards":[]}
{"upwards":["Transfer principle"],"downwards":["History","Transfer principle for the hyperreals","Generalizations of the concept of number","Differences between R and *R","Constructions of the hyperreals","Statement","Three examples"]}
{"upwards":["Transfer principle","History"],"downwards":[]}
{"upwards":["Transfer principle","Transfer principle for the hyperreals"],"downwards":["Example"]}
{"upwards":["Transfer principle","Transfer principle for the hyperreals","Example"],"downwards":[]}
{"upwards":["Transfer principle","Generalizations of the concept of number"],"downwards":[]}
{"upwards":["Transfer principle","Differences between R and *R"],"downwards":[]}
{"upwards":["Transfer principle","Constructions of the hyperreals"],"downwards":[]}
{"upwards":["Transfer principle","Statement"],"downwards":[]}
{"upwards":["Transfer principle","Three examples"],"downwards":[]}
{"upwards":["Truth predicate"],"downwards":["Languages which allow a truth predicate"]}
{"upwards":["Truth predicate","Languages which allow a truth predicate"],"downwards":[]}
{"upwards":["T-schema"],"downwards":["The inductive definition","Natural languages"]}
{"upwards":["T-schema","The inductive definition"],"downwards":[]}
{"upwards":["T-schema","Natural languages"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Type (model theory)"],"downwards":["Formal definition","Examples of types","Stone spaces","Omitting types theorem","References"]}
{"upwards":["Type (model theory)","Formal definition"],"downwards":[]}
{"upwards":["Type (model theory)","Examples of types"],"downwards":[]}
{"upwards":["Type (model theory)","Stone spaces"],"downwards":[]}
{"upwards":["Type (model theory)","Omitting types theorem"],"downwards":[]}
{"upwards":["Type (model theory)","References"],"downwards":[]}
{"upwards":["P versus NP problem"],"downwards":["Example","History","Context","NP-completeness","Harder problems","Problems in NP not known to be in P or NP-complete","Does P mean \"easy\"?","Reasons to believe P \u2260 NP or P = NP","Consequences of solution","Results about difficulty of proof","Logical characterizations","Polynomial-time algorithms","Formal definitions","Claimed solutions","Popular culture"]}
{"upwards":["P versus NP problem","Example"],"downwards":[]}
{"upwards":["P versus NP problem","History"],"downwards":[]}
{"upwards":["P versus NP problem","Context"],"downwards":[]}
{"upwards":["P versus NP problem","NP-completeness"],"downwards":[]}
{"upwards":["P versus NP problem","Harder problems"],"downwards":[]}
{"upwards":["P versus NP problem","Problems in NP not known to be in P or NP-complete"],"downwards":[]}
{"upwards":["P versus NP problem","Does P mean \"easy\"?"],"downwards":[]}
{"upwards":["P versus NP problem","Reasons to believe P \u2260 NP or P = NP"],"downwards":[]}
{"upwards":["P versus NP problem","Consequences of solution"],"downwards":["P = NP","P \u2260 NP"]}
{"upwards":["P versus NP problem","Consequences of solution","P = NP"],"downwards":[]}
{"upwards":["P versus NP problem","Consequences of solution","P \u2260 NP"],"downwards":[]}
{"upwards":["P versus NP problem","Results about difficulty of proof"],"downwards":[]}
{"upwards":["P versus NP problem","Logical characterizations"],"downwards":[]}
{"upwards":["P versus NP problem","Polynomial-time algorithms"],"downwards":[]}
{"upwards":["P versus NP problem","Formal definitions"],"downwards":["P and NP","NP-completeness"]}
{"upwards":["P versus NP problem","Formal definitions","P and NP"],"downwards":["Example"]}
{"upwards":["P versus NP problem","Formal definitions","P and NP","Example"],"downwards":[]}
{"upwards":["P versus NP problem","Formal definitions","NP-completeness"],"downwards":[]}
{"upwards":["P versus NP problem","Claimed solutions"],"downwards":[]}
{"upwards":["P versus NP problem","Popular culture"],"downwards":[]}
{"upwards":["Kolmogorov complexity"],"downwards":["Definition","Invariance theorem","History and context","Basic results","Compression","Chaitin's incompleteness theorem","Minimum message length","Kolmogorov randomness","Relation to entropy","Conditional versions"]}
{"upwards":["Kolmogorov complexity","Definition"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Invariance theorem"],"downwards":["Informal treatment","A more formal treatment"]}
{"upwards":["Kolmogorov complexity","Invariance theorem","Informal treatment"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Invariance theorem","A more formal treatment"],"downwards":[]}
{"upwards":["Kolmogorov complexity","History and context"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Basic results"],"downwards":["Uncomputability of Kolmogorov complexity","Chain rule for Kolmogorov complexity"]}
{"upwards":["Kolmogorov complexity","Basic results","Uncomputability of Kolmogorov complexity"],"downwards":["A naive attempt at a program to compute K","Formal proof of uncomputability of K"]}
{"upwards":["Kolmogorov complexity","Basic results","Uncomputability of Kolmogorov complexity","A naive attempt at a program to compute K"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Basic results","Uncomputability of Kolmogorov complexity","Formal proof of uncomputability of K"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Basic results","Chain rule for Kolmogorov complexity"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Compression"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Chaitin's incompleteness theorem"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Minimum message length"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Kolmogorov randomness"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Relation to entropy"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Conditional versions"],"downwards":[]}
{"upwards":["Computably enumerable set"],"downwards":["Formal definition","Equivalent formulations","Examples","Properties","Remarks"]}
{"upwards":["Computably enumerable set","Formal definition"],"downwards":[]}
{"upwards":["Computably enumerable set","Equivalent formulations"],"downwards":[]}
{"upwards":["Computably enumerable set","Examples"],"downwards":[]}
{"upwards":["Computably enumerable set","Properties"],"downwards":[]}
{"upwards":["Computably enumerable set","Remarks"],"downwards":[]}
{"upwards":["NP (complexity)"],"downwards":["Formal definition","Background","Properties","Why some NP problems are hard to solve","Equivalence of definitions","Relationship to other classes","Other characterizations","Examples"]}
{"upwards":["NP (complexity)","Formal definition"],"downwards":[]}
{"upwards":["NP (complexity)","Background"],"downwards":["Verifier-based definition","Machine-definition"]}
{"upwards":["NP (complexity)","Background","Verifier-based definition"],"downwards":[]}
{"upwards":["NP (complexity)","Background","Machine-definition"],"downwards":[]}
{"upwards":["NP (complexity)","Properties"],"downwards":[]}
{"upwards":["NP (complexity)","Why some NP problems are hard to solve"],"downwards":[]}
{"upwards":["NP (complexity)","Equivalence of definitions"],"downwards":[]}
{"upwards":["NP (complexity)","Relationship to other classes"],"downwards":[]}
{"upwards":["NP (complexity)","Other characterizations"],"downwards":[]}
{"upwards":["NP (complexity)","Examples"],"downwards":["P","Integer factorization","NP-complete problems"]}
{"upwards":["NP (complexity)","Examples","P"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","Integer factorization"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","NP-complete problems"],"downwards":["Boolean satisfiability","Travelling salesman","Subgraph isomorphism"]}
{"upwards":["NP (complexity)","Examples","NP-complete problems","Boolean satisfiability"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","NP-complete problems","Travelling salesman"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","NP-complete problems","Subgraph isomorphism"],"downwards":[]}
{"upwards":["Church encoding"],"downwards":["Use","Church numerals","Church Booleans","Predicates","Church pairs","List encodings"]}
{"upwards":["Church encoding","Use"],"downwards":[]}
{"upwards":["Church encoding","Church numerals"],"downwards":["Calculation with Church numerals","Table of functions on Church numerals","Derivation of predecessor function","Division","Signed numbers","Plus and minus","Multiply and  divide","Rational and real numbers","Translation with other representations"]}
{"upwards":["Church encoding","Church numerals","Calculation with Church numerals"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Table of functions on Church numerals"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Derivation of predecessor function"],"downwards":["Another way of defining pred"]}
{"upwards":["Church encoding","Church numerals","Derivation of predecessor function","Another way of defining pred"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Division"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Signed numbers"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Plus and minus"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Multiply and  divide"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Rational and real numbers"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Translation with other representations"],"downwards":[]}
{"upwards":["Church encoding","Church Booleans"],"downwards":[]}
{"upwards":["Church encoding","Predicates"],"downwards":[]}
{"upwards":["Church encoding","Church pairs"],"downwards":[]}
{"upwards":["Church encoding","List encodings"],"downwards":["Two pairs as a list node","One pair as a list node","Represent the list using right fold","Represent the list using Scott encoding"]}
{"upwards":["Church encoding","List encodings","Two pairs as a list node"],"downwards":[]}
{"upwards":["Church encoding","List encodings","One pair as a list node"],"downwards":[]}
{"upwards":["Church encoding","List encodings","Represent the list using right fold"],"downwards":[]}
{"upwards":["Church encoding","List encodings","Represent the list using Scott encoding"],"downwards":[]}
{"upwards":["Ultraproduct"],"downwards":["Definition","Examples","\u0141o\u015b's theorem","Direct limits of ultrapowers (ultralimits)","Ultraproduct monad"]}
{"upwards":["Ultraproduct","Definition"],"downwards":["Ultrapower"]}
{"upwards":["Ultraproduct","Definition","Ultrapower"],"downwards":[]}
{"upwards":["Ultraproduct","Examples"],"downwards":[]}
{"upwards":["Ultraproduct","\u0141o\u015b's theorem"],"downwards":["Examples"]}
{"upwards":["Ultraproduct","\u0141o\u015b's theorem","Examples"],"downwards":[]}
{"upwards":["Ultraproduct","Direct limits of ultrapowers (ultralimits)"],"downwards":[]}
{"upwards":["Ultraproduct","Ultraproduct monad"],"downwards":[]}
{"upwards":["Computable function"],"downwards":["Definition","Characteristics of computable functions","Computable sets and relations","Formal languages","Examples","Church\u2013Turing thesis","Provability","Uncomputable functions and unsolvable problems","Extensions of computability"]}
{"upwards":["Computable function","Definition"],"downwards":[]}
{"upwards":["Computable function","Characteristics of computable functions"],"downwards":[]}
{"upwards":["Computable function","Computable sets and relations"],"downwards":[]}
{"upwards":["Computable function","Formal languages"],"downwards":[]}
{"upwards":["Computable function","Examples"],"downwards":[]}
{"upwards":["Computable function","Church\u2013Turing thesis"],"downwards":[]}
{"upwards":["Computable function","Provability"],"downwards":["Relation to recursively defined functions","Total functions that are not provably total"]}
{"upwards":["Computable function","Provability","Relation to recursively defined functions"],"downwards":[]}
{"upwards":["Computable function","Provability","Total functions that are not provably total"],"downwards":[]}
{"upwards":["Computable function","Uncomputable functions and unsolvable problems"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability"],"downwards":["Relative computability","Higher recursion theory","Hyper-computation"]}
{"upwards":["Computable function","Extensions of computability","Relative computability"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability","Higher recursion theory"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability","Hyper-computation"],"downwards":[]}
{"upwards":["Computability theory"],"downwards":["Introduction","Turing computability","Areas of research","Relationships between definability, proof and computability","Name","Professional organizations"]}
{"upwards":["Computability theory","Introduction"],"downwards":[]}
{"upwards":["Computability theory","Turing computability"],"downwards":[]}
{"upwards":["Computability theory","Areas of research"],"downwards":["Relative computability and the Turing degrees","Other reducibilities","Rice's theorem and the arithmetical hierarchy","Reverse mathematics","Numberings","The priority method","The lattice of computably enumerable sets","Automorphism problems","Kolmogorov complexity","Frequency computation","Inductive inference","Generalizations of Turing computability","Continuous computability theory"]}
{"upwards":["Computability theory","Areas of research","Relative computability and the Turing degrees"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Other reducibilities"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Rice's theorem and the arithmetical hierarchy"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Reverse mathematics"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Numberings"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","The priority method"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","The lattice of computably enumerable sets"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Automorphism problems"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Kolmogorov complexity"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Frequency computation"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Inductive inference"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Generalizations of Turing computability"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Continuous computability theory"],"downwards":[]}
{"upwards":["Computability theory","Relationships between definability, proof and computability"],"downwards":[]}
{"upwards":["Computability theory","Name"],"downwards":[]}
{"upwards":["Computability theory","Professional organizations"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis"],"downwards":["Statement in Church's and Turing's words","History","Success of the thesis","Informal usage in proofs","Variations","Philosophical implications","Non-computable functions"]}
{"upwards":["Church\u2013Turing thesis","Statement in Church's and Turing's words"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History"],"downwards":["Circa 1930\u20131952","Later developments","The thesis as a definition"]}
{"upwards":["Church\u2013Turing thesis","History","Circa 1930\u20131952"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History","Later developments"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History","The thesis as a definition"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Success of the thesis"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Informal usage in proofs"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Variations"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Philosophical implications"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Non-computable functions"],"downwards":[]}
{"upwards":["Decidability (logic)"],"downwards":["Decidability of a logical system","Decidability of a theory","Some decidable theories","Some undecidable theories","Semidecidability","Relationship with completeness","Relationship to computability","In context of games"]}
{"upwards":["Decidability (logic)","Decidability of a logical system"],"downwards":[]}
{"upwards":["Decidability (logic)","Decidability of a theory"],"downwards":[]}
{"upwards":["Decidability (logic)","Some decidable theories"],"downwards":[]}
{"upwards":["Decidability (logic)","Some undecidable theories"],"downwards":[]}
{"upwards":["Decidability (logic)","Semidecidability"],"downwards":[]}
{"upwards":["Decidability (logic)","Relationship with completeness"],"downwards":[]}
{"upwards":["Decidability (logic)","Relationship to computability"],"downwards":[]}
{"upwards":["Decidability (logic)","In context of games"],"downwards":[]}
{"upwards":["Undecidable problem"],"downwards":["Background","Example: the halting problem in computability theory","Relationship with G\u00f6del's incompleteness theorem","Examples of undecidable problems","Examples of undecidable statements"]}
{"upwards":["Undecidable problem","Background"],"downwards":[]}
{"upwards":["Undecidable problem","Example: the halting problem in computability theory"],"downwards":[]}
{"upwards":["Undecidable problem","Relationship with G\u00f6del's incompleteness theorem"],"downwards":[]}
{"upwards":["Undecidable problem","Examples of undecidable problems"],"downwards":[]}
{"upwards":["Undecidable problem","Examples of undecidable statements"],"downwards":[]}
{"upwards":["Primitive recursive function"],"downwards":["Definition","Examples","Use in first-order Peano arithmetic","Elimination of parameters","Relationship to recursive functions","Limitations","Variants","Finitism and consistency results","History"]}
{"upwards":["Primitive recursive function","Definition"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples"],"downwards":["Addition","Doubling","Multiplication","Predecessor","Truncated subtraction","Converting predicates to numeric functions","Predicate \"Is zero\"","Predicate \"Less or equal\"","Predicate \"Greater or equal\"","If-then-else","Junctors","Equality predicate","Other operations on natural numbers","Operations on integers and rational numbers","Some common primitive recursive functions"]}
{"upwards":["Primitive recursive function","Examples","Addition"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Doubling"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Multiplication"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predecessor"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Truncated subtraction"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Converting predicates to numeric functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predicate \"Is zero\""],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predicate \"Less or equal\""],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predicate \"Greater or equal\""],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","If-then-else"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Junctors"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Equality predicate"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Other operations on natural numbers"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Operations on integers and rational numbers"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Some common primitive recursive functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Use in first-order Peano arithmetic"],"downwards":[]}
{"upwards":["Primitive recursive function","Elimination of parameters"],"downwards":[]}
{"upwards":["Primitive recursive function","Relationship to recursive functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Limitations"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants"],"downwards":["Constant functions","Weak primitive recursion","Iterative functions","Additional primitive recursive forms","Computer language definition"]}
{"upwards":["Primitive recursive function","Variants","Constant functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Weak primitive recursion"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Iterative functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Additional primitive recursive forms"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Computer language definition"],"downwards":[]}
{"upwards":["Primitive recursive function","Finitism and consistency results"],"downwards":[]}
{"upwards":["Primitive recursive function","History"],"downwards":[]}
{"upwards":["Lambda calculus"],"downwards":["Explanation and applications","History","Informal description","Formal definition","Reduction","Normal forms and confluence","Encoding datatypes","Additional programming techniques","Typed lambda calculus","Reduction strategies","Computability","Complexity","Lambda calculus and programming languages","Semantics","Variations and extensions"]}
{"upwards":["Lambda calculus","Explanation and applications"],"downwards":[]}
{"upwards":["Lambda calculus","History"],"downwards":["Origin of the \u03bb symbol"]}
{"upwards":["Lambda calculus","History","Origin of the \u03bb symbol"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description"],"downwards":["Motivation","The lambda calculus"]}
{"upwards":["Lambda calculus","Informal description","Motivation"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus"],"downwards":["Lambda terms","Functions that operate on functions","Alpha equivalence","Free variables","Capture-avoiding substitutions","\u03b2-reduction"]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Lambda terms"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Functions that operate on functions"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Alpha equivalence"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Free variables"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Capture-avoiding substitutions"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","\u03b2-reduction"],"downwards":[]}
{"upwards":["Lambda calculus","Formal definition"],"downwards":["Definition","Notation","Free and bound variables"]}
{"upwards":["Lambda calculus","Formal definition","Definition"],"downwards":[]}
{"upwards":["Lambda calculus","Formal definition","Notation"],"downwards":[]}
{"upwards":["Lambda calculus","Formal definition","Free and bound variables"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction"],"downwards":["\u03b1-conversion","\u03b2-reduction","\u03b7-reduction"]}
{"upwards":["Lambda calculus","Reduction","\u03b1-conversion"],"downwards":["Substitution"]}
{"upwards":["Lambda calculus","Reduction","\u03b1-conversion","Substitution"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction","\u03b2-reduction"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction","\u03b7-reduction"],"downwards":[]}
{"upwards":["Lambda calculus","Normal forms and confluence"],"downwards":[]}
{"upwards":["Lambda calculus","Encoding datatypes"],"downwards":["Arithmetic in lambda calculus","Logic and predicates","Pairs"]}
{"upwards":["Lambda calculus","Encoding datatypes","Arithmetic in lambda calculus"],"downwards":[]}
{"upwards":["Lambda calculus","Encoding datatypes","Logic and predicates"],"downwards":[]}
{"upwards":["Lambda calculus","Encoding datatypes","Pairs"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques"],"downwards":["Named constants","Recursion and fixed points","Standard terms","Abstraction elimination"]}
{"upwards":["Lambda calculus","Additional programming techniques","Named constants"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques","Recursion and fixed points"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques","Standard terms"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques","Abstraction elimination"],"downwards":[]}
{"upwards":["Lambda calculus","Typed lambda calculus"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction strategies"],"downwards":[]}
{"upwards":["Lambda calculus","Computability"],"downwards":[]}
{"upwards":["Lambda calculus","Complexity"],"downwards":[]}
{"upwards":["Lambda calculus","Lambda calculus and programming languages"],"downwards":["Anonymous functions","Parallelism and concurrency"]}
{"upwards":["Lambda calculus","Lambda calculus and programming languages","Anonymous functions"],"downwards":[]}
{"upwards":["Lambda calculus","Lambda calculus and programming languages","Parallelism and concurrency"],"downwards":[]}
{"upwards":["Lambda calculus","Semantics"],"downwards":[]}
{"upwards":["Lambda calculus","Variations and extensions"],"downwards":[]}
{"upwards":["Recursion"],"downwards":["Formal definitions","Informal definition","In language","In mathematics","In computer science","In biology","In business","In art","In culture"]}
{"upwards":["Recursion","Formal definitions"],"downwards":[]}
{"upwards":["Recursion","Informal definition"],"downwards":[]}
{"upwards":["Recursion","In language"],"downwards":["Recursive humor"]}
{"upwards":["Recursion","In language","Recursive humor"],"downwards":[]}
{"upwards":["Recursion","In mathematics"],"downwards":["Recursively defined sets","Finite subdivision rules","Functional recursion","Proofs involving recursive definitions","Recursive optimization","The recursion theorem"]}
{"upwards":["Recursion","In mathematics","Recursively defined sets"],"downwards":["Example: the natural numbers","Example: Proof procedure"]}
{"upwards":["Recursion","In mathematics","Recursively defined sets","Example: the natural numbers"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Recursively defined sets","Example: Proof procedure"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Finite subdivision rules"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Functional recursion"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Proofs involving recursive definitions"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Recursive optimization"],"downwards":[]}
{"upwards":["Recursion","In mathematics","The recursion theorem"],"downwards":["Proof of uniqueness"]}
{"upwards":["Recursion","In mathematics","The recursion theorem","Proof of uniqueness"],"downwards":[]}
{"upwards":["Recursion","In computer science"],"downwards":[]}
{"upwards":["Recursion","In biology"],"downwards":[]}
{"upwards":["Recursion","In business"],"downwards":[]}
{"upwards":["Recursion","In art"],"downwards":[]}
{"upwards":["Recursion","In culture"],"downwards":[]}
{"upwards":["Turing machine"],"downwards":["Overview","Description","Formal definition","Additional details required to visualize or implement Turing machines","Equivalent models","Choice c-machines, oracle o-machines","Universal Turing machines","Comparison with real machines","History"]}
{"upwards":["Turing machine","Overview"],"downwards":["Physical description"]}
{"upwards":["Turing machine","Overview","Physical description"],"downwards":[]}
{"upwards":["Turing machine","Description"],"downwards":[]}
{"upwards":["Turing machine","Formal definition"],"downwards":[]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines"],"downwards":["Alternative definitions","The \"state\"","\"State\" diagrams"]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines","Alternative definitions"],"downwards":[]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines","The \"state\""],"downwards":[]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines","\"State\" diagrams"],"downwards":[]}
{"upwards":["Turing machine","Equivalent models"],"downwards":[]}
{"upwards":["Turing machine","Choice c-machines, oracle o-machines"],"downwards":[]}
{"upwards":["Turing machine","Universal Turing machines"],"downwards":[]}
{"upwards":["Turing machine","Comparison with real machines"],"downwards":["Limitations"]}
{"upwards":["Turing machine","Comparison with real machines","Limitations"],"downwards":["Computational complexity theory","Interaction"]}
{"upwards":["Turing machine","Comparison with real machines","Limitations","Computational complexity theory"],"downwards":[]}
{"upwards":["Turing machine","Comparison with real machines","Limitations","Interaction"],"downwards":[]}
{"upwards":["Turing machine","History"],"downwards":["Historical background: computational machinery","The Entscheidungsproblem (the \"decision problem\"): Hilbert's tenth question of 1900","Alan Turing's a-machine","1937\u20131970: The \"digital computer\", the birth of \"computer science\"","1970\u2013present: as a model of computation"]}
{"upwards":["Turing machine","History","Historical background: computational machinery"],"downwards":[]}
{"upwards":["Turing machine","History","The Entscheidungsproblem (the \"decision problem\"): Hilbert's tenth question of 1900"],"downwards":[]}
{"upwards":["Turing machine","History","Alan Turing's a-machine"],"downwards":[]}
{"upwards":["Turing machine","History","1937\u20131970: The \"digital computer\", the birth of \"computer science\""],"downwards":[]}
{"upwards":["Turing machine","History","1970\u2013present: as a model of computation"],"downwards":[]}
{"upwards":["Axiom of adjunction"],"downwards":["Interpretability of arithmetic","References"]}
{"upwards":["Axiom of adjunction","Interpretability of arithmetic"],"downwards":[]}
{"upwards":["Axiom of adjunction","References"],"downwards":[]}
{"upwards":["Timeline of mathematical logic"],"downwards":["19th century","20th century"]}
{"upwards":["Timeline of mathematical logic","19th century"],"downwards":[]}
{"upwards":["Timeline of mathematical logic","20th century"],"downwards":["1950-1999"]}
{"upwards":["Timeline of mathematical logic","20th century","1950-1999"],"downwards":[]}
{"upwards":["Category (mathematics)"],"downwards":["Definition","Small and large categories","Examples","Construction of new categories","Types of morphisms","Types of categories"]}
{"upwards":["Category (mathematics)","Definition"],"downwards":[]}
{"upwards":["Category (mathematics)","Small and large categories"],"downwards":[]}
{"upwards":["Category (mathematics)","Examples"],"downwards":[]}
{"upwards":["Category (mathematics)","Construction of new categories"],"downwards":["Dual category","Product categories"]}
{"upwards":["Category (mathematics)","Construction of new categories","Dual category"],"downwards":[]}
{"upwards":["Category (mathematics)","Construction of new categories","Product categories"],"downwards":[]}
{"upwards":["Category (mathematics)","Types of morphisms"],"downwards":[]}
{"upwards":["Category (mathematics)","Types of categories"],"downwards":[]}
{"upwards":["Supertask"],"downwards":["History","Prominent supertasks"]}
{"upwards":["Supertask","History"],"downwards":["Zeno","Thomson","Benacerraf","Modern literature"]}
{"upwards":["Supertask","History","Zeno"],"downwards":["Motion","Achilles and the tortoise"]}
{"upwards":["Supertask","History","Zeno","Motion"],"downwards":[]}
{"upwards":["Supertask","History","Zeno","Achilles and the tortoise"],"downwards":[]}
{"upwards":["Supertask","History","Thomson"],"downwards":[]}
{"upwards":["Supertask","History","Benacerraf"],"downwards":[]}
{"upwards":["Supertask","History","Modern literature"],"downwards":["Philosophy of mathematics","Physical possibility","Super Turing machines"]}
{"upwards":["Supertask","History","Modern literature","Philosophy of mathematics"],"downwards":[]}
{"upwards":["Supertask","History","Modern literature","Physical possibility"],"downwards":[]}
{"upwards":["Supertask","History","Modern literature","Super Turing machines"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks"],"downwards":["Ross\u2013Littlewood paradox","Benardete's paradox","Grim Reaper paradox","Davies' super-machine"]}
{"upwards":["Supertask","Prominent supertasks","Ross\u2013Littlewood paradox"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks","Benardete's paradox"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks","Grim Reaper paradox"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks","Davies' super-machine"],"downwards":[]}
{"upwards":["History of logic"],"downwards":["Logic in the East","Logic in the West","Medieval logic","Traditional logic","Rise of modern logic","Modern logic"]}
{"upwards":["History of logic","Logic in the East"],"downwards":["Logic in India","Hindu logic","Origin","Before Gautama","Dattatreya","Medhatithi Gautama","Panini","Nyaya-Vaisheshika","Jain Logic","Buddhist logic","Nagarjuna","Dignaga","Syllogism and influence","Logic in China"]}
{"upwards":["History of logic","Logic in the East","Logic in India"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Hindu logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Origin"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Before Gautama"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Dattatreya"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Medhatithi Gautama"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Panini"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Nyaya-Vaisheshika"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Jain Logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Buddhist logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Nagarjuna"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Dignaga"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Syllogism and influence"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Logic in China"],"downwards":[]}
{"upwards":["History of logic","Logic in the West"],"downwards":["Prehistory of logic","Ancient Greece before Aristotle","Aristotle","Stoics"]}
{"upwards":["History of logic","Logic in the West","Prehistory of logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle"],"downwards":["Thales","Pythagoras","Heraclitus and Parmenides","Plato"]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Thales"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Pythagoras"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Heraclitus and Parmenides"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Plato"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Aristotle"],"downwards":["The Organon"]}
{"upwards":["History of logic","Logic in the West","Aristotle","The Organon"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Stoics"],"downwards":[]}
{"upwards":["History of logic","Medieval logic"],"downwards":["Logic in the Middle East","Logic in medieval Europe"]}
{"upwards":["History of logic","Medieval logic","Logic in the Middle East"],"downwards":[]}
{"upwards":["History of logic","Medieval logic","Logic in medieval Europe"],"downwards":[]}
{"upwards":["History of logic","Traditional logic"],"downwards":["The textbook tradition","Logic in Hegel's philosophy","Logic and psychology"]}
{"upwards":["History of logic","Traditional logic","The textbook tradition"],"downwards":[]}
{"upwards":["History of logic","Traditional logic","Logic in Hegel's philosophy"],"downwards":[]}
{"upwards":["History of logic","Traditional logic","Logic and psychology"],"downwards":[]}
{"upwards":["History of logic","Rise of modern logic"],"downwards":[]}
{"upwards":["History of logic","Modern logic"],"downwards":["Embryonic period","Algebraic period","Logicist period","Metamathematical period","Logic after WWII"]}
{"upwards":["History of logic","Modern logic","Embryonic period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Algebraic period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Logicist period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Metamathematical period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Logic after WWII"],"downwards":[]}
{"upwards":["Philosophy of mathematics"],"downwards":["History","Major themes","Contemporary schools of thought","Arguments","Aesthetics","Journals"]}
{"upwards":["Philosophy of mathematics","History"],"downwards":["Contemporary philosophy"]}
{"upwards":["Philosophy of mathematics","History","Contemporary philosophy"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Major themes"],"downwards":["Mathematical realism","Mathematical anti-realism"]}
{"upwards":["Philosophy of mathematics","Major themes","Mathematical realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Major themes","Mathematical anti-realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought"],"downwards":["Artistic","Platonism","Mathematicism","Logicism","Formalism","Conventionalism","Intuitionism","Structuralism","Embodied mind theories","Fictionalism","Social constructivism","Beyond the traditional schools"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Artistic"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Platonism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Mathematicism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Logicism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Formalism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Conventionalism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Intuitionism"],"downwards":["Constructivism","Finitism"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Intuitionism","Constructivism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Intuitionism","Finitism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Structuralism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories"],"downwards":["Aristotelian realism","Psychologism","Empiricism"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories","Aristotelian realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories","Psychologism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories","Empiricism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Fictionalism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Social constructivism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools"],"downwards":["Unreasonable effectiveness","Popper's two senses of number statements","Philosophy of language"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools","Unreasonable effectiveness"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools","Popper's two senses of number statements"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools","Philosophy of language"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Arguments"],"downwards":["Indispensability argument for realism","Epistemic argument against realism"]}
{"upwards":["Philosophy of mathematics","Arguments","Indispensability argument for realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Arguments","Epistemic argument against realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Aesthetics"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Journals"],"downwards":[]}
{"upwards":["Abstract logic"],"downwards":[]}
{"upwards":["Axiom of countable choice"],"downwards":["Overview","Use","References"]}
{"upwards":["Axiom of countable choice","Overview"],"downwards":[]}
{"upwards":["Axiom of countable choice","Use"],"downwards":[]}
{"upwards":["Axiom of countable choice","References"],"downwards":[]}
{"upwards":["Concrete category"],"downwards":["Definition","Remarks","Further examples","Counter-examples","Implicit structure of concrete categories","Relative concreteness","Notes","References"]}
{"upwards":["Concrete category","Definition"],"downwards":[]}
{"upwards":["Concrete category","Remarks"],"downwards":[]}
{"upwards":["Concrete category","Further examples"],"downwards":[]}
{"upwards":["Concrete category","Counter-examples"],"downwards":[]}
{"upwards":["Concrete category","Implicit structure of concrete categories"],"downwards":[]}
{"upwards":["Concrete category","Relative concreteness"],"downwards":[]}
{"upwards":["Concrete category","Notes"],"downwards":[]}
{"upwards":["Concrete category","References"],"downwards":[]}
{"upwards":["Category theory"],"downwards":["Categories, objects, and morphisms","Functors","Natural transformations","Other concepts","Historical notes"]}
{"upwards":["Category theory","Categories, objects, and morphisms"],"downwards":["Categories","Morphisms"]}
{"upwards":["Category theory","Categories, objects, and morphisms","Categories"],"downwards":[]}
{"upwards":["Category theory","Categories, objects, and morphisms","Morphisms"],"downwards":[]}
{"upwards":["Category theory","Functors"],"downwards":[]}
{"upwards":["Category theory","Natural transformations"],"downwards":[]}
{"upwards":["Category theory","Other concepts"],"downwards":["Universal constructions, limits, and colimits","Equivalent categories","Further concepts and results","Higher-dimensional categories"]}
{"upwards":["Category theory","Other concepts","Universal constructions, limits, and colimits"],"downwards":[]}
{"upwards":["Category theory","Other concepts","Equivalent categories"],"downwards":[]}
{"upwards":["Category theory","Other concepts","Further concepts and results"],"downwards":[]}
{"upwards":["Category theory","Other concepts","Higher-dimensional categories"],"downwards":[]}
{"upwards":["Category theory","Historical notes"],"downwards":[]}
{"upwards":["Mathematical logic"],"downwards":["Subfields and scope","History","Formal logical systems","Set theory","Model theory","Recursion theory","Proof theory and constructive mathematics","Applications","Connections with computer science","Foundations of mathematics"]}
{"upwards":["Mathematical logic","Subfields and scope"],"downwards":[]}
{"upwards":["Mathematical logic","History"],"downwards":["Early history","19th century","20th century"]}
{"upwards":["Mathematical logic","History","Early history"],"downwards":[]}
{"upwards":["Mathematical logic","History","19th century"],"downwards":["Foundational theories"]}
{"upwards":["Mathematical logic","History","19th century","Foundational theories"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century"],"downwards":["Set theory and paradoxes","Symbolic logic","Beginnings of the other branches"]}
{"upwards":["Mathematical logic","History","20th century","Set theory and paradoxes"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Symbolic logic"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Beginnings of the other branches"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems"],"downwards":["First-order logic","Other classical logics","Nonclassical and modal logic","Algebraic logic"]}
{"upwards":["Mathematical logic","Formal logical systems","First-order logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Other classical logics"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Nonclassical and modal logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Algebraic logic"],"downwards":[]}
{"upwards":["Mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Mathematical logic","Model theory"],"downwards":[]}
{"upwards":["Mathematical logic","Recursion theory"],"downwards":["Algorithmically unsolvable problems"]}
{"upwards":["Mathematical logic","Recursion theory","Algorithmically unsolvable problems"],"downwards":[]}
{"upwards":["Mathematical logic","Proof theory and constructive mathematics"],"downwards":[]}
{"upwards":["Mathematical logic","Applications"],"downwards":[]}
{"upwards":["Mathematical logic","Connections with computer science"],"downwards":[]}
{"upwards":["Mathematical logic","Foundations of mathematics"],"downwards":[]}
{"upwards":["Logicism"],"downwards":["Overview","Origin of the name 'logicism'","Intent, or goal, of logicism","Epistemology, ontology and logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","The unit class, impredicativity, and the vicious circle principle","Neo-logicism"]}
{"upwards":["Logicism","Overview"],"downwards":[]}
{"upwards":["Logicism","Origin of the name 'logicism'"],"downwards":[]}
{"upwards":["Logicism","Intent, or goal, of logicism"],"downwards":["History"]}
{"upwards":["Logicism","Intent, or goal, of logicism","History"],"downwards":[]}
{"upwards":["Logicism","Epistemology, ontology and logicism"],"downwards":[]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia"],"downwards":["Preliminaries","The definition of the natural numbers","Criticism"]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","Preliminaries"],"downwards":[]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","The definition of the natural numbers"],"downwards":[]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","Criticism"],"downwards":[]}
{"upwards":["Logicism","The unit class, impredicativity, and the vicious circle principle"],"downwards":["A solution to impredicativity: a hierarchy of types","G\u00f6del's criticism and suggestions"]}
{"upwards":["Logicism","The unit class, impredicativity, and the vicious circle principle","A solution to impredicativity: a hierarchy of types"],"downwards":[]}
{"upwards":["Logicism","The unit class, impredicativity, and the vicious circle principle","G\u00f6del's criticism and suggestions"],"downwards":[]}
{"upwards":["Logicism","Neo-logicism"],"downwards":[]}
{"upwards":["Decision problem"],"downwards":["Definition","Examples","Decidability","Complete problems","Function problems","Optimization problems"]}
{"upwards":["Decision problem","Definition"],"downwards":[]}
{"upwards":["Decision problem","Examples"],"downwards":[]}
{"upwards":["Decision problem","Decidability"],"downwards":[]}
{"upwards":["Decision problem","Complete problems"],"downwards":[]}
{"upwards":["Decision problem","Function problems"],"downwards":[]}
{"upwards":["Decision problem","Optimization problems"],"downwards":[]}
{"upwards":["P (complexity)"],"downwards":["Definition","Notable problems in P","Relationships to other classes","Properties","Pure existence proofs of polynomial-time algorithms","Alternative characterizations","History","Notes","References","External links"]}
{"upwards":["P (complexity)","Definition"],"downwards":[]}
{"upwards":["P (complexity)","Notable problems in P"],"downwards":[]}
{"upwards":["P (complexity)","Relationships to other classes"],"downwards":[]}
{"upwards":["P (complexity)","Properties"],"downwards":[]}
{"upwards":["P (complexity)","Pure existence proofs of polynomial-time algorithms"],"downwards":[]}
{"upwards":["P (complexity)","Alternative characterizations"],"downwards":[]}
{"upwards":["P (complexity)","History"],"downwards":[]}
{"upwards":["P (complexity)","Notes"],"downwards":[]}
{"upwards":["P (complexity)","References"],"downwards":[]}
{"upwards":["P (complexity)","External links"],"downwards":[]}
{"upwards":["Axiom of global choice"],"downwards":["Statement","References"]}
{"upwards":["Axiom of global choice","Statement"],"downwards":[]}
{"upwards":["Axiom of global choice","References"],"downwards":[]}
{"upwards":["Axiom of dependent choice"],"downwards":["Formal statement","Use","Equivalent statements","Relation with other axioms","Notes","References"]}
{"upwards":["Axiom of dependent choice","Formal statement"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Use"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Equivalent statements"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Relation with other axioms"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Notes"],"downwards":[]}
{"upwards":["Axiom of dependent choice","References"],"downwards":[]}
{"upwards":["Computable set"],"downwards":["Formal definition","Examples and non-examples","Properties"]}
{"upwards":["Computable set","Formal definition"],"downwards":[]}
{"upwards":["Computable set","Examples and non-examples"],"downwards":[]}
{"upwards":["Computable set","Properties"],"downwards":[]}
{"upwards":["Axiom of infinity"],"downwards":["Formal statement","Interpretation and consequences","Extracting the natural numbers from the infinite set","An apparently weaker version","Independence"]}
{"upwards":["Axiom of infinity","Formal statement"],"downwards":[]}
{"upwards":["Axiom of infinity","Interpretation and consequences"],"downwards":[]}
{"upwards":["Axiom of infinity","Extracting the natural numbers from the infinite set"],"downwards":["Alternative method"]}
{"upwards":["Axiom of infinity","Extracting the natural numbers from the infinite set","Alternative method"],"downwards":[]}
{"upwards":["Axiom of infinity","An apparently weaker version"],"downwards":[]}
{"upwards":["Axiom of infinity","Independence"],"downwards":[]}
{"upwards":["Axiom of power set"],"downwards":["Consequences","Limitations","References"]}
{"upwards":["Axiom of power set","Consequences"],"downwards":[]}
{"upwards":["Axiom of power set","Limitations"],"downwards":[]}
{"upwards":["Axiom of power set","References"],"downwards":[]}
{"upwards":["Axiom of extensionality"],"downwards":["Formal statement","Interpretation","In predicate logic without equality","In set theory with ur-elements"]}
{"upwards":["Axiom of extensionality","Formal statement"],"downwards":[]}
{"upwards":["Axiom of extensionality","Interpretation"],"downwards":[]}
{"upwards":["Axiom of extensionality","In predicate logic without equality"],"downwards":[]}
{"upwards":["Axiom of extensionality","In set theory with ur-elements"],"downwards":[]}
{"upwards":["Axiom of pairing"],"downwards":["Formal statement","Consequences","Alternatives","References"]}
{"upwards":["Axiom of pairing","Formal statement"],"downwards":[]}
{"upwards":["Axiom of pairing","Consequences"],"downwards":[]}
{"upwards":["Axiom of pairing","Alternatives"],"downwards":["Non-independence","Weaker","Stronger"]}
{"upwards":["Axiom of pairing","Alternatives","Non-independence"],"downwards":[]}
{"upwards":["Axiom of pairing","Alternatives","Weaker"],"downwards":[]}
{"upwards":["Axiom of pairing","Alternatives","Stronger"],"downwards":[]}
{"upwards":["Axiom of pairing","References"],"downwards":[]}
{"upwards":["Axiom of union"],"downwards":["Formal statement","Relation to Pairing","Relation to Replacement","Relation to Separation","Relation to Intersection","References","Further reading"]}
{"upwards":["Axiom of union","Formal statement"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Pairing"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Replacement"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Separation"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Intersection"],"downwards":[]}
{"upwards":["Axiom of union","References"],"downwards":[]}
{"upwards":["Axiom of union","Further reading"],"downwards":[]}
{"upwards":["Axiom of limitation of size"],"downwards":["Formal statement","Implications of the axiom","Zermelo's models and the axiom of limitation of size","Limitation of size doctrine","History","Notes","References","Bibliography"]}
{"upwards":["Axiom of limitation of size","Formal statement"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Implications of the axiom"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Zermelo's models and the axiom of limitation of size"],"downwards":["The model V\u03c9","The models V\u03ba where \u03ba is a strongly inaccessible cardinal"]}
{"upwards":["Axiom of limitation of size","Zermelo's models and the axiom of limitation of size","The model V\u03c9"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Zermelo's models and the axiom of limitation of size","The models V\u03ba where \u03ba is a strongly inaccessible cardinal"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Limitation of size doctrine"],"downwards":[]}
{"upwards":["Axiom of limitation of size","History"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Notes"],"downwards":[]}
{"upwards":["Axiom of limitation of size","References"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Bibliography"],"downwards":[]}
{"upwards":["Axiom of regularity"],"downwards":["Elementary implications of regularity","The axiom of dependent choice and no infinite descending sequence of sets implies regularity","Regularity and the rest of ZF(C) axioms","Regularity and Russell's paradox","Regularity, the cumulative hierarchy, and types","History","Regularity in the presence of urelements"]}
{"upwards":["Axiom of regularity","Elementary implications of regularity"],"downwards":["No set is an element of itself","No infinite descending sequence of sets exists","Simpler set-theoretic definition of the ordered pair","Every set has an ordinal rank","For every two sets, only one can be an element of the other"]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","No set is an element of itself"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","No infinite descending sequence of sets exists"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","Simpler set-theoretic definition of the ordered pair"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","Every set has an ordinal rank"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","For every two sets, only one can be an element of the other"],"downwards":[]}
{"upwards":["Axiom of regularity","The axiom of dependent choice and no infinite descending sequence of sets implies regularity"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity and the rest of ZF(C) axioms"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity and Russell's paradox"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity, the cumulative hierarchy, and types"],"downwards":[]}
{"upwards":["Axiom of regularity","History"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity in the presence of urelements"],"downwards":[]}
{"upwards":["Axiom of determinacy"],"downwards":["Types of game that are determined","Incompatibility of the axiom of determinacy with the axiom of choice","Infinitary logic and the axiom of determinacy","Large cardinals and the axiom of determinacy","Projective ordinals"]}
{"upwards":["Axiom of determinacy","Types of game that are determined"],"downwards":[]}
{"upwards":["Axiom of determinacy","Incompatibility of the axiom of determinacy with the axiom of choice"],"downwards":[]}
{"upwards":["Axiom of determinacy","Infinitary logic and the axiom of determinacy"],"downwards":[]}
{"upwards":["Axiom of determinacy","Large cardinals and the axiom of determinacy"],"downwards":[]}
{"upwards":["Axiom of determinacy","Projective ordinals"],"downwards":[]}
{"upwards":["Axiom of constructibility"],"downwards":["Implications","In arithmetic","Significance","Statements true in L","References","External links"]}
{"upwards":["Axiom of constructibility","Implications"],"downwards":[]}
{"upwards":["Axiom of constructibility","In arithmetic"],"downwards":[]}
{"upwards":["Axiom of constructibility","Significance"],"downwards":[]}
{"upwards":["Axiom of constructibility","Statements true in L"],"downwards":[]}
{"upwards":["Axiom of constructibility","References"],"downwards":[]}
{"upwards":["Axiom of constructibility","External links"],"downwards":[]}
{"upwards":["Argument"],"downwards":["Etymology","Formal and informal","Standard logical account of argument types","Defeasible arguments and argumentation schemes","By analogy","Other kinds","Explanations","Fallacies and non-arguments","Elliptical or ethymematic arguments","Argument mining"]}
{"upwards":["Argument","Etymology"],"downwards":[]}
{"upwards":["Argument","Formal and informal"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types"],"downwards":["Deductive arguments","Inductive arguments"]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments"],"downwards":["Validity","Soundness"]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments","Validity"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments","Soundness"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types","Inductive arguments"],"downwards":[]}
{"upwards":["Argument","Defeasible arguments and argumentation schemes"],"downwards":[]}
{"upwards":["Argument","By analogy"],"downwards":[]}
{"upwards":["Argument","Other kinds"],"downwards":["World-disclosing"]}
{"upwards":["Argument","Other kinds","World-disclosing"],"downwards":[]}
{"upwards":["Argument","Explanations"],"downwards":[]}
{"upwards":["Argument","Fallacies and non-arguments"],"downwards":[]}
{"upwards":["Argument","Elliptical or ethymematic arguments"],"downwards":[]}
{"upwards":["Argument","Argument mining"],"downwards":[]}
{"upwards":["Logical equality"],"downwards":["Definition","Alternative descriptions"]}
{"upwards":["Logical equality","Definition"],"downwards":[]}
{"upwards":["Logical equality","Alternative descriptions"],"downwards":[]}
{"upwards":["Almost"],"downwards":[]}
{"upwards":["Filter (set theory)"],"downwards":["Preliminaries, notation, and basic notions","Filters and prefilters","Set theoretic properties and constructions","Filters and nets"]}
{"upwards":["Filter (set theory)","Preliminaries, notation, and basic notions"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters"],"downwards":["Basic examples","Ultrafilters","Kernels","Finer\/coarser, subordination, and meshing"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Basic examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Ultrafilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels"],"downwards":["Classifying families by their kernels","Characterizing fixed ultra prefilters"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Classifying families by their kernels"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Characterizing fixed ultra prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing"],"downwards":["Equivalent families of sets"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing","Equivalent families of sets"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions"],"downwards":["Trace and meshing","Images and preimages under functions","Products of prefilters","Set subtraction and some examples"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Trace and meshing"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions"],"downwards":["Subordination is preserved by images and preimages"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions","Subordination is preserved by images and preimages"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Products of prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Set subtraction and some examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets"],"downwards":["Nets to prefilters","Prefilters to nets","Subordinate filters and subnets"]}
{"upwards":["Filter (set theory)","Filters and nets","Nets to prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets"],"downwards":["Partially ordered net"]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets","Partially ordered net"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets"],"downwards":["Non\u2013equivalence of subnets and subordinate filters"]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets","Non\u2013equivalence of subnets and subordinate filters"],"downwards":[]}
{"upwards":["Hereditarily finite set"],"downwards":["Formal definition","Discussion","Axiomatizations","Graph models"]}
{"upwards":["Hereditarily finite set","Formal definition"],"downwards":[]}
{"upwards":["Hereditarily finite set","Discussion"],"downwards":["Ackermann coding","Representation"]}
{"upwards":["Hereditarily finite set","Discussion","Ackermann coding"],"downwards":[]}
{"upwards":["Hereditarily finite set","Discussion","Representation"],"downwards":[]}
{"upwards":["Hereditarily finite set","Axiomatizations"],"downwards":["Theories of finite sets","ZF"]}
{"upwards":["Hereditarily finite set","Axiomatizations","Theories of finite sets"],"downwards":[]}
{"upwards":["Hereditarily finite set","Axiomatizations","ZF"],"downwards":[]}
{"upwards":["Hereditarily finite set","Graph models"],"downwards":[]}
{"upwards":["Amorphous set"],"downwards":["Existence","Additional properties","Variations","References"]}
{"upwards":["Amorphous set","Existence"],"downwards":[]}
{"upwards":["Amorphous set","Additional properties"],"downwards":[]}
{"upwards":["Amorphous set","Variations"],"downwards":[]}
{"upwards":["Amorphous set","References"],"downwards":[]}
{"upwards":["Disjoint union"],"downwards":["Example","Set theory definition","Category theory point of view"]}
{"upwards":["Disjoint union","Example"],"downwards":[]}
{"upwards":["Disjoint union","Set theory definition"],"downwards":[]}
{"upwards":["Disjoint union","Category theory point of view"],"downwards":[]}
{"upwards":["Transfinite induction"],"downwards":["Induction by cases","Transfinite recursion","Relationship to the axiom of choice"]}
{"upwards":["Transfinite induction","Induction by cases"],"downwards":[]}
{"upwards":["Transfinite induction","Transfinite recursion"],"downwards":[]}
{"upwards":["Transfinite induction","Relationship to the axiom of choice"],"downwards":[]}
{"upwards":["Cardinal number"],"downwards":["History","Motivation","Formal definition","Cardinal arithmetic","The continuum hypothesis"]}
{"upwards":["Cardinal number","History"],"downwards":[]}
{"upwards":["Cardinal number","Motivation"],"downwards":[]}
{"upwards":["Cardinal number","Formal definition"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic"],"downwards":["Successor cardinal","Cardinal addition","Cardinal multiplication","Cardinal exponentiation"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Successor cardinal"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal addition"],"downwards":["Subtraction"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal addition","Subtraction"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal multiplication"],"downwards":["Division"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal multiplication","Division"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal exponentiation"],"downwards":["Roots","Logarithms"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal exponentiation","Roots"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal exponentiation","Logarithms"],"downwards":[]}
{"upwards":["Cardinal number","The continuum hypothesis"],"downwards":[]}
{"upwards":["Axiom schema of replacement"],"downwards":["Statement","Applications","Relation to other axiom schemas","History","References"]}
{"upwards":["Axiom schema of replacement","Statement"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Applications"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas"],"downwards":["Collection","Separation"]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas","Collection"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas","Separation"],"downwards":[]}
{"upwards":["Axiom schema of replacement","History"],"downwards":[]}
{"upwards":["Axiom schema of replacement","References"],"downwards":[]}
{"upwards":["Martin's axiom"],"downwards":["Statement","Equivalent forms of MA(\ud835\udecb)","Consequences","Further development","References","Further reading"]}
{"upwards":["Martin's axiom","Statement"],"downwards":[]}
{"upwards":["Martin's axiom","Equivalent forms of MA(\ud835\udecb)"],"downwards":[]}
{"upwards":["Martin's axiom","Consequences"],"downwards":[]}
{"upwards":["Martin's axiom","Further development"],"downwards":[]}
{"upwards":["Martin's axiom","References"],"downwards":[]}
{"upwards":["Martin's axiom","Further reading"],"downwards":[]}
{"upwards":["Axiom schema of specification"],"downwards":["Statement","Relation to the axiom schema of replacement","Unrestricted comprehension","In NBG class theory","In higher-order settings","In Quine's New Foundations","References"]}
{"upwards":["Axiom schema of specification","Statement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Relation to the axiom schema of replacement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Unrestricted comprehension"],"downwards":[]}
{"upwards":["Axiom schema of specification","In NBG class theory"],"downwards":[]}
{"upwards":["Axiom schema of specification","In higher-order settings"],"downwards":[]}
{"upwards":["Axiom schema of specification","In Quine's New Foundations"],"downwards":[]}
{"upwards":["Axiom schema of specification","References"],"downwards":["Citations"]}
{"upwards":["Axiom schema of specification","References","Citations"],"downwards":[]}
{"upwards":["Filter (set theory)"],"downwards":["Preliminaries, notation, and basic notions","Filters and prefilters","Set theoretic properties and constructions","Filters and nets"]}
{"upwards":["Filter (set theory)","Preliminaries, notation, and basic notions"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters"],"downwards":["Basic examples","Ultrafilters","Kernels","Finer\/coarser, subordination, and meshing"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Basic examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Ultrafilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels"],"downwards":["Classifying families by their kernels","Characterizing fixed ultra prefilters"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Classifying families by their kernels"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Characterizing fixed ultra prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing"],"downwards":["Equivalent families of sets"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing","Equivalent families of sets"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions"],"downwards":["Trace and meshing","Images and preimages under functions","Products of prefilters","Set subtraction and some examples"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Trace and meshing"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions"],"downwards":["Subordination is preserved by images and preimages"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions","Subordination is preserved by images and preimages"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Products of prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Set subtraction and some examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets"],"downwards":["Nets to prefilters","Prefilters to nets","Subordinate filters and subnets"]}
{"upwards":["Filter (set theory)","Filters and nets","Nets to prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets"],"downwards":["Partially ordered net"]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets","Partially ordered net"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets"],"downwards":["Non\u2013equivalence of subnets and subordinate filters"]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets","Non\u2013equivalence of subnets and subordinate filters"],"downwards":[]}
{"upwards":["Many-valued logic"],"downwards":["History","Examples","Relation to classical logic","Functional completeness of many-valued logics","Applications","Research venues"]}
{"upwards":["Many-valued logic","History"],"downwards":[]}
{"upwards":["Many-valued logic","Examples"],"downwards":["Kleene (strong) K3 and Priest logic P3","Bochvar's internal three-valued logic","Belnap logic (B4)","G\u00f6del logics Gk and G\u221e","\u0141ukasiewicz logics Lv and L\u221e","Product logic \u03a0","Post logics Pm","Rose logics"]}
{"upwards":["Many-valued logic","Examples","Kleene (strong) K3 and Priest logic P3"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Bochvar's internal three-valued logic"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Belnap logic (B4)"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","G\u00f6del logics Gk and G\u221e"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","\u0141ukasiewicz logics Lv and L\u221e"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Product logic \u03a0"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Post logics Pm"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Rose logics"],"downwards":[]}
{"upwards":["Many-valued logic","Relation to classical logic"],"downwards":["Suszko's thesis"]}
{"upwards":["Many-valued logic","Relation to classical logic","Suszko's thesis"],"downwards":[]}
{"upwards":["Many-valued logic","Functional completeness of many-valued logics"],"downwards":[]}
{"upwards":["Many-valued logic","Applications"],"downwards":[]}
{"upwards":["Many-valued logic","Research venues"],"downwards":[]}
{"upwards":["Square of opposition"],"downwards":["Summary","The problem of existential import","Modern squares of opposition","Logical hexagons and other bi-simplexes","Square of opposition (or logical square) and modal logic","Set-theoretical interpretation of categorical statements"]}
{"upwards":["Square of opposition","Summary"],"downwards":[]}
{"upwards":["Square of opposition","The problem of existential import"],"downwards":[]}
{"upwards":["Square of opposition","Modern squares of opposition"],"downwards":[]}
{"upwards":["Square of opposition","Logical hexagons and other bi-simplexes"],"downwards":[]}
{"upwards":["Square of opposition","Square of opposition (or logical square) and modal logic"],"downwards":[]}
{"upwards":["Square of opposition","Set-theoretical interpretation of categorical statements"],"downwards":[]}
{"upwards":["Boolean algebra"],"downwards":["History","Values","Operations","Laws","Diagrammatic representations","Boolean algebras","Axiomatizing Boolean algebra","Propositional logic","Applications"]}
{"upwards":["Boolean algebra","History"],"downwards":[]}
{"upwards":["Boolean algebra","Values"],"downwards":[]}
{"upwards":["Boolean algebra","Operations"],"downwards":["Basic operations","Secondary operations"]}
{"upwards":["Boolean algebra","Operations","Basic operations"],"downwards":[]}
{"upwards":["Boolean algebra","Operations","Secondary operations"],"downwards":[]}
{"upwards":["Boolean algebra","Laws"],"downwards":["Monotone laws","Nonmonotone laws","Completeness","Duality principle"]}
{"upwards":["Boolean algebra","Laws","Monotone laws"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Nonmonotone laws"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Completeness"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Duality principle"],"downwards":[]}
{"upwards":["Boolean algebra","Diagrammatic representations"],"downwards":["Venn diagrams","Digital logic gates"]}
{"upwards":["Boolean algebra","Diagrammatic representations","Venn diagrams"],"downwards":[]}
{"upwards":["Boolean algebra","Diagrammatic representations","Digital logic gates"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras"],"downwards":["Concrete Boolean algebras","Subsets as bit vectors","The prototypical Boolean algebra","Boolean algebras: the definition","Representable Boolean algebras"]}
{"upwards":["Boolean algebra","Boolean algebras","Concrete Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Subsets as bit vectors"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","The prototypical Boolean algebra"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Boolean algebras: the definition"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Representable Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebra","Axiomatizing Boolean algebra"],"downwards":[]}
{"upwards":["Boolean algebra","Propositional logic"],"downwards":["Applications","Deductive systems for propositional logic"]}
{"upwards":["Boolean algebra","Propositional logic","Applications"],"downwards":[]}
{"upwards":["Boolean algebra","Propositional logic","Deductive systems for propositional logic"],"downwards":["Sequent calculus"]}
{"upwards":["Boolean algebra","Propositional logic","Deductive systems for propositional logic","Sequent calculus"],"downwards":[]}
{"upwards":["Boolean algebra","Applications"],"downwards":["Computers","Two-valued logic","Boolean operations"]}
{"upwards":["Boolean algebra","Applications","Computers"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Two-valued logic"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations"],"downwards":["Natural language","Digital logic","Naive set theory","Video cards","Modeling and CAD","Boolean searches"]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Natural language"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Digital logic"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Naive set theory"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Video cards"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Modeling and CAD"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Boolean searches"],"downwards":[]}
{"upwards":["Validity (logic)"],"downwards":["Arguments","Valid formula","Statements","Soundness","Satisfiability","Preservation"]}
{"upwards":["Validity (logic)","Arguments"],"downwards":[]}
{"upwards":["Validity (logic)","Valid formula"],"downwards":[]}
{"upwards":["Validity (logic)","Statements"],"downwards":[]}
{"upwards":["Validity (logic)","Soundness"],"downwards":[]}
{"upwards":["Validity (logic)","Satisfiability"],"downwards":[]}
{"upwards":["Validity (logic)","Preservation"],"downwards":[]}
{"upwards":["Propositional calculus"],"downwards":["Explanation","History","Terminology","Basic concepts","Generic description of a propositional calculus","Example 1. Simple axiom system","Example 2. Natural deduction system","Basic and derived argument forms","Proofs in propositional calculus","Soundness and completeness of the rules","Interpretation of a truth-functional propositional calculus","Alternative calculus","Equivalence to equational logics","Graphical calculi","Other logical calculi","Solvers"]}
{"upwards":["Propositional calculus","Explanation"],"downwards":[]}
{"upwards":["Propositional calculus","History"],"downwards":[]}
{"upwards":["Propositional calculus","Terminology"],"downwards":[]}
{"upwards":["Propositional calculus","Basic concepts"],"downwards":["Closure under operations","Argument"]}
{"upwards":["Propositional calculus","Basic concepts","Closure under operations"],"downwards":[]}
{"upwards":["Propositional calculus","Basic concepts","Argument"],"downwards":[]}
{"upwards":["Propositional calculus","Generic description of a propositional calculus"],"downwards":[]}
{"upwards":["Propositional calculus","Example 1. Simple axiom system"],"downwards":[]}
{"upwards":["Propositional calculus","Example 2. Natural deduction system"],"downwards":[]}
{"upwards":["Propositional calculus","Basic and derived argument forms"],"downwards":[]}
{"upwards":["Propositional calculus","Proofs in propositional calculus"],"downwards":["Example of a proof in natural deduction system","Example of a proof in a classical propositional calculus system"]}
{"upwards":["Propositional calculus","Proofs in propositional calculus","Example of a proof in natural deduction system"],"downwards":[]}
{"upwards":["Propositional calculus","Proofs in propositional calculus","Example of a proof in a classical propositional calculus system"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules"],"downwards":["Sketch of a soundness proof","Sketch of completeness proof","Another outline for a completeness proof"]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of a soundness proof"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof"],"downwards":["Example","Verifying completeness for the classical propositional calculus system"]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof","Example"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof","Verifying completeness for the classical propositional calculus system"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Another outline for a completeness proof"],"downwards":[]}
{"upwards":["Propositional calculus","Interpretation of a truth-functional propositional calculus"],"downwards":["Interpretation of a sentence of truth-functional propositional logic"]}
{"upwards":["Propositional calculus","Interpretation of a truth-functional propositional calculus","Interpretation of a sentence of truth-functional propositional logic"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus"],"downwards":["Axioms","Inference rule","Meta-inference rule","Example of a proof"]}
{"upwards":["Propositional calculus","Alternative calculus","Axioms"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Inference rule"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Meta-inference rule"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Example of a proof"],"downwards":[]}
{"upwards":["Propositional calculus","Equivalence to equational logics"],"downwards":[]}
{"upwards":["Propositional calculus","Graphical calculi"],"downwards":[]}
{"upwards":["Propositional calculus","Other logical calculi"],"downwards":[]}
{"upwards":["Propositional calculus","Solvers"],"downwards":[]}
{"upwards":["Syllogism"],"downwards":["Early history","Modern history","Basic structure","Types","Terms in syllogism","Existential import","Syllogistic fallacies","Other types of syllogism"]}
{"upwards":["Syllogism","Early history"],"downwards":["Aristotle","Medieval syllogism"]}
{"upwards":["Syllogism","Early history","Aristotle"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism"],"downwards":["Boethius","Peter Abelard","Jean Buridan"]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Boethius"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Peter Abelard"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Jean Buridan"],"downwards":[]}
{"upwards":["Syllogism","Modern history"],"downwards":["Boole's acceptance of Aristotle"]}
{"upwards":["Syllogism","Modern history","Boole's acceptance of Aristotle"],"downwards":[]}
{"upwards":["Syllogism","Basic structure"],"downwards":["Polysyllogism"]}
{"upwards":["Syllogism","Basic structure","Polysyllogism"],"downwards":[]}
{"upwards":["Syllogism","Types"],"downwards":["Examples","Table of all syllogisms"]}
{"upwards":["Syllogism","Types","Examples"],"downwards":["Barbara (AAA-1)","Celarent (EAE-1)","Darii (AII-1)","Ferio (EIO-1)","Baroco (AOO-2)","Bocardo (OAO-3)","Barbari (AAI-1)","Celaront (EAO-1)","Camestros (AEO-2)","Felapton (EAO-3)","Darapti (AAI-3)"]}
{"upwards":["Syllogism","Types","Examples","Barbara (AAA-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Celarent (EAE-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Darii (AII-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Ferio (EIO-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Baroco (AOO-2)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Bocardo (OAO-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Barbari (AAI-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Celaront (EAO-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Camestros (AEO-2)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Felapton (EAO-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Darapti (AAI-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Table of all syllogisms"],"downwards":[]}
{"upwards":["Syllogism","Terms in syllogism"],"downwards":[]}
{"upwards":["Syllogism","Existential import"],"downwards":[]}
{"upwards":["Syllogism","Syllogistic fallacies"],"downwards":[]}
{"upwards":["Syllogism","Other types of syllogism"],"downwards":[]}
{"upwards":["Logical connective"],"downwards":["Overview","Natural language","Properties","Order of precedence","Computer science","Table and Hasse diagram"]}
{"upwards":["Logical connective","Overview"],"downwards":["List of common logical connectives","History of notations","Redundancy"]}
{"upwards":["Logical connective","Overview","List of common logical connectives"],"downwards":[]}
{"upwards":["Logical connective","Overview","History of notations"],"downwards":[]}
{"upwards":["Logical connective","Overview","Redundancy"],"downwards":[]}
{"upwards":["Logical connective","Natural language"],"downwards":[]}
{"upwards":["Logical connective","Properties"],"downwards":[]}
{"upwards":["Logical connective","Order of precedence"],"downwards":[]}
{"upwards":["Logical connective","Computer science"],"downwards":[]}
{"upwards":["Logical connective","Table and Hasse diagram"],"downwards":[]}
{"upwards":["Soundness"],"downwards":["Definition","Use in mathematical logic"]}
{"upwards":["Soundness","Definition"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic"],"downwards":["Logical systems","Relation to completeness"]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems"],"downwards":["Soundness","Strong soundness","Arithmetic soundness"]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Strong soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Arithmetic soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Relation to completeness"],"downwards":[]}
{"upwards":["Three-valued logic"],"downwards":["Pre-discovery","Representation of values","Logics","Applications"]}
{"upwards":["Three-valued logic","Pre-discovery"],"downwards":[]}
{"upwards":["Three-valued logic","Representation of values"],"downwards":[]}
{"upwards":["Three-valued logic","Logics"],"downwards":["Kleene and Priest logics","\u0141ukasiewicz logic","HT logic","Bochvar logic","Ternary Post logic","Modular algebras"]}
{"upwards":["Three-valued logic","Logics","Kleene and Priest logics"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","\u0141ukasiewicz logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","HT logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","Bochvar logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","Ternary Post logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","Modular algebras"],"downwards":[]}
{"upwards":["Three-valued logic","Applications"],"downwards":["SQL"]}
{"upwards":["Three-valued logic","Applications","SQL"],"downwards":[]}
{"upwards":["Propositional formula"],"downwards":["Propositions","An algebra of propositions, the propositional calculus","Propositional connectives","More complex formulas","Inductive definition","Parsing formulas","Well-formed formulas (wffs)","Reduced sets of connectives","Normal forms","Impredicative propositions","Propositional formula with \"feedback\"","Historical development","Footnotes","Citations","References","External links"]}
{"upwards":["Propositional formula","Propositions"],"downwards":["Relationship between propositional and predicate formulas","Identity"]}
{"upwards":["Propositional formula","Propositions","Relationship between propositional and predicate formulas"],"downwards":[]}
{"upwards":["Propositional formula","Propositions","Identity"],"downwards":[]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus"],"downwards":["Usefulness of propositional formulas","Propositional variables","Truth-value assignments, formula evaluations"]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus","Usefulness of propositional formulas"],"downwards":[]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus","Propositional variables"],"downwards":[]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus","Truth-value assignments, formula evaluations"],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives"],"downwards":["Connectives of rhetoric, philosophy and mathematics","Engineering connectives","CASE connective: IF ... THEN ... ELSE ...","IDENTITY and evaluation"]}
{"upwards":["Propositional formula","Propositional connectives","Connectives of rhetoric, philosophy and mathematics"],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives","Engineering connectives"],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives","CASE connective: IF ... THEN ... ELSE ..."],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives","IDENTITY and evaluation"],"downwards":[]}
{"upwards":["Propositional formula","More complex formulas"],"downwards":["Definitions","Axiom and definition schemas","Substitution versus replacement"]}
{"upwards":["Propositional formula","More complex formulas","Definitions"],"downwards":[]}
{"upwards":["Propositional formula","More complex formulas","Axiom and definition schemas"],"downwards":[]}
{"upwards":["Propositional formula","More complex formulas","Substitution versus replacement"],"downwards":[]}
{"upwards":["Propositional formula","Inductive definition"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas"],"downwards":["Connective seniority (symbol rank)","Commutative and associative laws","Distributive laws","De Morgan's laws","Laws of absorption","Laws of evaluation: Identity, nullity, and complement","Double negative (involution)"]}
{"upwards":["Propositional formula","Parsing formulas","Connective seniority (symbol rank)"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Commutative and associative laws"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Distributive laws"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","De Morgan's laws"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Laws of absorption"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Laws of evaluation: Identity, nullity, and complement"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Double negative (involution)"],"downwards":[]}
{"upwards":["Propositional formula","Well-formed formulas (wffs)"],"downwards":["Well-formed formulas versus valid formulas in inferences"]}
{"upwards":["Propositional formula","Well-formed formulas (wffs)","Well-formed formulas versus valid formulas in inferences"],"downwards":[]}
{"upwards":["Propositional formula","Reduced sets of connectives"],"downwards":["The stroke (NAND)","IF ... THEN ... ELSE"]}
{"upwards":["Propositional formula","Reduced sets of connectives","The stroke (NAND)"],"downwards":[]}
{"upwards":["Propositional formula","Reduced sets of connectives","IF ... THEN ... ELSE"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms"],"downwards":["Reduction to normal form","Reduction by use of the map method (Veitch, Karnaugh)"]}
{"upwards":["Propositional formula","Normal forms","Reduction to normal form"],"downwards":["Literal, term and alterm","Minterms"]}
{"upwards":["Propositional formula","Normal forms","Reduction to normal form","Literal, term and alterm"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction to normal form","Minterms"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)"],"downwards":["Produce the formula's truth table","Create the formula's Karnaugh map","Reduce minterms","Verify reduction with a truth table"]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Produce the formula's truth table"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Create the formula's Karnaugh map"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Reduce minterms"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Verify reduction with a truth table"],"downwards":[]}
{"upwards":["Propositional formula","Impredicative propositions"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\""],"downwards":["Oscillation","Memory"]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Oscillation"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory"],"downwards":["Once-flip memory","Flip-flop memory","Clocked flip-flop memory"]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory","Once-flip memory"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory","Flip-flop memory"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory","Clocked flip-flop memory"],"downwards":[]}
{"upwards":["Propositional formula","Historical development"],"downwards":[]}
{"upwards":["Propositional formula","Footnotes"],"downwards":[]}
{"upwards":["Propositional formula","Citations"],"downwards":[]}
{"upwards":["Propositional formula","References"],"downwards":[]}
{"upwards":["Propositional formula","External links"],"downwards":[]}
{"upwards":["Monadic second-order logic"],"downwards":["Variants","Computational complexity of evaluation","Decidability and complexity of satisfiability"]}
{"upwards":["Monadic second-order logic","Variants"],"downwards":[]}
{"upwards":["Monadic second-order logic","Computational complexity of evaluation"],"downwards":[]}
{"upwards":["Monadic second-order logic","Decidability and complexity of satisfiability"],"downwards":["Use of satisfiability of MSO on trees in verification"]}
{"upwards":["Monadic second-order logic","Decidability and complexity of satisfiability","Use of satisfiability of MSO on trees in verification"],"downwards":[]}
{"upwards":["Truth table"],"downwards":["Unary operations","Binary operations","Size of truth tables","Applications","History"]}
{"upwards":["Truth table","Unary operations"],"downwards":["Logical true","Logical false","Logical identity","Logical negation"]}
{"upwards":["Truth table","Unary operations","Logical true"],"downwards":[]}
{"upwards":["Truth table","Unary operations","Logical false"],"downwards":[]}
{"upwards":["Truth table","Unary operations","Logical identity"],"downwards":[]}
{"upwards":["Truth table","Unary operations","Logical negation"],"downwards":[]}
{"upwards":["Truth table","Binary operations"],"downwards":["Truth table for all binary logical operators","Logical conjunction (AND)","Logical disjunction (OR)","Logical implication","Logical equality","Exclusive disjunction","Logical NAND","Logical NOR"]}
{"upwards":["Truth table","Binary operations","Truth table for all binary logical operators"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical conjunction (AND)"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical disjunction (OR)"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical implication"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical equality"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Exclusive disjunction"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical NAND"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical NOR"],"downwards":[]}
{"upwards":["Truth table","Size of truth tables"],"downwards":[]}
{"upwards":["Truth table","Applications"],"downwards":["Truth table for most commonly used logical operators","Condensed truth tables for binary operators","Truth tables in digital logic","Applications of truth tables in digital electronics"]}
{"upwards":["Truth table","Applications","Truth table for most commonly used logical operators"],"downwards":[]}
{"upwards":["Truth table","Applications","Condensed truth tables for binary operators"],"downwards":[]}
{"upwards":["Truth table","Applications","Truth tables in digital logic"],"downwards":[]}
{"upwards":["Truth table","Applications","Applications of truth tables in digital electronics"],"downwards":[]}
{"upwards":["Truth table","History"],"downwards":[]}
{"upwards":["Quantifier (logic)"],"downwards":["Relations to logical conjunction and disjunction","Algebraic approaches to quantification","Notation","Order of quantifiers (nesting)","Equivalent expressions","Range of quantification","Formal semantics","Paucal, multal and other degree quantifiers","Other quantifiers","History"]}
{"upwards":["Quantifier (logic)","Relations to logical conjunction and disjunction"],"downwards":["Infinite domain of discourse"]}
{"upwards":["Quantifier (logic)","Relations to logical conjunction and disjunction","Infinite domain of discourse"],"downwards":[]}
{"upwards":["Quantifier (logic)","Algebraic approaches to quantification"],"downwards":[]}
{"upwards":["Quantifier (logic)","Notation"],"downwards":[]}
{"upwards":["Quantifier (logic)","Order of quantifiers (nesting)"],"downwards":[]}
{"upwards":["Quantifier (logic)","Equivalent expressions"],"downwards":["Quantifier elimination"]}
{"upwards":["Quantifier (logic)","Equivalent expressions","Quantifier elimination"],"downwards":[]}
{"upwards":["Quantifier (logic)","Range of quantification"],"downwards":[]}
{"upwards":["Quantifier (logic)","Formal semantics"],"downwards":[]}
{"upwards":["Quantifier (logic)","Paucal, multal and other degree quantifiers"],"downwards":[]}
{"upwards":["Quantifier (logic)","Other quantifiers"],"downwards":[]}
{"upwards":["Quantifier (logic)","History"],"downwards":[]}
{"upwards":["Boolean function"],"downwards":["Examples","Representation","Analysis","Real polynomial form","Applications"]}
{"upwards":["Boolean function","Examples"],"downwards":[]}
{"upwards":["Boolean function","Representation"],"downwards":[]}
{"upwards":["Boolean function","Analysis"],"downwards":["Properties","Derived functions","Cryptographic analysis"]}
{"upwards":["Boolean function","Analysis","Properties"],"downwards":[]}
{"upwards":["Boolean function","Analysis","Derived functions"],"downwards":[]}
{"upwards":["Boolean function","Analysis","Cryptographic analysis"],"downwards":["Linear approximation table"]}
{"upwards":["Boolean function","Analysis","Cryptographic analysis","Linear approximation table"],"downwards":[]}
{"upwards":["Boolean function","Real polynomial form"],"downwards":["On the unit hypercube","On the symmetric hypercube"]}
{"upwards":["Boolean function","Real polynomial form","On the unit hypercube"],"downwards":[]}
{"upwards":["Boolean function","Real polynomial form","On the symmetric hypercube"],"downwards":[]}
{"upwards":["Boolean function","Applications"],"downwards":[]}
{"upwards":["Free logic"],"downwards":["Explanation","Interpretation"]}
{"upwards":["Free logic","Explanation"],"downwards":[]}
{"upwards":["Free logic","Interpretation"],"downwards":[]}
{"upwards":["Higher-order logic"],"downwards":["Quantification scope","Semantics","Properties"]}
{"upwards":["Higher-order logic","Quantification scope"],"downwards":[]}
{"upwards":["Higher-order logic","Semantics"],"downwards":[]}
{"upwards":["Higher-order logic","Properties"],"downwards":[]}
{"upwards":["Second-order logic"],"downwards":["Examples","Syntax and fragments","Semantics","Expressive power","Deductive systems","Non-reducibility to first-order logic","Metalogical results","History and disputed value","Relation to computational complexity"]}
{"upwards":["Second-order logic","Examples"],"downwards":[]}
{"upwards":["Second-order logic","Syntax and fragments"],"downwards":[]}
{"upwards":["Second-order logic","Semantics"],"downwards":[]}
{"upwards":["Second-order logic","Expressive power"],"downwards":[]}
{"upwards":["Second-order logic","Deductive systems"],"downwards":[]}
{"upwards":["Second-order logic","Non-reducibility to first-order logic"],"downwards":[]}
{"upwards":["Second-order logic","Metalogical results"],"downwards":[]}
{"upwards":["Second-order logic","History and disputed value"],"downwards":[]}
{"upwards":["Second-order logic","Relation to computational complexity"],"downwards":[]}
{"upwards":["First-order logic"],"downwards":["Introduction","Syntax","Semantics","Deductive systems","Equality and its axioms","Metalogical properties","Limitations","Restrictions, extensions, and variations","Automated theorem proving and formal methods"]}
{"upwards":["First-order logic","Introduction"],"downwards":[]}
{"upwards":["First-order logic","Syntax"],"downwards":["Alphabet","Formation rules","Free and bound variables","Example: ordered abelian groups"]}
{"upwards":["First-order logic","Syntax","Alphabet"],"downwards":["Logical symbols","Non-logical symbols"]}
{"upwards":["First-order logic","Syntax","Alphabet","Logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Alphabet","Non-logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules"],"downwards":["Terms","Formulas","Notational conventions"]}
{"upwards":["First-order logic","Syntax","Formation rules","Terms"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Formulas"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Notational conventions"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Free and bound variables"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Example: ordered abelian groups"],"downwards":[]}
{"upwards":["First-order logic","Semantics"],"downwards":["First-order structures","Evaluation of truth values","Validity, satisfiability, and logical consequence","Algebraizations","First-order theories, models, and elementary classes","Empty domains"]}
{"upwards":["First-order logic","Semantics","First-order structures"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Evaluation of truth values"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Validity, satisfiability, and logical consequence"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Algebraizations"],"downwards":[]}
{"upwards":["First-order logic","Semantics","First-order theories, models, and elementary classes"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Empty domains"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems"],"downwards":["Rules of inference","Hilbert-style systems and natural deduction","Sequent calculus","Tableaux method","Resolution","Provable identities"]}
{"upwards":["First-order logic","Deductive systems","Rules of inference"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Hilbert-style systems and natural deduction"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Sequent calculus"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Tableaux method"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Resolution"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Provable identities"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms"],"downwards":["First-order logic without equality","Defining equality within a theory"]}
{"upwards":["First-order logic","Equality and its axioms","First-order logic without equality"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms","Defining equality within a theory"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties"],"downwards":["Completeness and undecidability","The L\u00f6wenheim\u2013Skolem theorem","The compactness theorem","Lindstr\u00f6m's theorem"]}
{"upwards":["First-order logic","Metalogical properties","Completeness and undecidability"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The compactness theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","Lindstr\u00f6m's theorem"],"downwards":[]}
{"upwards":["First-order logic","Limitations"],"downwards":["Expressiveness","Formalizing natural languages"]}
{"upwards":["First-order logic","Limitations","Expressiveness"],"downwards":[]}
{"upwards":["First-order logic","Limitations","Formalizing natural languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations"],"downwards":["Restricted languages","Many-sorted logic","Additional quantifiers","Infinitary logics","Non-classical and modal logics","Fixpoint logic","Higher-order logics"]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Restricted languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Many-sorted logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Additional quantifiers"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Infinitary logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Non-classical and modal logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Fixpoint logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Higher-order logics"],"downwards":[]}
{"upwards":["First-order logic","Automated theorem proving and formal methods"],"downwards":[]}
{"upwards":["Hereditary set"],"downwards":["Examples","In formulations of set theory","Assumptions"]}
{"upwards":["Hereditary set","Examples"],"downwards":[]}
{"upwards":["Hereditary set","In formulations of set theory"],"downwards":[]}
{"upwards":["Hereditary set","Assumptions"],"downwards":[]}
{"upwards":["Finite-valued logic"],"downwards":["History","Examples"]}
{"upwards":["Finite-valued logic","History"],"downwards":[]}
{"upwards":["Finite-valued logic","Examples"],"downwards":[]}
{"upwards":["Monadic predicate calculus"],"downwards":["Expressiveness","Relationship with term logic","Variants","Footnotes"]}
{"upwards":["Monadic predicate calculus","Expressiveness"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Relationship with term logic"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Variants"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Footnotes"],"downwards":[]}
{"upwards":["Free content"],"downwards":["Definition","Legal matters","Usage","Open content"]}
{"upwards":["Free content","Definition"],"downwards":[]}
{"upwards":["Free content","Legal matters"],"downwards":["Copyright","Public domain","Copyleft"]}
{"upwards":["Free content","Legal matters","Copyright"],"downwards":[]}
{"upwards":["Free content","Legal matters","Public domain"],"downwards":[]}
{"upwards":["Free content","Legal matters","Copyleft"],"downwards":[]}
{"upwards":["Free content","Usage"],"downwards":["Media","Software","Engineering and technology","Academia","Legislation"]}
{"upwards":["Free content","Usage","Media"],"downwards":[]}
{"upwards":["Free content","Usage","Software"],"downwards":[]}
{"upwards":["Free content","Usage","Engineering and technology"],"downwards":[]}
{"upwards":["Free content","Usage","Academia"],"downwards":[]}
{"upwards":["Free content","Usage","Legislation"],"downwards":[]}
{"upwards":["Free content","Open content"],"downwards":["History","\"Open content\" definition","Open access","Open content and education","Textbooks","Licenses"]}
{"upwards":["Free content","Open content","History"],"downwards":[]}
{"upwards":["Free content","Open content","\"Open content\" definition"],"downwards":[]}
{"upwards":["Free content","Open content","Open access"],"downwards":[]}
{"upwards":["Free content","Open content","Open content and education"],"downwards":[]}
{"upwards":["Free content","Open content","Textbooks"],"downwards":[]}
{"upwards":["Free content","Open content","Licenses"],"downwards":[]}
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{"upwards":["Wikipedia","Access to content","Chinese access"],"downwards":[]}
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{"upwards":["Michoac\u00e1n","Infrastructure","Media"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Education"],"downwards":[]}
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{"upwards":["Michoac\u00e1n","Crime"],"downwards":["Killings","Kidnappings and Ransoms","Drug Cartels","Police and Militia"]}
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{"upwards":["John von Neumann"],"downwards":["Life and education","Career and private life","Mathematics","Physics","Economics","Computer science","Defense work","Personality","Recognition","Legacy","Selected works"]}
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{"upwards":["John von Neumann","Mathematics","Set theory"],"downwards":["Von Neumann paradox"]}
{"upwards":["John von Neumann","Mathematics","Set theory","Von Neumann paradox"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Proof theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Ergodic theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Measure theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Topological groups"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Functional analysis"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Operator algebras"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Lattice theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Mathematical statistics"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Other work"],"downwards":[]}
{"upwards":["John von Neumann","Physics"],"downwards":["Quantum mechanics","Fluid dynamics","Other work"]}
{"upwards":["John von Neumann","Physics","Quantum mechanics"],"downwards":["Von Neumann entropy","Quantum mutual information","Density matrix","Von Neumann measurement scheme","Quantum logic"]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Von Neumann entropy"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Quantum mutual information"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Density matrix"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Von Neumann measurement scheme"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Quantum logic"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Fluid dynamics"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Other work"],"downwards":[]}
{"upwards":["John von Neumann","Economics"],"downwards":["Game theory","Mathematical economics","Linear programming"]}
{"upwards":["John von Neumann","Economics","Game theory"],"downwards":[]}
{"upwards":["John von Neumann","Economics","Mathematical economics"],"downwards":[]}
{"upwards":["John von Neumann","Economics","Linear programming"],"downwards":[]}
{"upwards":["John von Neumann","Computer science"],"downwards":["Cellular automata, DNA and the universal constructor","Scientific computing and numerical analysis","Weather systems and global warming","Technological singularity hypothesis"]}
{"upwards":["John von Neumann","Computer science","Cellular automata, DNA and the universal constructor"],"downwards":[]}
{"upwards":["John von Neumann","Computer science","Scientific computing and numerical analysis"],"downwards":[]}
{"upwards":["John von Neumann","Computer science","Weather systems and global warming"],"downwards":[]}
{"upwards":["John von Neumann","Computer science","Technological singularity hypothesis"],"downwards":[]}
{"upwards":["John von Neumann","Defense work"],"downwards":["Manhattan Project","Post war","Atomic Energy Commission","Mutual assured destruction","Consultancies"]}
{"upwards":["John von Neumann","Defense work","Manhattan Project"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Post war"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Atomic Energy Commission"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Mutual assured destruction"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Consultancies"],"downwards":[]}
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{"upwards":["John von Neumann","Personality","Mathematical style"],"downwards":[]}
{"upwards":["John von Neumann","Recognition"],"downwards":["Cognitive abilities","Eidetic memory"]}
{"upwards":["John von Neumann","Recognition","Cognitive abilities"],"downwards":[]}
{"upwards":["John von Neumann","Recognition","Eidetic memory"],"downwards":[]}
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{"upwards":["John von Neumann","Selected works","Books authored \/ coauthored"],"downwards":[]}
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{"upwards":["John von Neumann","Selected works","Popular articles"],"downwards":[]}
{"upwards":["John von Neumann","Selected works","Collected works"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
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{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["Paul Bernays"],"downwards":["Biography","Mathematical work","Publications","Notes","References","External links"]}
{"upwards":["Paul Bernays","Biography"],"downwards":[]}
{"upwards":["Paul Bernays","Mathematical work"],"downwards":[]}
{"upwards":["Paul Bernays","Publications"],"downwards":[]}
{"upwards":["Paul Bernays","Notes"],"downwards":[]}
{"upwards":["Paul Bernays","References"],"downwards":[]}
{"upwards":["Paul Bernays","External links"],"downwards":[]}
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{"upwards":["Landfall","Tropical cyclone"],"downwards":[]}
{"upwards":["Landfall","Tornado or waterspout"],"downwards":[]}
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{"upwards":["Mudflow","Mudflows and landslides"],"downwards":[]}
{"upwards":["Mudflow","Mudflow geography"],"downwards":[]}
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{"upwards":["English language","Grammar","Syntax","Auxiliary verb constructions"],"downwards":[]}
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{"upwards":["English language","Grammar","Syntax","Discourse level syntax"],"downwards":[]}
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{"upwards":["English language","Vocabulary","English loanwords and calques in other languages"],"downwards":[]}
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{"upwards":["English language","Sample text"],"downwards":[]}
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{"upwards":["Abraham Fraenkel"],"downwards":["Biography","Mathematician","Awards","Published works"]}
{"upwards":["Abraham Fraenkel","Biography"],"downwards":[]}
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{"upwards":["Mathematics","Symbolic notation and terminology"],"downwards":[]}
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{"upwards":["Mathematics","Relationship with sciences","Specific sciences"],"downwards":["Physics","Computing","Biology and chemistry","Earth sciences","Social sciences"]}
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{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Computing"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Biology and chemistry"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Earth sciences"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Social sciences"],"downwards":[]}
{"upwards":["Mathematics","Relationship with astrology and esotericism"],"downwards":[]}
{"upwards":["Mathematics","Philosophy"],"downwards":["Reality","Proposed definitions","Rigor"]}
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{"upwards":["Mathematics","Philosophy","Proposed definitions"],"downwards":[]}
{"upwards":["Mathematics","Philosophy","Rigor"],"downwards":[]}
{"upwards":["Mathematics","Training and practice"],"downwards":["Education","Psychology (aesthetic, creativity and intuition)"]}
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{"upwards":["Mathematics","Cultural impact"],"downwards":["Artistic expression","Popularization"]}
{"upwards":["Mathematics","Cultural impact","Artistic expression"],"downwards":[]}
{"upwards":["Mathematics","Cultural impact","Popularization"],"downwards":[]}
{"upwards":["Mathematics","Awards and prize problems"],"downwards":[]}
{"upwards":["Tropical cyclone scales"],"downwards":["Background","Atlantic, Eastern and Central Pacific","Western Pacific","North Indian Ocean","South-West Indian Ocean","Australia and Fiji","Alternative scales"]}
{"upwards":["Tropical cyclone scales","Background"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Atlantic, Eastern and Central Pacific"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Western Pacific"],"downwards":[]}
{"upwards":["Tropical cyclone scales","North Indian Ocean"],"downwards":[]}
{"upwards":["Tropical cyclone scales","South-West Indian Ocean"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Australia and Fiji"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Alternative scales"],"downwards":[]}
{"upwards":["Bar (unit)"],"downwards":["Definition and conversion","Origin","Usage"]}
{"upwards":["Bar (unit)","Definition and conversion"],"downwards":[]}
{"upwards":["Bar (unit)","Origin"],"downwards":[]}
{"upwards":["Bar (unit)","Usage"],"downwards":[]}
{"upwards":["Tropical wave"],"downwards":["Characteristics","Atlantic","East Pacific","Screaming eagle waves"]}
{"upwards":["Tropical wave","Characteristics"],"downwards":[]}
{"upwards":["Tropical wave","Atlantic"],"downwards":[]}
{"upwards":["Tropical wave","East Pacific"],"downwards":[]}
{"upwards":["Tropical wave","Screaming eagle waves"],"downwards":[]}
{"upwards":["Encyclopedia"],"downwards":["Etymology","Characteristics","Pre-modern encyclopedias","Printed encyclopedias","Digital encyclopedias"]}
{"upwards":["Encyclopedia","Etymology"],"downwards":[]}
{"upwards":["Encyclopedia","Characteristics"],"downwards":["Four major elements","Encyclopedic dictionaries","Differences between encyclopedias and dictionaries"]}
{"upwards":["Encyclopedia","Characteristics","Four major elements"],"downwards":[]}
{"upwards":["Encyclopedia","Characteristics","Encyclopedic dictionaries"],"downwards":[]}
{"upwards":["Encyclopedia","Characteristics","Differences between encyclopedias and dictionaries"],"downwards":[]}
{"upwards":["Encyclopedia","Pre-modern encyclopedias"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias"],"downwards":["Encyclop\u00e9die","Encyclop\u00e6dia Britannica","Brockhaus Enzyklop\u00e4die","Encyclopedias in the US"]}
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{"upwards":["Encyclopedia","Printed encyclopedias","Encyclop\u00e6dia Britannica"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias","Brockhaus Enzyklop\u00e4die"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias","Encyclopedias in the US"],"downwards":[]}
{"upwards":["Encyclopedia","Digital encyclopedias"],"downwards":["Physical media","Online"]}
{"upwards":["Encyclopedia","Digital encyclopedias","Physical media"],"downwards":[]}
{"upwards":["Encyclopedia","Digital encyclopedias","Online"],"downwards":["Free encyclopedias"]}
{"upwards":["Encyclopedia","Digital encyclopedias","Online","Free encyclopedias"],"downwards":[]}
{"upwards":["Kurt G\u00f6del"],"downwards":["Early life and education","Career","Awards and honours","Later life and death","Religious views","Legacy","Bibliography"]}
{"upwards":["Kurt G\u00f6del","Early life and education"],"downwards":["Childhood","Studies in Vienna"]}
{"upwards":["Kurt G\u00f6del","Early life and education","Childhood"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Early life and education","Studies in Vienna"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Career"],"downwards":["Incompleteness theorems","Mid-1930s: further work and U.S. visits","Princeton, Einstein, U.S. citizenship"]}
{"upwards":["Kurt G\u00f6del","Career","Incompleteness theorems"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Career","Mid-1930s: further work and U.S. visits"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Career","Princeton, Einstein, U.S. citizenship"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Awards and honours"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Later life and death"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Religious views"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Legacy"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Bibliography"],"downwards":["Important publications"]}
{"upwards":["Kurt G\u00f6del","Bibliography","Important publications"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season"],"downwards":["Seasonal forecasts","Seasonal summary","Systems","Storm names","Season effects"]}
{"upwards":["2018 Pacific hurricane season","Seasonal forecasts"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Seasonal summary"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems"],"downwards":["Tropical Depression One-E","Hurricane Aletta","Hurricane Bud","Tropical Storm Carlotta","Tropical Storm Daniel","Tropical Storm Emilia","Hurricane Fabio","Tropical Storm Gilma","Tropical Depression Nine-E","Hurricane Hector","Tropical Storm Ileana","Hurricane John","Tropical Storm Kristy","Hurricane Lane","Hurricane Miriam","Hurricane Norman","Hurricane Olivia","Tropical Storm Paul","Tropical Depression Nineteen-E","Hurricane Rosa","Hurricane Sergio","Hurricane Walaka","Tropical Storm Tara","Tropical Storm Vicente","Hurricane Willa","Tropical Storm Xavier","Other systems"]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Depression One-E"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Aletta"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Bud"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Carlotta"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Daniel"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Emilia"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Fabio"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Gilma"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Depression Nine-E"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Hector"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Ileana"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane John"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Kristy"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Lane"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Miriam"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Norman"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Olivia"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Paul"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Depression Nineteen-E"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Rosa"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Sergio"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Walaka"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Tara"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Vicente"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Willa"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Xavier"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Other systems"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Storm names"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Season effects"],"downwards":[]}
{"upwards":["Richard Dedekind"],"downwards":["Life","Work","Bibliography"]}
{"upwards":["Richard Dedekind","Life"],"downwards":[]}
{"upwards":["Richard Dedekind","Work"],"downwards":[]}
{"upwards":["Richard Dedekind","Bibliography"],"downwards":[]}
{"upwards":["Tropical cyclone"],"downwards":["Background","Intensity","Classification and naming","Structure","Movement","Formation regions and warning centers","Preparations","Effects","Response","Climatology","Observation and forecasting","Related cyclone types"]}
{"upwards":["Tropical cyclone","Background"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity"],"downwards":["Factors that influence intensity","Formation","Rapid intensification","Dissipation","Methods for assessing intensity","Intensity metrics"]}
{"upwards":["Tropical cyclone","Intensity","Factors that influence intensity"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Formation"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Rapid intensification"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Dissipation"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Methods for assessing intensity"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Intensity metrics"],"downwards":[]}
{"upwards":["Tropical cyclone","Classification and naming"],"downwards":["Classification","Naming"]}
{"upwards":["Tropical cyclone","Classification and naming","Classification"],"downwards":[]}
{"upwards":["Tropical cyclone","Classification and naming","Naming"],"downwards":[]}
{"upwards":["Tropical cyclone","Structure"],"downwards":["Eye and center","Size"]}
{"upwards":["Tropical cyclone","Structure","Eye and center"],"downwards":[]}
{"upwards":["Tropical cyclone","Structure","Size"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement"],"downwards":["Environmental steering","Beta drift","Multiple storm interaction","Interaction with the mid-latitude westerlies"]}
{"upwards":["Tropical cyclone","Movement","Environmental steering"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement","Beta drift"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement","Multiple storm interaction"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement","Interaction with the mid-latitude westerlies"],"downwards":[]}
{"upwards":["Tropical cyclone","Formation regions and warning centers"],"downwards":[]}
{"upwards":["Tropical cyclone","Preparations"],"downwards":[]}
{"upwards":["Tropical cyclone","Effects"],"downwards":["Natural phenomena caused or worsened by tropical cyclones","Effect on property and human life","Environmental effects"]}
{"upwards":["Tropical cyclone","Effects","Natural phenomena caused or worsened by tropical cyclones"],"downwards":[]}
{"upwards":["Tropical cyclone","Effects","Effect on property and human life"],"downwards":[]}
{"upwards":["Tropical cyclone","Effects","Environmental effects"],"downwards":[]}
{"upwards":["Tropical cyclone","Response"],"downwards":[]}
{"upwards":["Tropical cyclone","Climatology"],"downwards":["Influence of climate change"]}
{"upwards":["Tropical cyclone","Climatology","Influence of climate change"],"downwards":[]}
{"upwards":["Tropical cyclone","Observation and forecasting"],"downwards":["Observation","Forecasting","Geopotential height"]}
{"upwards":["Tropical cyclone","Observation and forecasting","Observation"],"downwards":[]}
{"upwards":["Tropical cyclone","Observation and forecasting","Forecasting"],"downwards":[]}
{"upwards":["Tropical cyclone","Observation and forecasting","Geopotential height"],"downwards":[]}
{"upwards":["Tropical cyclone","Related cyclone types"],"downwards":[]}
{"upwards":["Academic publishing"],"downwards":["History","Publishers and business aspects","Scholarly paper","Peer review","Publishing process","Citations","Publishing by discipline","Open access journals","Growth","Role for publishers in scholarly communication"]}
{"upwards":["Academic publishing","History"],"downwards":[]}
{"upwards":["Academic publishing","Publishers and business aspects"],"downwards":["Crisis","Academic journal publishing reform"]}
{"upwards":["Academic publishing","Publishers and business aspects","Crisis"],"downwards":[]}
{"upwards":["Academic publishing","Publishers and business aspects","Academic journal publishing reform"],"downwards":[]}
{"upwards":["Academic publishing","Scholarly paper"],"downwards":["Categories of papers"]}
{"upwards":["Academic publishing","Scholarly paper","Categories of papers"],"downwards":[]}
{"upwards":["Academic publishing","Peer review"],"downwards":["Rejection rate"]}
{"upwards":["Academic publishing","Peer review","Rejection rate"],"downwards":[]}
{"upwards":["Academic publishing","Publishing process"],"downwards":[]}
{"upwards":["Academic publishing","Citations"],"downwards":[]}
{"upwards":["Academic publishing","Publishing by discipline"],"downwards":["Natural sciences","Social sciences","Humanities"]}
{"upwards":["Academic publishing","Publishing by discipline","Natural sciences"],"downwards":[]}
{"upwards":["Academic publishing","Publishing by discipline","Social sciences"],"downwards":[]}
{"upwards":["Academic publishing","Publishing by discipline","Humanities"],"downwards":[]}
{"upwards":["Academic publishing","Open access journals"],"downwards":[]}
{"upwards":["Academic publishing","Growth"],"downwards":[]}
{"upwards":["Academic publishing","Role for publishers in scholarly communication"],"downwards":[]}
{"upwards":["Thoralf Skolem"],"downwards":["Life","Mathematics","Completeness"]}
{"upwards":["Thoralf Skolem","Life"],"downwards":[]}
{"upwards":["Thoralf Skolem","Mathematics"],"downwards":[]}
{"upwards":["Thoralf Skolem","Completeness"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
{"upwards":["Subset","Definition"],"downwards":[]}
{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["Galileo's paradox"],"downwards":["Galileo on infinite sets"]}
{"upwards":["Galileo's paradox","Galileo on infinite sets"],"downwards":[]}
{"upwards":["Burali-Forti paradox"],"downwards":["Stated in terms of von Neumann ordinals","Stated more generally","Resolutions of the paradox"]}
{"upwards":["Burali-Forti paradox","Stated in terms of von Neumann ordinals"],"downwards":[]}
{"upwards":["Burali-Forti paradox","Stated more generally"],"downwards":[]}
{"upwards":["Burali-Forti paradox","Resolutions of the paradox"],"downwards":[]}
{"upwards":["Dedekind-infinite set"],"downwards":["Comparison with the usual definition of infinite set","Dedekind-infinite sets in ZF","History","Relation to the axiom of choice","Proof of equivalence to infinity, assuming axiom of countable choice","Generalizations","Notes","References"]}
{"upwards":["Dedekind-infinite set","Comparison with the usual definition of infinite set"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Dedekind-infinite sets in ZF"],"downwards":[]}
{"upwards":["Dedekind-infinite set","History"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Relation to the axiom of choice"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Proof of equivalence to infinity, assuming axiom of countable choice"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Generalizations"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Notes"],"downwards":[]}
{"upwards":["Dedekind-infinite set","References"],"downwards":[]}
{"upwards":["Thomas Jech"],"downwards":["Life","Work","Bibliography","References","External links"]}
{"upwards":["Thomas Jech","Life"],"downwards":[]}
{"upwards":["Thomas Jech","Work"],"downwards":[]}
{"upwards":["Thomas Jech","Bibliography"],"downwards":[]}
{"upwards":["Thomas Jech","References"],"downwards":[]}
{"upwards":["Thomas Jech","External links"],"downwards":[]}
{"upwards":["Ernst Zermelo"],"downwards":["Life","Research in set theory","Zermelo's navigation problem","Publications"]}
{"upwards":["Ernst Zermelo","Life"],"downwards":[]}
{"upwards":["Ernst Zermelo","Research in set theory"],"downwards":[]}
{"upwards":["Ernst Zermelo","Zermelo's navigation problem"],"downwards":[]}
{"upwards":["Ernst Zermelo","Publications"],"downwards":[]}
{"upwards":["Suslin's problem"],"downwards":["Formulation","Implications"]}
{"upwards":["Suslin's problem","Formulation"],"downwards":[]}
{"upwards":["Suslin's problem","Implications"],"downwards":[]}
{"upwards":["Tropical Storm Vicente (2018)"],"downwards":["Meteorological history","Impact"]}
{"upwards":["Tropical Storm Vicente (2018)","Meteorological history"],"downwards":[]}
{"upwards":["Tropical Storm Vicente (2018)","Impact"],"downwards":["Michoac\u00e1n","Oaxaca","Veracruz","Elsewhere"]}
{"upwards":["Tropical Storm Vicente (2018)","Impact","Michoac\u00e1n"],"downwards":[]}
{"upwards":["Tropical Storm Vicente (2018)","Impact","Oaxaca"],"downwards":[]}
{"upwards":["Tropical Storm Vicente (2018)","Impact","Veracruz"],"downwards":[]}
{"upwards":["Tropical Storm Vicente (2018)","Impact","Elsewhere"],"downwards":[]}
{"upwards":["Willard Van Orman Quine"],"downwards":["Biography","Work","In popular culture","Bibliography","Filmography"]}
{"upwards":["Willard Van Orman Quine","Biography"],"downwards":["Education","World War II","Personal","Harvard"]}
{"upwards":["Willard Van Orman Quine","Biography","Education"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","World War II"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","Personal"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","Harvard"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work"],"downwards":["Logic","Set theory","Metaphysics","Epistemology"]}
{"upwards":["Willard Van Orman Quine","Work","Logic"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Set theory"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics"],"downwards":["Rejection of the analytic\u2013synthetic distinction","Confirmation holism and ontological relativity","Existence and its contrary","Ontological commitment","Indispensability argument for mathematical realism"]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Rejection of the analytic\u2013synthetic distinction"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Confirmation holism and ontological relativity"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Existence and its contrary"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Ontological commitment"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Indispensability argument for mathematical realism"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Epistemology"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","In popular culture"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Bibliography"],"downwards":["Selected books","Important articles"]}
{"upwards":["Willard Van Orman Quine","Bibliography","Selected books"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Bibliography","Important articles"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Filmography"],"downwards":[]}
{"upwards":["Filter (set theory)"],"downwards":["Preliminaries, notation, and basic notions","Filters and prefilters","Set theoretic properties and constructions","Filters and nets"]}
{"upwards":["Filter (set theory)","Preliminaries, notation, and basic notions"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters"],"downwards":["Basic examples","Ultrafilters","Kernels","Finer\/coarser, subordination, and meshing"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Basic examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Ultrafilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels"],"downwards":["Classifying families by their kernels","Characterizing fixed ultra prefilters"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Classifying families by their kernels"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Characterizing fixed ultra prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing"],"downwards":["Equivalent families of sets"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing","Equivalent families of sets"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions"],"downwards":["Trace and meshing","Images and preimages under functions","Products of prefilters","Set subtraction and some examples"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Trace and meshing"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions"],"downwards":["Subordination is preserved by images and preimages"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions","Subordination is preserved by images and preimages"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Products of prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Set subtraction and some examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets"],"downwards":["Nets to prefilters","Prefilters to nets","Subordinate filters and subnets"]}
{"upwards":["Filter (set theory)","Filters and nets","Nets to prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets"],"downwards":["Partially ordered net"]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets","Partially ordered net"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets"],"downwards":["Non\u2013equivalence of subnets and subordinate filters"]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets","Non\u2013equivalence of subnets and subordinate filters"],"downwards":[]}
{"upwards":["List of first-order theories"],"downwards":["Preliminaries","Pure identity theories","Unary relations","Equivalence relations","Orders","Lattices","Graphs","Boolean algebras","Groups","Rings and fields","Geometry","Differential algebra","Addition","Arithmetic","Second order arithmetic","Set theories"]}
{"upwards":["List of first-order theories","Preliminaries"],"downwards":[]}
{"upwards":["List of first-order theories","Pure identity theories"],"downwards":[]}
{"upwards":["List of first-order theories","Unary relations"],"downwards":[]}
{"upwards":["List of first-order theories","Equivalence relations"],"downwards":[]}
{"upwards":["List of first-order theories","Orders"],"downwards":[]}
{"upwards":["List of first-order theories","Lattices"],"downwards":[]}
{"upwards":["List of first-order theories","Graphs"],"downwards":[]}
{"upwards":["List of first-order theories","Boolean algebras"],"downwards":[]}
{"upwards":["List of first-order theories","Groups"],"downwards":[]}
{"upwards":["List of first-order theories","Rings and fields"],"downwards":[]}
{"upwards":["List of first-order theories","Geometry"],"downwards":[]}
{"upwards":["List of first-order theories","Differential algebra"],"downwards":[]}
{"upwards":["List of first-order theories","Addition"],"downwards":[]}
{"upwards":["List of first-order theories","Arithmetic"],"downwards":[]}
{"upwards":["List of first-order theories","Second order arithmetic"],"downwards":[]}
{"upwards":["List of first-order theories","Set theories"],"downwards":[]}
{"upwards":["Ordered pair"],"downwards":["Generalities","Informal and formal definitions","Defining the ordered pair using set theory","Category theory"]}
{"upwards":["Ordered pair","Generalities"],"downwards":[]}
{"upwards":["Ordered pair","Informal and formal definitions"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory"],"downwards":["Wiener's definition","Hausdorff's definition","Kuratowski's definition","Quine\u2013Rosser definition","Cantor\u2013Frege definition","Morse definition","Axiomatic definition"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Wiener's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Hausdorff's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition"],"downwards":["Variants","Proving that definitions satisfy the characteristic property"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Variants"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Proving that definitions satisfy the characteristic property"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Quine\u2013Rosser definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Cantor\u2013Frege definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Morse definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Axiomatic definition"],"downwards":[]}
{"upwards":["Ordered pair","Category theory"],"downwards":[]}
{"upwards":["Grothendieck universe"],"downwards":["Properties","Grothendieck universes and inaccessible cardinals"]}
{"upwards":["Grothendieck universe","Properties"],"downwards":[]}
{"upwards":["Grothendieck universe","Grothendieck universes and inaccessible cardinals"],"downwards":[]}
{"upwards":["Forcing (mathematics)"],"downwards":["Intuition","Forcing conditions and forcing posets","Generic filters","P-names and interpretations","Forcing","Consistency","Cohen forcing","The countable chain condition","Easton forcing","Random reals","Boolean-valued models","Meta-mathematical explanation","Logical explanation"]}
{"upwards":["Forcing (mathematics)","Intuition"],"downwards":["The role of the model"]}
{"upwards":["Forcing (mathematics)","Intuition","The role of the model"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Forcing conditions and forcing posets"],"downwards":["Examples"]}
{"upwards":["Forcing (mathematics)","Forcing conditions and forcing posets","Examples"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Generic filters"],"downwards":[]}
{"upwards":["Forcing (mathematics)","P-names and interpretations"],"downwards":["Rigorous definitions"]}
{"upwards":["Forcing (mathematics)","P-names and interpretations","Rigorous definitions"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Forcing"],"downwards":["Internal definition"]}
{"upwards":["Forcing (mathematics)","Forcing","Internal definition"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Consistency"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Cohen forcing"],"downwards":[]}
{"upwards":["Forcing (mathematics)","The countable chain condition"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Easton forcing"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Random reals"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Boolean-valued models"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Meta-mathematical explanation"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Logical explanation"],"downwards":[]}
{"upwards":["Urelement"],"downwards":["Theory","Urelements in set theory","Quine atoms","References","External links"]}
{"upwards":["Urelement","Theory"],"downwards":[]}
{"upwards":["Urelement","Urelements in set theory"],"downwards":[]}
{"upwards":["Urelement","Quine atoms"],"downwards":[]}
{"upwards":["Urelement","References"],"downwards":[]}
{"upwards":["Urelement","External links"],"downwards":[]}
{"upwards":["Extensionality"],"downwards":["Example","In mathematics"]}
{"upwards":["Extensionality","Example"],"downwards":[]}
{"upwards":["Extensionality","In mathematics"],"downwards":[]}
{"upwards":["Predicate (mathematical logic)"],"downwards":["Predicates in different systems"]}
{"upwards":["Predicate (mathematical logic)","Predicates in different systems"],"downwards":[]}
{"upwards":["Infinite-valued logic"],"downwards":["History","Examples"]}
{"upwards":["Infinite-valued logic","History"],"downwards":[]}
{"upwards":["Infinite-valued logic","Examples"],"downwards":[]}
{"upwards":["Ordinal number"],"downwards":["Ordinals extend the natural numbers","Definitions","Transfinite sequence","Transfinite induction","Arithmetic of ordinals","Ordinals and cardinals","Some \"large\" countable ordinals","Topology and ordinals","History"]}
{"upwards":["Ordinal number","Ordinals extend the natural numbers"],"downwards":[]}
{"upwards":["Ordinal number","Definitions"],"downwards":["Well-ordered sets","Definition of an ordinal as an equivalence class","Von Neumann definition of ordinals","Other definitions"]}
{"upwards":["Ordinal number","Definitions","Well-ordered sets"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Definition of an ordinal as an equivalence class"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Von Neumann definition of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Other definitions"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite sequence"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction"],"downwards":["Transfinite recursion","Successor and limit ordinals","Indexing classes of ordinals","Closed unbounded sets and classes"]}
{"upwards":["Ordinal number","Transfinite induction","Transfinite recursion"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Successor and limit ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Indexing classes of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Closed unbounded sets and classes"],"downwards":[]}
{"upwards":["Ordinal number","Arithmetic of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals"],"downwards":["Initial ordinal of a cardinal","Cofinality"]}
{"upwards":["Ordinal number","Ordinals and cardinals","Initial ordinal of a cardinal"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals","Cofinality"],"downwards":[]}
{"upwards":["Ordinal number","Some \"large\" countable ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Topology and ordinals"],"downwards":[]}
{"upwards":["Ordinal number","History"],"downwards":[]}
{"upwards":["Equivalence relation"],"downwards":["Notation","Definition","Examples","Connections to other relations","Well-definedness under an equivalence relation","Related important definitions","Fundamental theorem of equivalence relations","Comparing equivalence relations","Generating equivalence relations","Algebraic structure","Equivalence relations and mathematical logic"]}
{"upwards":["Equivalence relation","Notation"],"downwards":[]}
{"upwards":["Equivalence relation","Definition"],"downwards":["Alternative definition using relational algebra"]}
{"upwards":["Equivalence relation","Definition","Alternative definition using relational algebra"],"downwards":[]}
{"upwards":["Equivalence relation","Examples"],"downwards":["Simple example","Equivalence relations","Relations that are not equivalences"]}
{"upwards":["Equivalence relation","Examples","Simple example"],"downwards":[]}
{"upwards":["Equivalence relation","Examples","Equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Examples","Relations that are not equivalences"],"downwards":[]}
{"upwards":["Equivalence relation","Connections to other relations"],"downwards":[]}
{"upwards":["Equivalence relation","Well-definedness under an equivalence relation"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions"],"downwards":["Equivalence class","Quotient set","Projection","Equivalence kernel","Partition"]}
{"upwards":["Equivalence relation","Related important definitions","Equivalence class"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Quotient set"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Projection"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Equivalence kernel"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Partition"],"downwards":["Counting partitions"]}
{"upwards":["Equivalence relation","Related important definitions","Partition","Counting partitions"],"downwards":[]}
{"upwards":["Equivalence relation","Fundamental theorem of equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Comparing equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Generating equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure"],"downwards":["Group theory","Categories and groupoids","Lattices"]}
{"upwards":["Equivalence relation","Algebraic structure","Group theory"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure","Categories and groupoids"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure","Lattices"],"downwards":[]}
{"upwards":["Equivalence relation","Equivalence relations and mathematical logic"],"downwards":[]}
{"upwards":["Finite set"],"downwards":["Definition and terminology","Basic properties","Necessary and sufficient conditions for finiteness","Foundational issues","Set-theoretic definitions of finiteness"]}
{"upwards":["Finite set","Definition and terminology"],"downwards":[]}
{"upwards":["Finite set","Basic properties"],"downwards":[]}
{"upwards":["Finite set","Necessary and sufficient conditions for finiteness"],"downwards":[]}
{"upwards":["Finite set","Foundational issues"],"downwards":[]}
{"upwards":["Finite set","Set-theoretic definitions of finiteness"],"downwards":["Other concepts of finiteness"]}
{"upwards":["Finite set","Set-theoretic definitions of finiteness","Other concepts of finiteness"],"downwards":[]}
{"upwards":["Transitive set"],"downwards":["Examples","Properties","Transitive closure","Transitive models of set theory"]}
{"upwards":["Transitive set","Examples"],"downwards":[]}
{"upwards":["Transitive set","Properties"],"downwards":[]}
{"upwards":["Transitive set","Transitive closure"],"downwards":[]}
{"upwards":["Transitive set","Transitive models of set theory"],"downwards":[]}
{"upwards":["Inhabited set"],"downwards":["Comparison with nonempty sets","Example"]}
{"upwards":["Inhabited set","Comparison with nonempty sets"],"downwards":[]}
{"upwards":["Inhabited set","Example"],"downwards":[]}
{"upwards":["Universal set"],"downwards":["Reasons for nonexistence","Theories of universality"]}
{"upwards":["Universal set","Reasons for nonexistence"],"downwards":["Regularity","Russell's paradox","Cantor's theorem"]}
{"upwards":["Universal set","Reasons for nonexistence","Regularity"],"downwards":[]}
{"upwards":["Universal set","Reasons for nonexistence","Russell's paradox"],"downwards":[]}
{"upwards":["Universal set","Reasons for nonexistence","Cantor's theorem"],"downwards":[]}
{"upwards":["Universal set","Theories of universality"],"downwards":["Restricted comprehension","Universal objects that are not sets"]}
{"upwards":["Universal set","Theories of universality","Restricted comprehension"],"downwards":[]}
{"upwards":["Universal set","Theories of universality","Universal objects that are not sets"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)"],"downwards":["Definitions","Generalization to ultra prefilters","Characterizations","Examples, properties, and sufficient conditions","The ultrafilter lemma","Completeness","Ordering on ultrafilters","Ultrafilters on \u2118(\u03c9)"]}
{"upwards":["Ultrafilter (set theory)","Definitions"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Generalization to ultra prefilters"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Characterizations"],"downwards":["Grills and filter-grills","Free or principal"]}
{"upwards":["Ultrafilter (set theory)","Characterizations","Grills and filter-grills"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Characterizations","Free or principal"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Examples, properties, and sufficient conditions"],"downwards":["Monad structure"]}
{"upwards":["Ultrafilter (set theory)","Examples, properties, and sufficient conditions","Monad structure"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma"],"downwards":["Relationships to other statements under ZF"]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF"],"downwards":["Statements that cannot be deduced","Equivalent statements","Weaker statements"]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Statements that cannot be deduced"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Equivalent statements"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Weaker statements"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Completeness"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Ordering on ultrafilters"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Ultrafilters on \u2118(\u03c9)"],"downwards":[]}
{"upwards":["Computable set"],"downwards":["Formal definition","Examples and non-examples","Properties"]}
{"upwards":["Computable set","Formal definition"],"downwards":[]}
{"upwards":["Computable set","Examples and non-examples"],"downwards":[]}
{"upwards":["Computable set","Properties"],"downwards":[]}
{"upwards":["Inch of mercury"],"downwards":["Applications"]}
{"upwards":["Inch of mercury","Applications"],"downwards":["Aircraft and automobiles","Cooling systems","Vacuum brakes"]}
{"upwards":["Inch of mercury","Applications","Aircraft and automobiles"],"downwards":[]}
{"upwards":["Inch of mercury","Applications","Cooling systems"],"downwards":[]}
{"upwards":["Inch of mercury","Applications","Vacuum brakes"],"downwards":[]}
{"upwards":["Von Neumann universe"],"downwards":["Definition","Applications and interpretations","History","Philosophical perspectives"]}
{"upwards":["Von Neumann universe","Definition"],"downwards":["Finite and low cardinality stages of the hierarchy"]}
{"upwards":["Von Neumann universe","Definition","Finite and low cardinality stages of the hierarchy"],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations"],"downwards":["Applications of V as models for set theories","Interpretation of V as the \"set of all sets\"","V and the axiom of regularity","The existential status of V"]}
{"upwards":["Von Neumann universe","Applications and interpretations","Applications of V as models for set theories"],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations","Interpretation of V as the \"set of all sets\""],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations","V and the axiom of regularity"],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations","The existential status of V"],"downwards":[]}
{"upwards":["Von Neumann universe","History"],"downwards":[]}
{"upwards":["Von Neumann universe","Philosophical perspectives"],"downwards":[]}
{"upwards":["Image (mathematics)"],"downwards":["Definition","Inverse image","Notation for image and inverse image","Examples","Properties"]}
{"upwards":["Image (mathematics)","Definition"],"downwards":["Image of an element","Image of a subset","Image of a function","Generalization to binary relations"]}
{"upwards":["Image (mathematics)","Definition","Image of an element"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a subset"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a function"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Generalization to binary relations"],"downwards":[]}
{"upwards":["Image (mathematics)","Inverse image"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image"],"downwards":["Arrow notation","Star notation","Other terminology"]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Arrow notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Star notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Other terminology"],"downwards":[]}
{"upwards":["Image (mathematics)","Examples"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties"],"downwards":["General","Multiple functions","Multiple subsets of domain or codomain"]}
{"upwards":["Image (mathematics)","Properties","General"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple functions"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple subsets of domain or codomain"],"downwards":[]}
{"upwards":["Injective function"],"downwards":["Definition","Examples","Injections can be undone","Injections may be made invertible","Other properties","Proving that functions are injective","Gallery"]}
{"upwards":["Injective function","Definition"],"downwards":[]}
{"upwards":["Injective function","Examples"],"downwards":[]}
{"upwards":["Injective function","Injections can be undone"],"downwards":[]}
{"upwards":["Injective function","Injections may be made invertible"],"downwards":[]}
{"upwards":["Injective function","Other properties"],"downwards":[]}
{"upwards":["Injective function","Proving that functions are injective"],"downwards":[]}
{"upwards":["Injective function","Gallery"],"downwards":[]}
{"upwards":["Constructible universe"],"downwards":["What L is","Additional facts about the sets L\u03b1","L is a standard inner model of ZFC","L is absolute and minimal","L can be well-ordered","L has a reflection principle","The generalized continuum hypothesis holds in L","Constructible sets are definable from the ordinals","Relative constructibility"]}
{"upwards":["Constructible universe","What L is"],"downwards":[]}
{"upwards":["Constructible universe","Additional facts about the sets L\u03b1"],"downwards":[]}
{"upwards":["Constructible universe","L is a standard inner model of ZFC"],"downwards":[]}
{"upwards":["Constructible universe","L is absolute and minimal"],"downwards":["L and large cardinals"]}
{"upwards":["Constructible universe","L is absolute and minimal","L and large cardinals"],"downwards":[]}
{"upwards":["Constructible universe","L can be well-ordered"],"downwards":[]}
{"upwards":["Constructible universe","L has a reflection principle"],"downwards":[]}
{"upwards":["Constructible universe","The generalized continuum hypothesis holds in L"],"downwards":[]}
{"upwards":["Constructible universe","Constructible sets are definable from the ordinals"],"downwards":[]}
{"upwards":["Constructible universe","Relative constructibility"],"downwards":[]}
{"upwards":["Playa Azul (Michoac\u00e1n)"],"downwards":["References"]}
{"upwards":["Playa Azul (Michoac\u00e1n)","References"],"downwards":[]}
{"upwards":["Oaxaca"],"downwards":["History","Geography","Demographics","Government and political geography","Economy","Transport","Media","Culture","Education","Health","Sports","Notable people from Oaxaca"]}
{"upwards":["Oaxaca","History"],"downwards":["Name","Prehistoric and pre-Hispanic period","Spanish colonization","Independence","Late 19th century to present"]}
{"upwards":["Oaxaca","History","Name"],"downwards":[]}
{"upwards":["Oaxaca","History","Prehistoric and pre-Hispanic period"],"downwards":[]}
{"upwards":["Oaxaca","History","Spanish colonization"],"downwards":[]}
{"upwards":["Oaxaca","History","Independence"],"downwards":[]}
{"upwards":["Oaxaca","History","Late 19th century to present"],"downwards":[]}
{"upwards":["Oaxaca","Geography"],"downwards":["Regions, districts and major communities","Climate","Nature and conservation"]}
{"upwards":["Oaxaca","Geography","Regions, districts and major communities"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Climate"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation"],"downwards":["Lagunas de Chacahua National Park","Benito Ju\u00e1rez National Park","Huatulco National Park","Tehuac\u00e1n-Cuicatl\u00e1n Biosphere Reserve"]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Lagunas de Chacahua National Park"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Benito Ju\u00e1rez National Park"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Huatulco National Park"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Tehuac\u00e1n-Cuicatl\u00e1n Biosphere Reserve"],"downwards":[]}
{"upwards":["Oaxaca","Demographics"],"downwards":["Largest cities","Overview","Indigenous peoples","Afro-Mexicans","Religion"]}
{"upwards":["Oaxaca","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Overview"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples"],"downwards":["Zapotecs","Mixtecs","Mazatecos","Mixe","Other"]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Zapotecs"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Mixtecs"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Mazatecos"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Mixe"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Other"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Afro-Mexicans"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Religion"],"downwards":[]}
{"upwards":["Oaxaca","Government and political geography"],"downwards":[]}
{"upwards":["Oaxaca","Economy"],"downwards":["Migration","Agriculture","Mining and industry","Commerce"]}
{"upwards":["Oaxaca","Economy","Migration"],"downwards":[]}
{"upwards":["Oaxaca","Economy","Agriculture"],"downwards":[]}
{"upwards":["Oaxaca","Economy","Mining and industry"],"downwards":[]}
{"upwards":["Oaxaca","Economy","Commerce"],"downwards":[]}
{"upwards":["Oaxaca","Transport"],"downwards":["Road, rail and sea","Air","Local transportation services"]}
{"upwards":["Oaxaca","Transport","Road, rail and sea"],"downwards":[]}
{"upwards":["Oaxaca","Transport","Air"],"downwards":[]}
{"upwards":["Oaxaca","Transport","Local transportation services"],"downwards":[]}
{"upwards":["Oaxaca","Media"],"downwards":[]}
{"upwards":["Oaxaca","Culture"],"downwards":["Arts","Food and drink","Landmarks and tourist attractions","Handicrafts","Symbols"]}
{"upwards":["Oaxaca","Culture","Arts"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Food and drink"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Landmarks and tourist attractions"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Handicrafts"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Symbols"],"downwards":["Flag","Coat of Arms","State emblem"]}
{"upwards":["Oaxaca","Culture","Symbols","Flag"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Symbols","Coat of Arms"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Symbols","State emblem"],"downwards":[]}
{"upwards":["Oaxaca","Education"],"downwards":["Primary education","Higher education"]}
{"upwards":["Oaxaca","Education","Primary education"],"downwards":[]}
{"upwards":["Oaxaca","Education","Higher education"],"downwards":[]}
{"upwards":["Oaxaca","Health"],"downwards":[]}
{"upwards":["Oaxaca","Sports"],"downwards":[]}
{"upwards":["Oaxaca","Notable people from Oaxaca"],"downwards":[]}
{"upwards":["Veracruz"],"downwards":["Etymology","Geography","History","Economy","Culture","Education","Demographics","Tourism","Archeological sites","Government","Infrastructure","Major communities","Notable People"]}
{"upwards":["Veracruz","Etymology"],"downwards":[]}
{"upwards":["Veracruz","Geography"],"downwards":["Political geography","Natural geography","Climate","Ecosystems"]}
{"upwards":["Veracruz","Geography","Political geography"],"downwards":[]}
{"upwards":["Veracruz","Geography","Natural geography"],"downwards":[]}
{"upwards":["Veracruz","Geography","Climate"],"downwards":[]}
{"upwards":["Veracruz","Geography","Ecosystems"],"downwards":[]}
{"upwards":["Veracruz","History"],"downwards":["Pre-Columbian","Colonial period, 1519\u20131821","Independence","20th century to the present"]}
{"upwards":["Veracruz","History","Pre-Columbian"],"downwards":[]}
{"upwards":["Veracruz","History","Colonial period, 1519\u20131821"],"downwards":[]}
{"upwards":["Veracruz","History","Independence"],"downwards":[]}
{"upwards":["Veracruz","History","20th century to the present"],"downwards":[]}
{"upwards":["Veracruz","Economy"],"downwards":["Agriculture","Natural resources","Golden Lane Oil Fields","Industry, transportation and commerce","Handcrafts"]}
{"upwards":["Veracruz","Economy","Agriculture"],"downwards":[]}
{"upwards":["Veracruz","Economy","Natural resources"],"downwards":[]}
{"upwards":["Veracruz","Economy","Golden Lane Oil Fields"],"downwards":[]}
{"upwards":["Veracruz","Economy","Industry, transportation and commerce"],"downwards":[]}
{"upwards":["Veracruz","Economy","Handcrafts"],"downwards":[]}
{"upwards":["Veracruz","Culture"],"downwards":["Gastronomy","Museums","Fairs and festivals","Dance and music","Art and architecture","Literature","Religion"]}
{"upwards":["Veracruz","Culture","Gastronomy"],"downwards":[]}
{"upwards":["Veracruz","Culture","Museums"],"downwards":[]}
{"upwards":["Veracruz","Culture","Fairs and festivals"],"downwards":[]}
{"upwards":["Veracruz","Culture","Dance and music"],"downwards":[]}
{"upwards":["Veracruz","Culture","Art and architecture"],"downwards":[]}
{"upwards":["Veracruz","Culture","Literature"],"downwards":[]}
{"upwards":["Veracruz","Culture","Religion"],"downwards":[]}
{"upwards":["Veracruz","Education"],"downwards":[]}
{"upwards":["Veracruz","Demographics"],"downwards":["Largest cities"]}
{"upwards":["Veracruz","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Veracruz","Tourism"],"downwards":[]}
{"upwards":["Veracruz","Archeological sites"],"downwards":[]}
{"upwards":["Veracruz","Government"],"downwards":[]}
{"upwards":["Veracruz","Infrastructure"],"downwards":["Transport","Media"]}
{"upwards":["Veracruz","Infrastructure","Transport"],"downwards":[]}
{"upwards":["Veracruz","Infrastructure","Media"],"downwards":[]}
{"upwards":["Veracruz","Major communities"],"downwards":[]}
{"upwards":["Veracruz","Notable People"],"downwards":[]}
{"upwards":["Hidalgo (state)"],"downwards":["History","Demographics","Culture","Native gastronomy","Tourism","Geography","Ecology","Economy","Education","Infrastructure","Major communities"]}
{"upwards":["Hidalgo (state)","History"],"downwards":[]}
{"upwards":["Hidalgo (state)","Demographics"],"downwards":[]}
{"upwards":["Hidalgo (state)","Culture"],"downwards":[]}
{"upwards":["Hidalgo (state)","Native gastronomy"],"downwards":[]}
{"upwards":["Hidalgo (state)","Tourism"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography"],"downwards":["Mountains","Valleys","Water","Climate"]}
{"upwards":["Hidalgo (state)","Geography","Mountains"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography","Valleys"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography","Water"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography","Climate"],"downwards":[]}
{"upwards":["Hidalgo (state)","Ecology"],"downwards":[]}
{"upwards":["Hidalgo (state)","Economy"],"downwards":[]}
{"upwards":["Hidalgo (state)","Education"],"downwards":[]}
{"upwards":["Hidalgo (state)","Infrastructure"],"downwards":[]}
{"upwards":["Hidalgo (state)","Major communities"],"downwards":[]}
{"upwards":["Jalisco"],"downwards":["Geography","Demographics","Government and regions","History","Tourism","Culture","Economy","Education","Communication","Infrastructure","Sports","Twinning and covenants"]}
{"upwards":["Jalisco","Geography"],"downwards":["Climate"]}
{"upwards":["Jalisco","Geography","Climate"],"downwards":[]}
{"upwards":["Jalisco","Demographics"],"downwards":["Largest cities"]}
{"upwards":["Jalisco","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Jalisco","Government and regions"],"downwards":[]}
{"upwards":["Jalisco","History"],"downwards":["Nomenclature and seal","Pre-Hispanic period","Colonial period","Independence","Mexican Revolution to present"]}
{"upwards":["Jalisco","History","Nomenclature and seal"],"downwards":[]}
{"upwards":["Jalisco","History","Pre-Hispanic period"],"downwards":[]}
{"upwards":["Jalisco","History","Colonial period"],"downwards":[]}
{"upwards":["Jalisco","History","Independence"],"downwards":[]}
{"upwards":["Jalisco","History","Mexican Revolution to present"],"downwards":[]}
{"upwards":["Jalisco","Tourism"],"downwards":[]}
{"upwards":["Jalisco","Culture"],"downwards":["\"Jalisco is Mexico\"","Mariachi and other music","Traditional clothing and dance","Tequila","Cuisine","Catholic faith"]}
{"upwards":["Jalisco","Culture","\"Jalisco is Mexico\""],"downwards":[]}
{"upwards":["Jalisco","Culture","Mariachi and other music"],"downwards":[]}
{"upwards":["Jalisco","Culture","Traditional clothing and dance"],"downwards":["Dresses"]}
{"upwards":["Jalisco","Culture","Traditional clothing and dance","Dresses"],"downwards":[]}
{"upwards":["Jalisco","Culture","Tequila"],"downwards":[]}
{"upwards":["Jalisco","Culture","Cuisine"],"downwards":[]}
{"upwards":["Jalisco","Culture","Catholic faith"],"downwards":[]}
{"upwards":["Jalisco","Economy"],"downwards":[]}
{"upwards":["Jalisco","Education"],"downwards":[]}
{"upwards":["Jalisco","Communication"],"downwards":[]}
{"upwards":["Jalisco","Infrastructure"],"downwards":["Media","Transport"]}
{"upwards":["Jalisco","Infrastructure","Media"],"downwards":[]}
{"upwards":["Jalisco","Infrastructure","Transport"],"downwards":[]}
{"upwards":["Jalisco","Sports"],"downwards":[]}
{"upwards":["Jalisco","Twinning and covenants"],"downwards":[]}
{"upwards":["Colima"],"downwards":["Geography","Demographics","Principal communities","Economy","Tourist attractions","Culture","Communications and transportation","History","Archaeology","Education","Sister cities"]}
{"upwards":["Colima","Geography"],"downwards":[]}
{"upwards":["Colima","Demographics"],"downwards":[]}
{"upwards":["Colima","Principal communities"],"downwards":[]}
{"upwards":["Colima","Economy"],"downwards":[]}
{"upwards":["Colima","Tourist attractions"],"downwards":[]}
{"upwards":["Colima","Culture"],"downwards":[]}
{"upwards":["Colima","Communications and transportation"],"downwards":["Transport","Media"]}
{"upwards":["Colima","Communications and transportation","Transport"],"downwards":[]}
{"upwards":["Colima","Communications and transportation","Media"],"downwards":[]}
{"upwards":["Colima","History"],"downwards":["Name and seal","Pre-Hispanic period","Colonial era","Independence to the present"]}
{"upwards":["Colima","History","Name and seal"],"downwards":[]}
{"upwards":["Colima","History","Pre-Hispanic period"],"downwards":[]}
{"upwards":["Colima","History","Colonial era"],"downwards":[]}
{"upwards":["Colima","History","Independence to the present"],"downwards":[]}
{"upwards":["Colima","Archaeology"],"downwards":[]}
{"upwards":["Colima","Education"],"downwards":[]}
{"upwards":["Colima","Sister cities"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign"],"downwards":["Background","Planning","Assault","Build-up","Breakout and pursuit","Outcome","Notes","References"]}
{"upwards":["British logistics in the Normandy campaign","Background"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Planning"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Assault"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up"],"downwards":["Organisation","Mulberry harbour","Minor ports","Ordnance","Subsistence","Construction","Salvage","Casualties","Provost"]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Organisation"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Mulberry harbour"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Minor ports"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Ordnance"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Subsistence"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Construction"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Salvage"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Casualties"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Build-up","Provost"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Breakout and pursuit"],"downwards":["Transport","Ports","Salvage","Prisoners"]}
{"upwards":["British logistics in the Normandy campaign","Breakout and pursuit","Transport"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Breakout and pursuit","Ports"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Breakout and pursuit","Salvage"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Breakout and pursuit","Prisoners"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Outcome"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","Notes"],"downwards":[]}
{"upwards":["British logistics in the Normandy campaign","References"],"downwards":[]}
{"upwards":["Guerrero"],"downwards":["History","Demographics","Culture","Tourism","Handcrafts","Politics","Geography","Economy","Education","Infrastructure"]}
{"upwards":["Guerrero","History"],"downwards":["Prehistoric and Pre-Columbian periods","Colonial period","Independence and the later nineteenth century","Mexican Revolution to the present"]}
{"upwards":["Guerrero","History","Prehistoric and Pre-Columbian periods"],"downwards":[]}
{"upwards":["Guerrero","History","Colonial period"],"downwards":[]}
{"upwards":["Guerrero","History","Independence and the later nineteenth century"],"downwards":[]}
{"upwards":["Guerrero","History","Mexican Revolution to the present"],"downwards":[]}
{"upwards":["Guerrero","Demographics"],"downwards":["Major communities"]}
{"upwards":["Guerrero","Demographics","Major communities"],"downwards":[]}
{"upwards":["Guerrero","Culture"],"downwards":[]}
{"upwards":["Guerrero","Tourism"],"downwards":["Triangle of the Sun","Archeological sites","Outdoor activities"]}
{"upwards":["Guerrero","Tourism","Triangle of the Sun"],"downwards":[]}
{"upwards":["Guerrero","Tourism","Archeological sites"],"downwards":[]}
{"upwards":["Guerrero","Tourism","Outdoor activities"],"downwards":[]}
{"upwards":["Guerrero","Handcrafts"],"downwards":[]}
{"upwards":["Guerrero","Politics"],"downwards":[]}
{"upwards":["Guerrero","Geography"],"downwards":[]}
{"upwards":["Guerrero","Economy"],"downwards":[]}
{"upwards":["Guerrero","Education"],"downwards":[]}
{"upwards":["Guerrero","Infrastructure"],"downwards":["Media","Transport"]}
{"upwards":["Guerrero","Infrastructure","Media"],"downwards":[]}
{"upwards":["Guerrero","Infrastructure","Transport"],"downwards":[]}
{"upwards":["Yiddish"],"downwards":["History","Phonology","Grammar","Writing system","Numbers of speakers","Status as a language","Influence on other languages","Language examples"]}
{"upwards":["Yiddish","History"],"downwards":["Origins","Written evidence","Printing","Secularization","20th century"]}
{"upwards":["Yiddish","History","Origins"],"downwards":[]}
{"upwards":["Yiddish","History","Written evidence"],"downwards":[]}
{"upwards":["Yiddish","History","Printing"],"downwards":[]}
{"upwards":["Yiddish","History","Secularization"],"downwards":[]}
{"upwards":["Yiddish","History","20th century"],"downwards":[]}
{"upwards":["Yiddish","Phonology"],"downwards":["Consonants","Vowels","Dialectal variation","Comparison with German","Comparison with Hebrew"]}
{"upwards":["Yiddish","Phonology","Consonants"],"downwards":[]}
{"upwards":["Yiddish","Phonology","Vowels"],"downwards":[]}
{"upwards":["Yiddish","Phonology","Dialectal variation"],"downwards":[]}
{"upwards":["Yiddish","Phonology","Comparison with German"],"downwards":[]}
{"upwards":["Yiddish","Phonology","Comparison with Hebrew"],"downwards":[]}
{"upwards":["Yiddish","Grammar"],"downwards":[]}
{"upwards":["Yiddish","Writing system"],"downwards":[]}
{"upwards":["Yiddish","Numbers of speakers"],"downwards":[]}
{"upwards":["Yiddish","Status as a language"],"downwards":["Israel and Zionism","South Africa","Former Soviet Union","Council of Europe","Sweden","United States","United Kingdom","Canada","Religious communities","Modern Yiddish education","Internet"]}
{"upwards":["Yiddish","Status as a language","Israel and Zionism"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","South Africa"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Former Soviet Union"],"downwards":["Russia","Ukraine"]}
{"upwards":["Yiddish","Status as a language","Former Soviet Union","Russia"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Former Soviet Union","Ukraine"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Council of Europe"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Sweden"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","United States"],"downwards":["Present U.S. speaker population"]}
{"upwards":["Yiddish","Status as a language","United States","Present U.S. speaker population"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","United Kingdom"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Canada"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Religious communities"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Modern Yiddish education"],"downwards":[]}
{"upwards":["Yiddish","Status as a language","Internet"],"downwards":[]}
{"upwards":["Yiddish","Influence on other languages"],"downwards":[]}
{"upwards":["Yiddish","Language examples"],"downwards":[]}
{"upwards":["Martinus (son of Heraclius)"],"downwards":["Life","References","Primary sources"]}
{"upwards":["Martinus (son of Heraclius)","Life"],"downwards":["Reign of Constantine III and Heraclonas","Valentinus's revolt and Martinus's death"]}
{"upwards":["Martinus (son of Heraclius)","Life","Reign of Constantine III and Heraclonas"],"downwards":[]}
{"upwards":["Martinus (son of Heraclius)","Life","Valentinus's revolt and Martinus's death"],"downwards":[]}
{"upwards":["Martinus (son of Heraclius)","References"],"downwards":["Citations","Bibliography"]}
{"upwards":["Martinus (son of Heraclius)","References","Citations"],"downwards":[]}
{"upwards":["Martinus (son of Heraclius)","References","Bibliography"],"downwards":[]}
{"upwards":["Martinus (son of Heraclius)","Primary sources"],"downwards":[]}
{"upwards":["Chris Kreider"],"downwards":["Early life","Playing career","International play","Personal life","Career statistics","Awards and honors","References","External links"]}
{"upwards":["Chris Kreider","Early life"],"downwards":[]}
{"upwards":["Chris Kreider","Playing career"],"downwards":["Amateur","Collegiate","Professional"]}
{"upwards":["Chris Kreider","Playing career","Amateur"],"downwards":[]}
{"upwards":["Chris Kreider","Playing career","Collegiate"],"downwards":[]}
{"upwards":["Chris Kreider","Playing career","Professional"],"downwards":["Early years (2012\u20132015)","Later years"]}
{"upwards":["Chris Kreider","Playing career","Professional","Early years (2012\u20132015)"],"downwards":[]}
{"upwards":["Chris Kreider","Playing career","Professional","Later years"],"downwards":[]}
{"upwards":["Chris Kreider","International play"],"downwards":[]}
{"upwards":["Chris Kreider","Personal life"],"downwards":[]}
{"upwards":["Chris Kreider","Career statistics"],"downwards":["Regular season and playoffs","International"]}
{"upwards":["Chris Kreider","Career statistics","Regular season and playoffs"],"downwards":[]}
{"upwards":["Chris Kreider","Career statistics","International"],"downwards":[]}
{"upwards":["Chris Kreider","Awards and honors"],"downwards":[]}
{"upwards":["Chris Kreider","References"],"downwards":[]}
{"upwards":["Chris Kreider","External links"],"downwards":[]}
{"upwards":["Der Bialistoker Shtern"],"downwards":["Founding","Editors and contributors","Journalistic profile","Orthography","Decline and outbreak of war","References"]}
{"upwards":["Der Bialistoker Shtern","Founding"],"downwards":[]}
{"upwards":["Der Bialistoker Shtern","Editors and contributors"],"downwards":[]}
{"upwards":["Der Bialistoker Shtern","Journalistic profile"],"downwards":[]}
{"upwards":["Der Bialistoker Shtern","Orthography"],"downwards":[]}
{"upwards":["Der Bialistoker Shtern","Decline and outbreak of war"],"downwards":[]}
{"upwards":["Der Bialistoker Shtern","References"],"downwards":[]}
{"upwards":["Live PD"],"downwards":["Overview","Episodes","Notable incidents","Cancellation","Associated programs"]}
{"upwards":["Live PD","Overview"],"downwards":[]}
{"upwards":["Live PD","Episodes"],"downwards":[]}
{"upwards":["Live PD","Notable incidents"],"downwards":[]}
{"upwards":["Live PD","Cancellation"],"downwards":[]}
{"upwards":["Live PD","Associated programs"],"downwards":["Syndication"]}
{"upwards":["Live PD","Associated programs","Syndication"],"downwards":[]}
{"upwards":["Zakir Husain"],"downwards":["Early life and family","Career","Death and legacy","Notes","References","External links"]}
{"upwards":["Zakir Husain","Early life and family"],"downwards":[]}
{"upwards":["Zakir Husain","Career"],"downwards":["Sheikh-ul-Jamia, Jamia Millia Islamia (1926\u20131948)","Basic National Education Committee (1937)","Vice-Chancellor, Aligarh Muslim University (1948\u20131956)","Governor of Bihar (1957\u20131962)","Vice President of India (1962\u20131967)","President of India (1967\u20131969)","Author"]}
{"upwards":["Zakir Husain","Career","Sheikh-ul-Jamia, Jamia Millia Islamia (1926\u20131948)"],"downwards":[]}
{"upwards":["Zakir Husain","Career","Basic National Education Committee (1937)"],"downwards":[]}
{"upwards":["Zakir Husain","Career","Vice-Chancellor, Aligarh Muslim University (1948\u20131956)"],"downwards":[]}
{"upwards":["Zakir Husain","Career","Governor of Bihar (1957\u20131962)"],"downwards":[]}
{"upwards":["Zakir Husain","Career","Vice President of India (1962\u20131967)"],"downwards":[]}
{"upwards":["Zakir Husain","Career","President of India (1967\u20131969)"],"downwards":[]}
{"upwards":["Zakir Husain","Career","Author"],"downwards":[]}
{"upwards":["Zakir Husain","Death and legacy"],"downwards":["Commemoration"]}
{"upwards":["Zakir Husain","Death and legacy","Commemoration"],"downwards":[]}
{"upwards":["Zakir Husain","Notes"],"downwards":[]}
{"upwards":["Zakir Husain","References"],"downwards":[]}
{"upwards":["Zakir Husain","External links"],"downwards":[]}
{"upwards":["Busby Babes"],"downwards":["History"]}
{"upwards":["Busby Babes","History"],"downwards":[]}
{"upwards":["Geoff Bent"],"downwards":["Early life","Manchester United career","Munich air disaster","Legacy","Notes, citations and sources","External links"]}
{"upwards":["Geoff Bent","Early life"],"downwards":[]}
{"upwards":["Geoff Bent","Manchester United career"],"downwards":[]}
{"upwards":["Geoff Bent","Munich air disaster"],"downwards":[]}
{"upwards":["Geoff Bent","Legacy"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources"],"downwards":["Notes","Citations","Sources"]}
{"upwards":["Geoff Bent","Notes, citations and sources","Notes"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Citations"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources"],"downwards":["Books","Newspapers","Online"]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources","Books"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources","Newspapers"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources","Online"],"downwards":[]}
{"upwards":["Geoff Bent","External links"],"downwards":[]}
{"upwards":["Manchester United F.C."],"downwards":["History","Crest and colours","Grounds","Support","Global brand","Ownership and finances","Players","Player awards","Coaching staff","Management","Honours","Manchester United Women","Footnotes","References","Further reading","External links"]}
{"upwards":["Manchester United F.C.","History"],"downwards":["Early years (1878\u20131945)","Busby years (1945\u20131969)","1969\u20131986","Ferguson years (1986\u20132013)","2013\u2013present"]}
{"upwards":["Manchester United F.C.","History","Early years (1878\u20131945)"],"downwards":[]}
{"upwards":["Manchester United F.C.","History","Busby years (1945\u20131969)"],"downwards":[]}
{"upwards":["Manchester United F.C.","History","1969\u20131986"],"downwards":[]}
{"upwards":["Manchester United F.C.","History","Ferguson years (1986\u20132013)"],"downwards":[]}
{"upwards":["Manchester United F.C.","History","2013\u2013present"],"downwards":[]}
{"upwards":["Manchester United F.C.","Crest and colours"],"downwards":[]}
{"upwards":["Manchester United F.C.","Grounds"],"downwards":["1878\u20131893: North Road","1893\u20131910: Bank Street","1910\u2013present: Old Trafford"]}
{"upwards":["Manchester United F.C.","Grounds","1878\u20131893: North Road"],"downwards":[]}
{"upwards":["Manchester United F.C.","Grounds","1893\u20131910: Bank Street"],"downwards":[]}
{"upwards":["Manchester United F.C.","Grounds","1910\u2013present: Old Trafford"],"downwards":[]}
{"upwards":["Manchester United F.C.","Support"],"downwards":["Rivalries"]}
{"upwards":["Manchester United F.C.","Support","Rivalries"],"downwards":[]}
{"upwards":["Manchester United F.C.","Global brand"],"downwards":["Sponsorship"]}
{"upwards":["Manchester United F.C.","Global brand","Sponsorship"],"downwards":[]}
{"upwards":["Manchester United F.C.","Ownership and finances"],"downwards":["Glazer ownership"]}
{"upwards":["Manchester United F.C.","Ownership and finances","Glazer ownership"],"downwards":[]}
{"upwards":["Manchester United F.C.","Players"],"downwards":["First-team squad","Reserves and academy"]}
{"upwards":["Manchester United F.C.","Players","First-team squad"],"downwards":["Out on loan"]}
{"upwards":["Manchester United F.C.","Players","First-team squad","Out on loan"],"downwards":[]}
{"upwards":["Manchester United F.C.","Players","Reserves and academy"],"downwards":["Out on loan"]}
{"upwards":["Manchester United F.C.","Players","Reserves and academy","Out on loan"],"downwards":[]}
{"upwards":["Manchester United F.C.","Player awards"],"downwards":[]}
{"upwards":["Manchester United F.C.","Coaching staff"],"downwards":["Managerial history"]}
{"upwards":["Manchester United F.C.","Coaching staff","Managerial history"],"downwards":[]}
{"upwards":["Manchester United F.C.","Management"],"downwards":["Manchester United Limited","Manchester United Football Club"]}
{"upwards":["Manchester United F.C.","Management","Manchester United Limited"],"downwards":[]}
{"upwards":["Manchester United F.C.","Management","Manchester United Football Club"],"downwards":[]}
{"upwards":["Manchester United F.C.","Honours"],"downwards":["Doubles and Trebles"]}
{"upwards":["Manchester United F.C.","Honours","Doubles and Trebles"],"downwards":[]}
{"upwards":["Manchester United F.C.","Manchester United Women"],"downwards":[]}
{"upwards":["Manchester United F.C.","Footnotes"],"downwards":[]}
{"upwards":["Manchester United F.C.","References"],"downwards":[]}
{"upwards":["Manchester United F.C.","Further reading"],"downwards":[]}
{"upwards":["Manchester United F.C.","External links"],"downwards":[]}
{"upwards":["Final form"],"downwards":["Hebrew","References"]}
{"upwards":["Final form","Hebrew"],"downwards":[]}
{"upwards":["Final form","References"],"downwards":[]}
{"upwards":["Sandy Koufax"],"downwards":["Early life","Professional career","Pitching style","Post-playing career","Legacy","Personal life"]}
{"upwards":["Sandy Koufax","Early life"],"downwards":["Major League tryouts"]}
{"upwards":["Sandy Koufax","Early life","Major League tryouts"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career"],"downwards":["Early years (1955\u20131960)","Domination (1961\u20131964)","Playing in pain (1965\u201366)","Holdout","Retirement","Career overall"]}
{"upwards":["Sandy Koufax","Professional career","Early years (1955\u20131960)"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Domination (1961\u20131964)"],"downwards":["1961 season","1962 season","1963 season","1964 season"]}
{"upwards":["Sandy Koufax","Professional career","Domination (1961\u20131964)","1961 season"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Domination (1961\u20131964)","1962 season"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Domination (1961\u20131964)","1963 season"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Domination (1961\u20131964)","1964 season"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Playing in pain (1965\u201366)"],"downwards":["1965 season","Perfection","World Series and Yom Kippur"]}
{"upwards":["Sandy Koufax","Professional career","Playing in pain (1965\u201366)","1965 season"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Playing in pain (1965\u201366)","Perfection"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Playing in pain (1965\u201366)","World Series and Yom Kippur"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Holdout"],"downwards":["1966 season"]}
{"upwards":["Sandy Koufax","Professional career","Holdout","1966 season"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Retirement"],"downwards":[]}
{"upwards":["Sandy Koufax","Professional career","Career overall"],"downwards":["Career statistics"]}
{"upwards":["Sandy Koufax","Professional career","Career overall","Career statistics"],"downwards":[]}
{"upwards":["Sandy Koufax","Pitching style"],"downwards":[]}
{"upwards":["Sandy Koufax","Post-playing career"],"downwards":[]}
{"upwards":["Sandy Koufax","Legacy"],"downwards":[]}
{"upwards":["Sandy Koufax","Personal life"],"downwards":[]}
{"upwards":["A&E Networks"],"downwards":["History","Units"]}
{"upwards":["A&E Networks","History"],"downwards":["A&E Networks","Post-Lifetime acquisition"]}
{"upwards":["A&E Networks","History","A&E Networks"],"downwards":[]}
{"upwards":["A&E Networks","History","Post-Lifetime acquisition"],"downwards":[]}
{"upwards":["A&E Networks","Units"],"downwards":["TV Channels","A&E Studios","A+E Networks International","A+E Networks Home Entertainment"]}
{"upwards":["A&E Networks","Units","TV Channels"],"downwards":[]}
{"upwards":["A&E Networks","Units","A&E Studios"],"downwards":["Filmography","A&E IndieFilms","History Films"]}
{"upwards":["A&E Networks","Units","A&E Studios","Filmography"],"downwards":[]}
{"upwards":["A&E Networks","Units","A&E Studios","A&E IndieFilms"],"downwards":[]}
{"upwards":["A&E Networks","Units","A&E Studios","History Films"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International"],"downwards":["Availability","A+E Networks Latin America","A+E Networks UK","A+E Networks Germany","A+E Networks Italy","The History Channel Iberia","A+E Networks Asia","A+E Networks India"]}
{"upwards":["A&E Networks","Units","A+E Networks International","Availability"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","A+E Networks Latin America"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","A+E Networks UK"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","A+E Networks Germany"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","A+E Networks Italy"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","The History Channel Iberia"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","A+E Networks Asia"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks International","A+E Networks India"],"downwards":[]}
{"upwards":["A&E Networks","Units","A+E Networks Home Entertainment"],"downwards":[]}
{"upwards":["Governor (India)"],"downwards":["Selection process","Appointment","Powers and functions","Contingency situation","Emoluments","Removal","Legal immunity","Analysis of role in Government"]}
{"upwards":["Governor (India)","Selection process"],"downwards":["Qualifications"]}
{"upwards":["Governor (India)","Selection process","Qualifications"],"downwards":[]}
{"upwards":["Governor (India)","Appointment"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions"],"downwards":["Executive powers","Legislative powers","Financial powers","Discretionary powers"]}
{"upwards":["Governor (India)","Powers and functions","Executive powers"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions","Legislative powers"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions","Financial powers"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions","Discretionary powers"],"downwards":[]}
{"upwards":["Governor (India)","Contingency situation"],"downwards":[]}
{"upwards":["Governor (India)","Emoluments"],"downwards":[]}
{"upwards":["Governor (India)","Removal"],"downwards":[]}
{"upwards":["Governor (India)","Legal immunity"],"downwards":[]}
{"upwards":["Governor (India)","Analysis of role in Government"],"downwards":[]}
{"upwards":["President of India"],"downwards":["Origin","Powers and duties","Selection process","Time of election","Emoluments","Impeachment","Succession","President vs Chief Justice","President vs Prime minister","Important presidential interventions in the past"]}
{"upwards":["President of India","Origin"],"downwards":[]}
{"upwards":["President of India","Powers and duties"],"downwards":["Duty","Legislative powers","Executive powers","Judicial powers","Appointment powers","Financial powers","Diplomatic powers","Military powers","Pardoning powers","Emergency powers"]}
{"upwards":["President of India","Powers and duties","Duty"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Legislative powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Executive powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Judicial powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Appointment powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Financial powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Diplomatic powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Military powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Pardoning powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Emergency powers"],"downwards":["National emergency","State emergency","Financial emergency"]}
{"upwards":["President of India","Powers and duties","Emergency powers","National emergency"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Emergency powers","State emergency"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Emergency powers","Financial emergency"],"downwards":[]}
{"upwards":["President of India","Selection process"],"downwards":["Eligibility"]}
{"upwards":["President of India","Selection process","Eligibility"],"downwards":[]}
{"upwards":["President of India","Time of election"],"downwards":["Conditions for the presidency","Election process","Oath or affirmation"]}
{"upwards":["President of India","Time of election","Conditions for the presidency"],"downwards":[]}
{"upwards":["President of India","Time of election","Election process"],"downwards":[]}
{"upwards":["President of India","Time of election","Oath or affirmation"],"downwards":[]}
{"upwards":["President of India","Emoluments"],"downwards":[]}
{"upwards":["President of India","Impeachment"],"downwards":[]}
{"upwards":["President of India","Succession"],"downwards":[]}
{"upwards":["President of India","President vs Chief Justice"],"downwards":[]}
{"upwards":["President of India","President vs Prime minister"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past"],"downwards":["Proving majority in the parliament","Proof of Majority to form a Government","Vetoing of a Bill","Rashtrapati Bhavan Communiqu\u00e9s","Return of a Bill","Sacking state governors"]}
{"upwards":["President of India","Important presidential interventions in the past","Proving majority in the parliament"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Proof of Majority to form a Government"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Vetoing of a Bill"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Rashtrapati Bhavan Communiqu\u00e9s"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Return of a Bill"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Sacking state governors"],"downwards":[]}
{"upwards":["William James Roe"],"downwards":["Early life","Business career","Writing career","Personal life","References","External links"]}
{"upwards":["William James Roe","Early life"],"downwards":[]}
{"upwards":["William James Roe","Business career"],"downwards":[]}
{"upwards":["William James Roe","Writing career"],"downwards":[]}
{"upwards":["William James Roe","Personal life"],"downwards":[]}
{"upwards":["William James Roe","References"],"downwards":[]}
{"upwards":["William James Roe","External links"],"downwards":[]}
{"upwards":["Republican Palace Museum"],"downwards":["Location","History","Functionality","Departments","Notes","References","External links"]}
{"upwards":["Republican Palace Museum","Location"],"downwards":["Opening hours"]}
{"upwards":["Republican Palace Museum","Location","Opening hours"],"downwards":[]}
{"upwards":["Republican Palace Museum","History"],"downwards":[]}
{"upwards":["Republican Palace Museum","Functionality"],"downwards":["Cultural and social activities","Educational activities"]}
{"upwards":["Republican Palace Museum","Functionality","Cultural and social activities"],"downwards":[]}
{"upwards":["Republican Palace Museum","Functionality","Educational activities"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments"],"downwards":["Automobile suite","Photographs and paintings suite","Honours suite","Gift suite","Suite of musical instruments, utensils and furniture","Documentation suite","Decoration suite","Weapons suite","Library"]}
{"upwards":["Republican Palace Museum","Departments","Automobile suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Photographs and paintings suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Honours suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Gift suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Suite of musical instruments, utensils and furniture"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Documentation suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Decoration suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Weapons suite"],"downwards":[]}
{"upwards":["Republican Palace Museum","Departments","Library"],"downwards":[]}
{"upwards":["Republican Palace Museum","Notes"],"downwards":[]}
{"upwards":["Republican Palace Museum","References"],"downwards":[]}
{"upwards":["Republican Palace Museum","External links"],"downwards":[]}
{"upwards":["Khartoum"],"downwards":["Etymology","History","Geography","Climate","Demographics","Economy","Education","Transportation","Architecture","Culture","References","Bibliography","External links"]}
{"upwards":["Khartoum","Etymology"],"downwards":[]}
{"upwards":["Khartoum","History"],"downwards":["19th century","20th century","21st century"]}
{"upwards":["Khartoum","History","19th century"],"downwards":[]}
{"upwards":["Khartoum","History","20th century"],"downwards":[]}
{"upwards":["Khartoum","History","21st century"],"downwards":[]}
{"upwards":["Khartoum","Geography"],"downwards":["Location"]}
{"upwards":["Khartoum","Geography","Location"],"downwards":[]}
{"upwards":["Khartoum","Climate"],"downwards":[]}
{"upwards":["Khartoum","Demographics"],"downwards":[]}
{"upwards":["Khartoum","Economy"],"downwards":["Retailing"]}
{"upwards":["Khartoum","Economy","Retailing"],"downwards":[]}
{"upwards":["Khartoum","Education"],"downwards":[]}
{"upwards":["Khartoum","Transportation"],"downwards":[]}
{"upwards":["Khartoum","Architecture"],"downwards":[]}
{"upwards":["Khartoum","Culture"],"downwards":["Museums","Botanical gardens","Clubs","Places of worship"]}
{"upwards":["Khartoum","Culture","Museums"],"downwards":[]}
{"upwards":["Khartoum","Culture","Botanical gardens"],"downwards":[]}
{"upwards":["Khartoum","Culture","Clubs"],"downwards":[]}
{"upwards":["Khartoum","Culture","Places of worship"],"downwards":[]}
{"upwards":["Khartoum","References"],"downwards":[]}
{"upwards":["Khartoum","Bibliography"],"downwards":[]}
{"upwards":["Khartoum","External links"],"downwards":[]}
{"upwards":["Balasore"],"downwards":["History","Geography","Demographics","Government and politics","Economy","Culture\/Cityscape","Transport","Education","Sports and entertainment","Notable people"]}
{"upwards":["Balasore","History"],"downwards":[]}
{"upwards":["Balasore","Geography"],"downwards":["Climate"]}
{"upwards":["Balasore","Geography","Climate"],"downwards":[]}
{"upwards":["Balasore","Demographics"],"downwards":["Languages"]}
{"upwards":["Balasore","Demographics","Languages"],"downwards":[]}
{"upwards":["Balasore","Government and politics"],"downwards":["Civic Utility \/ Amenities \/ Services"]}
{"upwards":["Balasore","Government and politics","Civic Utility \/ Amenities \/ Services"],"downwards":["Healthcare"]}
{"upwards":["Balasore","Government and politics","Civic Utility \/ Amenities \/ Services","Healthcare"],"downwards":[]}
{"upwards":["Balasore","Economy"],"downwards":["Industry"]}
{"upwards":["Balasore","Economy","Industry"],"downwards":[]}
{"upwards":["Balasore","Culture\/Cityscape"],"downwards":["Festivals","Places of interest"]}
{"upwards":["Balasore","Culture\/Cityscape","Festivals"],"downwards":[]}
{"upwards":["Balasore","Culture\/Cityscape","Places of interest"],"downwards":[]}
{"upwards":["Balasore","Transport"],"downwards":["Air","Rail","Road"]}
{"upwards":["Balasore","Transport","Air"],"downwards":[]}
{"upwards":["Balasore","Transport","Rail"],"downwards":[]}
{"upwards":["Balasore","Transport","Road"],"downwards":[]}
{"upwards":["Balasore","Education"],"downwards":["Technical colleges\/institutions","Universities\/colleges","Schools"]}
{"upwards":["Balasore","Education","Technical colleges\/institutions"],"downwards":[]}
{"upwards":["Balasore","Education","Universities\/colleges"],"downwards":[]}
{"upwards":["Balasore","Education","Schools"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment"],"downwards":["Permit Field","Reserved Police Line Ground","Zilla School Stadium","ITI Field","Entertainment"]}
{"upwards":["Balasore","Sports and entertainment","Permit Field"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","Reserved Police Line Ground"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","Zilla School Stadium"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","ITI Field"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","Entertainment"],"downwards":[]}
{"upwards":["Balasore","Notable people"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam"],"downwards":["Background","Destruction","Attribution","Consequences","Reactions"]}
{"upwards":["Destruction of the Kakhovka Dam","Background"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Destruction"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Attribution"],"downwards":["Claims of Russian responsibility","Claims of Ukrainian responsibility"]}
{"upwards":["Destruction of the Kakhovka Dam","Attribution","Claims of Russian responsibility"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Attribution","Claims of Ukrainian responsibility"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences"],"downwards":["Flooding and evacuations","Animals and environment","Counteroffensive","Water supply"]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Flooding and evacuations"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Animals and environment"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Counteroffensive"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Water supply"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions"],"downwards":["Ukraine","International community"]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions","Ukraine"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions","International community"],"downwards":[]}
{"upwards":["Gujarat Titans"],"downwards":["Franchise history","Team history","Team identity","Home ground","Kit manufacturers and sponsors","Seasons","Current squad","Administration and support staff","Statistics","References","External links"]}
{"upwards":["Gujarat Titans","Franchise history"],"downwards":[]}
{"upwards":["Gujarat Titans","Team history"],"downwards":["2022 IPL season"]}
{"upwards":["Gujarat Titans","Team history","2022 IPL season"],"downwards":[]}
{"upwards":["Gujarat Titans","Team identity"],"downwards":[]}
{"upwards":["Gujarat Titans","Home ground"],"downwards":[]}
{"upwards":["Gujarat Titans","Kit manufacturers and sponsors"],"downwards":[]}
{"upwards":["Gujarat Titans","Seasons"],"downwards":["Indian Premier League"]}
{"upwards":["Gujarat Titans","Seasons","Indian Premier League"],"downwards":[]}
{"upwards":["Gujarat Titans","Current squad"],"downwards":[]}
{"upwards":["Gujarat Titans","Administration and support staff"],"downwards":[]}
{"upwards":["Gujarat Titans","Statistics"],"downwards":["Overall results in the IPL","By opposition"]}
{"upwards":["Gujarat Titans","Statistics","Overall results in the IPL"],"downwards":[]}
{"upwards":["Gujarat Titans","Statistics","By opposition"],"downwards":[]}
{"upwards":["Gujarat Titans","References"],"downwards":[]}
{"upwards":["Gujarat Titans","External links"],"downwards":[]}
{"upwards":["2023 Indian Premier League"],"downwards":["Administration","Background","Participating teams","Personnel changes","Rules","Venues","Format","Points table","Match summary","League stage","Playoffs","Broadcasting","Criticism","Statistics and awards","Notes","References","External links"]}
{"upwards":["2023 Indian Premier League","Administration"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Background"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Participating teams"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Personnel changes"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Rules"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Venues"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Format"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Points table"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Match summary"],"downwards":[]}
{"upwards":["2023 Indian Premier League","League stage"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs"],"downwards":["Qualifier 1","Eliminator","Qualifier 2","Final"]}
{"upwards":["2023 Indian Premier League","Playoffs","Qualifier 1"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs","Eliminator"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs","Qualifier 2"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs","Final"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Broadcasting"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Criticism"],"downwards":["Slow over- rate","Impact player rule"]}
{"upwards":["2023 Indian Premier League","Criticism","Slow over- rate"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Criticism","Impact player rule"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Statistics and awards"],"downwards":["Most runs","Most wickets","End of the season awards"]}
{"upwards":["2023 Indian Premier League","Statistics and awards","Most runs"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Statistics and awards","Most wickets"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Statistics and awards","End of the season awards"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Notes"],"downwards":[]}
{"upwards":["2023 Indian Premier League","References"],"downwards":[]}
{"upwards":["2023 Indian Premier League","External links"],"downwards":[]}
{"upwards":["2023 Indianapolis 500"],"downwards":["Race background","Race schedules","Entry list","Testing and rookie orientation","Practice","Time trials","Post-qualifying practice","Starting grid","Race report","Box score","Broadcasting","References","External links"]}
{"upwards":["2023 Indianapolis 500","Race background"],"downwards":["Sponsorship","Rule changes","2023 IndyCar Series"]}
{"upwards":["2023 Indianapolis 500","Race background","Sponsorship"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race background","Rule changes"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race background","2023 IndyCar Series"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race schedules"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Entry list"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Testing and rookie orientation"],"downwards":["Open Test\u2014Thursday April 20","Open test\u2014Friday April 21"]}
{"upwards":["2023 Indianapolis 500","Testing and rookie orientation","Open Test\u2014Thursday April 20"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Testing and rookie orientation","Open test\u2014Friday April 21"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice"],"downwards":["Opening day\u2014Tuesday May 16","Wednesday May 17","Thursday May 18","Fast Friday\u2014Friday May 19"]}
{"upwards":["2023 Indianapolis 500","Practice","Opening day\u2014Tuesday May 16"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice","Wednesday May 17"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice","Thursday May 18"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice","Fast Friday\u2014Friday May 19"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials"],"downwards":["Saturday, May 20","Sunday, May 21"]}
{"upwards":["2023 Indianapolis 500","Time trials","Saturday, May 20"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21"],"downwards":["Top 12 qualifying","Last Chance Qualifying","Fast Six Qualifying"]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21","Top 12 qualifying"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21","Last Chance Qualifying"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21","Fast Six Qualifying"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice"],"downwards":["Post-qualifying practice\u2014Monday, May 22","Special test\u2014Thursday, May 25","Carb Day\u2014Friday, May 26","Pit Stop Challenge"]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Post-qualifying practice\u2014Monday, May 22"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Special test\u2014Thursday, May 25"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Carb Day\u2014Friday, May 26"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Pit Stop Challenge"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Starting grid"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report"],"downwards":["First half","Second half"]}
{"upwards":["2023 Indianapolis 500","Race report","First half"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report","Second half"],"downwards":["First red flag","Second red flag","Third red flag and finish"]}
{"upwards":["2023 Indianapolis 500","Race report","Second half","First red flag"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report","Second half","Second red flag"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report","Second half","Third red flag and finish"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Box score"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Broadcasting"],"downwards":["Television","Radio"]}
{"upwards":["2023 Indianapolis 500","Broadcasting","Television"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Broadcasting","Radio"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","References"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","External links"],"downwards":[]}
{"upwards":["Kakhovka Dam"],"downwards":["Dam","Russian invasion of Ukraine"]}
{"upwards":["Kakhovka Dam","Dam"],"downwards":[]}
{"upwards":["Kakhovka Dam","Russian invasion of Ukraine"],"downwards":["Destruction of the dam"]}
{"upwards":["Kakhovka Dam","Russian invasion of Ukraine","Destruction of the dam"],"downwards":[]}
{"upwards":["President of Turkey"],"downwards":["History","Leadership roles","Selection process","Incumbency","Duties and responsibilities","Acting President","Latest election","Timeline of presidents"]}
{"upwards":["President of Turkey","History"],"downwards":[]}
{"upwards":["President of Turkey","Leadership roles"],"downwards":["Head of state","Head of party"]}
{"upwards":["President of Turkey","Leadership roles","Head of state"],"downwards":[]}
{"upwards":["President of Turkey","Leadership roles","Head of party"],"downwards":[]}
{"upwards":["President of Turkey","Selection process"],"downwards":["Eligibility","Election","Inauguration"]}
{"upwards":["President of Turkey","Selection process","Eligibility"],"downwards":[]}
{"upwards":["President of Turkey","Selection process","Election"],"downwards":[]}
{"upwards":["President of Turkey","Selection process","Inauguration"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency"],"downwards":["Term limit","Vacancies and successions","Accountability and non-accountability","Residence"]}
{"upwards":["President of Turkey","Incumbency","Term limit"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Vacancies and successions"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Accountability and non-accountability"],"downwards":["After the 2017 constitutional amendment","Before the 2017 constitutional amendment"]}
{"upwards":["President of Turkey","Incumbency","Accountability and non-accountability","After the 2017 constitutional amendment"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Accountability and non-accountability","Before the 2017 constitutional amendment"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Residence"],"downwards":[]}
{"upwards":["President of Turkey","Duties and responsibilities"],"downwards":[]}
{"upwards":["President of Turkey","Acting President"],"downwards":["After the 2017 constitutional referendum","Before the 2017 constitutional referendum"]}
{"upwards":["President of Turkey","Acting President","After the 2017 constitutional referendum"],"downwards":[]}
{"upwards":["President of Turkey","Acting President","Before the 2017 constitutional referendum"],"downwards":[]}
{"upwards":["President of Turkey","Latest election"],"downwards":[]}
{"upwards":["President of Turkey","Timeline of presidents"],"downwards":[]}
{"upwards":["Chennai Super Kings"],"downwards":["History","Home ground","Players","Identity","Kit and sponsors","Support and fan following","Brand value","Controversies","Result summary","Statistics","Rivalries","Seasons","Current squad","Administration and support staff","In popular culture"]}
{"upwards":["Chennai Super Kings","History"],"downwards":["2008\u20132009: First seasons","2010: IPL and CLT20 double","2011\u20132015: Success and suspension","2018\u2013present"]}
{"upwards":["Chennai Super Kings","History","2008\u20132009: First seasons"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2010: IPL and CLT20 double"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2011\u20132015: Success and suspension"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present"],"downwards":["2018","2019","2020","2021","2022","2023"]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2018"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2019"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2020"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2021"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2022"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2023"],"downwards":[]}
{"upwards":["Chennai Super Kings","Home ground"],"downwards":[]}
{"upwards":["Chennai Super Kings","Players"],"downwards":[]}
{"upwards":["Chennai Super Kings","Identity"],"downwards":["Name, logo, crest and colors","Theme song"]}
{"upwards":["Chennai Super Kings","Identity","Name, logo, crest and colors"],"downwards":[]}
{"upwards":["Chennai Super Kings","Identity","Theme song"],"downwards":[]}
{"upwards":["Chennai Super Kings","Kit and sponsors"],"downwards":[]}
{"upwards":["Chennai Super Kings","Support and fan following"],"downwards":[]}
{"upwards":["Chennai Super Kings","Brand value"],"downwards":[]}
{"upwards":["Chennai Super Kings","Controversies"],"downwards":["Ownership of N. Srinivasan","2013 betting scandal","IPL suspension"]}
{"upwards":["Chennai Super Kings","Controversies","Ownership of N. Srinivasan"],"downwards":[]}
{"upwards":["Chennai Super Kings","Controversies","2013 betting scandal"],"downwards":[]}
{"upwards":["Chennai Super Kings","Controversies","IPL suspension"],"downwards":[]}
{"upwards":["Chennai Super Kings","Result summary"],"downwards":["Overall results","By opposition","Overall results in the Champions League T20"]}
{"upwards":["Chennai Super Kings","Result summary","Overall results"],"downwards":[]}
{"upwards":["Chennai Super Kings","Result summary","By opposition"],"downwards":[]}
{"upwards":["Chennai Super Kings","Result summary","Overall results in the Champions League T20"],"downwards":[]}
{"upwards":["Chennai Super Kings","Statistics"],"downwards":["Most runs","Most wickets"]}
{"upwards":["Chennai Super Kings","Statistics","Most runs"],"downwards":[]}
{"upwards":["Chennai Super Kings","Statistics","Most wickets"],"downwards":[]}
{"upwards":["Chennai Super Kings","Rivalries"],"downwards":["Rivalry with Mumbai Indians","Rivalry with Royal Challengers Bangalore"]}
{"upwards":["Chennai Super Kings","Rivalries","Rivalry with Mumbai Indians"],"downwards":[]}
{"upwards":["Chennai Super Kings","Rivalries","Rivalry with Royal Challengers Bangalore"],"downwards":[]}
{"upwards":["Chennai Super Kings","Seasons"],"downwards":["Indian Premier League","Champions League T20"]}
{"upwards":["Chennai Super Kings","Seasons","Indian Premier League"],"downwards":[]}
{"upwards":["Chennai Super Kings","Seasons","Champions League T20"],"downwards":[]}
{"upwards":["Chennai Super Kings","Current squad"],"downwards":[]}
{"upwards":["Chennai Super Kings","Administration and support staff"],"downwards":[]}
{"upwards":["Chennai Super Kings","In popular culture"],"downwards":[]}
{"upwards":["Josef Newgarden"],"downwards":["Career","Personal life","Media appearances","Racing record","References","External links"]}
{"upwards":["Josef Newgarden","Career"],"downwards":["Karting","Early formula car racing","IndyCar","Other racing"]}
{"upwards":["Josef Newgarden","Career","Karting"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","Early formula car racing"],"downwards":["Skip Barber","Formula Ford","Formula Palmer Audi","GP3 Series","Indy Lights"]}
{"upwards":["Josef Newgarden","Career","Early formula car racing","Skip Barber"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","Early formula car racing","Formula Ford"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","Early formula car racing","Formula Palmer Audi"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","Early formula car racing","GP3 Series"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","Early formula car racing","Indy Lights"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","IndyCar"],"downwards":["2012\u20132014: Sarah Fisher Hartman Racing","2015: CFH Racing","2016: Ed Carpenter Racing","2017\u2013present: Team Penske"]}
{"upwards":["Josef Newgarden","Career","IndyCar","2012\u20132014: Sarah Fisher Hartman Racing"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","IndyCar","2015: CFH Racing"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","IndyCar","2016: Ed Carpenter Racing"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","IndyCar","2017\u2013present: Team Penske"],"downwards":[]}
{"upwards":["Josef Newgarden","Career","Other racing"],"downwards":[]}
{"upwards":["Josef Newgarden","Personal life"],"downwards":[]}
{"upwards":["Josef Newgarden","Media appearances"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record"],"downwards":["Career summary","Complete GP3 Series results","American open\u2013wheel racing results","Superstar Racing Experience","Complete IMSA SportsCar Championship results"]}
{"upwards":["Josef Newgarden","Racing record","Career summary"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record","Complete GP3 Series results"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record","American open\u2013wheel racing results"],"downwards":["Indy Lights","IndyCar Series","Indianapolis 500"]}
{"upwards":["Josef Newgarden","Racing record","American open\u2013wheel racing results","Indy Lights"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record","American open\u2013wheel racing results","IndyCar Series"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record","American open\u2013wheel racing results","Indianapolis 500"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record","Superstar Racing Experience"],"downwards":[]}
{"upwards":["Josef Newgarden","Racing record","Complete IMSA SportsCar Championship results"],"downwards":[]}
{"upwards":["Josef Newgarden","References"],"downwards":[]}
{"upwards":["Josef Newgarden","External links"],"downwards":[]}
{"upwards":["2023 Turkish presidential election"],"downwards":["Background","Date","Electoral system","Candidates","Endorsements","Second round","Controversies","Opinion polls","Results","Aftermath"]}
{"upwards":["2023 Turkish presidential election","Background"],"downwards":["2018 elections"]}
{"upwards":["2023 Turkish presidential election","Background","2018 elections"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Date"],"downwards":["Discussion on snap elections","Discussion on postponing election date","Election calendar"]}
{"upwards":["2023 Turkish presidential election","Date","Discussion on snap elections"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Date","Discussion on postponing election date"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Date","Election calendar"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Electoral system"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Candidates"],"downwards":["Candidates","Nominations"]}
{"upwards":["2023 Turkish presidential election","Candidates","Candidates"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Candidates","Nominations"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Endorsements"],"downwards":["Parties and alliances"]}
{"upwards":["2023 Turkish presidential election","Endorsements","Parties and alliances"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round"],"downwards":["Negotiations by O\u011fan and \u00d6zda\u011f","Erdo\u011fan's strategy","K\u0131l\u0131\u00e7daro\u011flu's strategy","Potential debate"]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f"],"downwards":["Dissolution of ATA Alliance","Endorsement of Erdo\u011fan by O\u011fan","\u00d6zda\u011f's support for K\u0131l\u0131\u00e7daro\u011flu"]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f","Dissolution of ATA Alliance"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f","Endorsement of Erdo\u011fan by O\u011fan"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f","\u00d6zda\u011f's support for K\u0131l\u0131\u00e7daro\u011flu"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Erdo\u011fan's strategy"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","K\u0131l\u0131\u00e7daro\u011flu's strategy"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Potential debate"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies"],"downwards":["Erdo\u011fan's right to seek re-election","Political violence","Erdo\u011fan's photo on ballot paper","Fake videos","Block of Ek\u015fi S\u00f6zl\u00fck","Reporting of the results","Azerbaijani and Russian interference allegations"]}
{"upwards":["2023 Turkish presidential election","Controversies","Erdo\u011fan's right to seek re-election"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Political violence"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Erdo\u011fan's photo on ballot paper"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Fake videos"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Block of Ek\u015fi S\u00f6zl\u00fck"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Reporting of the results"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Azerbaijani and Russian interference allegations"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Opinion polls"],"downwards":["First round","Second round"]}
{"upwards":["2023 Turkish presidential election","Opinion polls","First round"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Opinion polls","Second round"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Results"],"downwards":["Analysis"]}
{"upwards":["2023 Turkish presidential election","Results","Analysis"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Aftermath"],"downwards":["Domestic"]}
{"upwards":["2023 Turkish presidential election","Aftermath","Domestic"],"downwards":["International"]}
{"upwards":["2023 Turkish presidential election","Aftermath","Domestic","International"],"downwards":[]}
{"upwards":["2023 Odisha train collision"],"downwards":["Crash","Emergency response","Victims","Investigation","Aftermath","Reactions"]}
{"upwards":["2023 Odisha train collision","Crash"],"downwards":[]}
{"upwards":["2023 Odisha train collision","Emergency response"],"downwards":[]}
{"upwards":["2023 Odisha train collision","Victims"],"downwards":[]}
{"upwards":["2023 Odisha train collision","Investigation"],"downwards":[]}
{"upwards":["2023 Odisha train collision","Aftermath"],"downwards":[]}
{"upwards":["2023 Odisha train collision","Reactions"],"downwards":[]}
{"upwards":["2023 Indian Premier League final"],"downwards":["Background","Road to the final","Match","References","Note","External links"]}
{"upwards":["2023 Indian Premier League final","Background"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Road to the final"],"downwards":["Group stage","League stage matches between finalists","Playoffs"]}
{"upwards":["2023 Indian Premier League final","Road to the final","Group stage"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Road to the final","League stage matches between finalists"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Road to the final","Playoffs"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Match"],"downwards":["Match officials","Summary","Gujarat Titans innings","Chennai Super Kings innings","Scorecard"]}
{"upwards":["2023 Indian Premier League final","Match","Match officials"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Match","Summary"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Match","Gujarat Titans innings"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Match","Chennai Super Kings innings"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Match","Scorecard"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","References"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","Note"],"downwards":[]}
{"upwards":["2023 Indian Premier League final","External links"],"downwards":[]}
{"upwards":["Cynthia Weil"],"downwards":["Life and career","Songs written by Barry Mann and Cynthia Weil","Bibliography","References","External links"]}
{"upwards":["Cynthia Weil","Life and career"],"downwards":[]}
{"upwards":["Cynthia Weil","Songs written by Barry Mann and Cynthia Weil"],"downwards":[]}
{"upwards":["Cynthia Weil","Bibliography"],"downwards":[]}
{"upwards":["Cynthia Weil","References"],"downwards":[]}
{"upwards":["Cynthia Weil","External links"],"downwards":[]}
{"upwards":["June 7"],"downwards":["Events","Births","Deaths","Holidays and observances","References","External links"]}
{"upwards":["June 7","Events"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 7","Events","Pre-1600"],"downwards":[]}
{"upwards":["June 7","Events","1601\u20131900"],"downwards":[]}
{"upwards":["June 7","Events","1901\u2013present"],"downwards":[]}
{"upwards":["June 7","Births"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 7","Births","Pre-1600"],"downwards":[]}
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{"upwards":["June 7","Deaths","Pre-1600"],"downwards":[]}
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{"upwards":["June 7","Deaths","1901\u2013present"],"downwards":[]}
{"upwards":["June 7","Holidays and observances"],"downwards":[]}
{"upwards":["June 7","References"],"downwards":[]}
{"upwards":["June 7","External links"],"downwards":[]}
{"upwards":["Ama Ata Aidoo"],"downwards":["Early life","Education","Career","Writings","Death","Honours and recognition","Selected works","References","Further reading","External links"]}
{"upwards":["Ama Ata Aidoo","Early life"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Education"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Career"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Writings"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Death"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Honours and recognition"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Selected works"],"downwards":["As editor"]}
{"upwards":["Ama Ata Aidoo","Selected works","As editor"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","References"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","Further reading"],"downwards":[]}
{"upwards":["Ama Ata Aidoo","External links"],"downwards":[]}
{"upwards":["Natasha Al-Maani"],"downwards":["Biography","Style and subject matter","References","External links"]}
{"upwards":["Natasha Al-Maani","Biography"],"downwards":[]}
{"upwards":["Natasha Al-Maani","Style and subject matter"],"downwards":[]}
{"upwards":["Natasha Al-Maani","References"],"downwards":[]}
{"upwards":["Natasha Al-Maani","External links"],"downwards":[]}
{"upwards":["Theodosius II"],"downwards":["Early life","Reign","Death"]}
{"upwards":["Theodosius II","Early life"],"downwards":[]}
{"upwards":["Theodosius II","Reign"],"downwards":["Early reign","University and Law Code","Famous \"apple incident\"","Wars with the Huns, Vandals, and Persians","Theological disputes"]}
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{"upwards":["Theodosius II","Reign","Famous \"apple incident\""],"downwards":[]}
{"upwards":["Theodosius II","Reign","Wars with the Huns, Vandals, and Persians"],"downwards":[]}
{"upwards":["Theodosius II","Reign","Theological disputes"],"downwards":[]}
{"upwards":["Theodosius II","Death"],"downwards":[]}
{"upwards":["Robert Hanssen"],"downwards":["Early life","FBI career and first espionage activities (1976\u20131981)","FBI counterintelligence unit, further espionage activities (1985\u20131991)","Later FBI career, continued espionage activities (1992\u20132001)","Investigation and arrest","Guilty plea and imprisonment","Modus operandi","Personal life","Death","In the media","Notes","References and further reading","External links"]}
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{"upwards":["Robert Hanssen","FBI counterintelligence unit, further espionage activities (1985\u20131991)"],"downwards":[]}
{"upwards":["Robert Hanssen","Later FBI career, continued espionage activities (1992\u20132001)"],"downwards":[]}
{"upwards":["Robert Hanssen","Investigation and arrest"],"downwards":[]}
{"upwards":["Robert Hanssen","Guilty plea and imprisonment"],"downwards":[]}
{"upwards":["Robert Hanssen","Modus operandi"],"downwards":[]}
{"upwards":["Robert Hanssen","Personal life"],"downwards":[]}
{"upwards":["Robert Hanssen","Death"],"downwards":[]}
{"upwards":["Robert Hanssen","In the media"],"downwards":[]}
{"upwards":["Robert Hanssen","Notes"],"downwards":[]}
{"upwards":["Robert Hanssen","References and further reading"],"downwards":[]}
{"upwards":["Robert Hanssen","External links"],"downwards":[]}
{"upwards":["2023 Sudan conflict"],"downwards":["Background","Timeline","Casualties","Foreign involvement","Evacuation of foreign nationals","Humanitarian impact","Peace efforts","Disinformation","Reactions","References"]}
{"upwards":["2023 Sudan conflict","Background"],"downwards":["Political context","Prelude"]}
{"upwards":["2023 Sudan conflict","Background","Political context"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Background","Prelude"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Timeline"],"downwards":["April","May"]}
{"upwards":["2023 Sudan conflict","Timeline","April"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Timeline","May"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Casualties"],"downwards":["By location","Foreign casualties","Casualties among humanitarian workers"]}
{"upwards":["2023 Sudan conflict","Casualties","By location"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Casualties","Foreign casualties"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Casualties","Casualties among humanitarian workers"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Foreign involvement"],"downwards":["Libya","Wagner Group","Egypt","Ethiopia","United Arab Emirates"]}
{"upwards":["2023 Sudan conflict","Foreign involvement","Libya"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Foreign involvement","Wagner Group"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Foreign involvement","Egypt"],"downwards":["Egyptian POWs"]}
{"upwards":["2023 Sudan conflict","Foreign involvement","Egypt","Egyptian POWs"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Foreign involvement","Ethiopia"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Foreign involvement","United Arab Emirates"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Evacuation of foreign nationals"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Humanitarian impact"],"downwards":["Refugees"]}
{"upwards":["2023 Sudan conflict","Humanitarian impact","Refugees"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Peace efforts"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Disinformation"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Reactions"],"downwards":["Domestic","International"]}
{"upwards":["2023 Sudan conflict","Reactions","Domestic"],"downwards":["Military","Civilian"]}
{"upwards":["2023 Sudan conflict","Reactions","Domestic","Military"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Reactions","Domestic","Civilian"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Reactions","International"],"downwards":["Countries","Organizations"]}
{"upwards":["2023 Sudan conflict","Reactions","International","Countries"],"downwards":[]}
{"upwards":["2023 Sudan conflict","Reactions","International","Organizations"],"downwards":[]}
{"upwards":["2023 Sudan conflict","References"],"downwards":[]}
{"upwards":["Kurt Widmer"],"downwards":["Career","Recordings","Publication","Awards","References","External links"]}
{"upwards":["Kurt Widmer","Career"],"downwards":[]}
{"upwards":["Kurt Widmer","Recordings"],"downwards":[]}
{"upwards":["Kurt Widmer","Publication"],"downwards":[]}
{"upwards":["Kurt Widmer","Awards"],"downwards":[]}
{"upwards":["Kurt Widmer","References"],"downwards":[]}
{"upwards":["Kurt Widmer","External links"],"downwards":[]}
{"upwards":["Jessie Maple"],"downwards":["Early life and education","Career","Death","Selected filmography","Books","Notes","References","External links"]}
{"upwards":["Jessie Maple","Early life and education"],"downwards":[]}
{"upwards":["Jessie Maple","Career"],"downwards":[]}
{"upwards":["Jessie Maple","Death"],"downwards":[]}
{"upwards":["Jessie Maple","Selected filmography"],"downwards":["Features","Documentaries"]}
{"upwards":["Jessie Maple","Selected filmography","Features"],"downwards":[]}
{"upwards":["Jessie Maple","Selected filmography","Documentaries"],"downwards":[]}
{"upwards":["Jessie Maple","Books"],"downwards":[]}
{"upwards":["Jessie Maple","Notes"],"downwards":[]}
{"upwards":["Jessie Maple","References"],"downwards":[]}
{"upwards":["Jessie Maple","External links"],"downwards":[]}
{"upwards":["Matt Landers"],"downwards":["Early life","College career","Professional career","References"]}
{"upwards":["Matt Landers","Early life"],"downwards":[]}
{"upwards":["Matt Landers","College career"],"downwards":[]}
{"upwards":["Matt Landers","Professional career"],"downwards":[]}
{"upwards":["Matt Landers","References"],"downwards":[]}
{"upwards":["421"],"downwards":["Events","Births","Deaths","References"]}
{"upwards":["421","Events"],"downwards":["By place"]}
{"upwards":["421","Events","By place"],"downwards":["Roman Empire","Europe","Persia"]}
{"upwards":["421","Events","By place","Roman Empire"],"downwards":[]}
{"upwards":["421","Events","By place","Europe"],"downwards":[]}
{"upwards":["421","Events","By place","Persia"],"downwards":[]}
{"upwards":["421","Births"],"downwards":[]}
{"upwards":["421","Deaths"],"downwards":[]}
{"upwards":["421","References"],"downwards":[]}
{"upwards":["Deaths in 2023"],"downwards":["June","Previous months","References","External links"]}
{"upwards":["Deaths in 2023","June"],"downwards":["7","6","5","4","3","2","1"]}
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{"upwards":["Deaths in 2023","June","5"],"downwards":[]}
{"upwards":["Deaths in 2023","June","4"],"downwards":[]}
{"upwards":["Deaths in 2023","June","3"],"downwards":[]}
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{"upwards":["Deaths in 2023","References"],"downwards":[]}
{"upwards":["Deaths in 2023","External links"],"downwards":[]}
{"upwards":["Structure (mathematical logic)"],"downwards":["Definition","Induced substructures and closed subsets","Homomorphisms and embeddings","Structures and first-order logic","Many-sorted structures","Other generalizations"]}
{"upwards":["Structure (mathematical logic)","Definition"],"downwards":["Domain","Signature","Interpretation function","Examples"]}
{"upwards":["Structure (mathematical logic)","Definition","Domain"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Signature"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Interpretation function"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets"],"downwards":["Examples"]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings"],"downwards":["Homomorphisms","Embeddings","Example","Homomorphism problem"]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphisms"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Embeddings"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Example"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphism problem"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic"],"downwards":["Satisfaction relation","Definable relations"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Satisfaction relation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations"],"downwards":["Definability with parameters","Implicit definability"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Definability with parameters"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Implicit definability"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Many-sorted structures"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations"],"downwards":["Partial algebras","Structures for typed languages","Higher-order languages","Structures that are proper classes"]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Partial algebras"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures for typed languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Higher-order languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures that are proper classes"],"downwards":[]}
{"upwards":["Keila"],"downwards":["History","Military base","Nature","Education","Population","Demographics","International relations","Notable residents"]}
{"upwards":["Keila","History"],"downwards":[]}
{"upwards":["Keila","Military base"],"downwards":[]}
{"upwards":["Keila","Nature"],"downwards":[]}
{"upwards":["Keila","Education"],"downwards":[]}
{"upwards":["Keila","Population"],"downwards":[]}
{"upwards":["Keila","Demographics"],"downwards":[]}
{"upwards":["Keila","International relations"],"downwards":["Twin towns \u2014 sister cities"]}
{"upwards":["Keila","International relations","Twin towns \u2014 sister cities"],"downwards":[]}
{"upwards":["Keila","Notable residents"],"downwards":[]}
{"upwards":["Larva"],"downwards":["Selected types of larvae","Insect larvae"]}
{"upwards":["Larva","Selected types of larvae"],"downwards":[]}
{"upwards":["Larva","Insect larvae"],"downwards":[]}
{"upwards":["Scrophularia"],"downwards":["Selected species","References"]}
{"upwards":["Scrophularia","Selected species"],"downwards":[]}
{"upwards":["Scrophularia","References"],"downwards":[]}
{"upwards":["Tenthredinidae"],"downwards":["Life cycle of Cladius difformis","Taxonomy","References","Bibliography","External links"]}
{"upwards":["Tenthredinidae","Life cycle of Cladius difformis"],"downwards":[]}
{"upwards":["Tenthredinidae","Taxonomy"],"downwards":["Subfamilies and genera","Phylogeny"]}
{"upwards":["Tenthredinidae","Taxonomy","Subfamilies and genera"],"downwards":[]}
{"upwards":["Tenthredinidae","Taxonomy","Phylogeny"],"downwards":[]}
{"upwards":["Tenthredinidae","References"],"downwards":[]}
{"upwards":["Tenthredinidae","Bibliography"],"downwards":[]}
{"upwards":["Tenthredinidae","External links"],"downwards":[]}
{"upwards":["June 8"],"downwards":["Events","Births","Deaths","Holidays and observances","References","External links"]}
{"upwards":["June 8","Events"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 8","Events","Pre-1600"],"downwards":[]}
{"upwards":["June 8","Events","1601\u20131900"],"downwards":[]}
{"upwards":["June 8","Events","1901\u2013present"],"downwards":[]}
{"upwards":["June 8","Births"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 8","Births","Pre-1600"],"downwards":[]}
{"upwards":["June 8","Births","1601\u20131900"],"downwards":[]}
{"upwards":["June 8","Births","1901\u2013present"],"downwards":[]}
{"upwards":["June 8","Deaths"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 8","Deaths","Pre-1600"],"downwards":[]}
{"upwards":["June 8","Deaths","1601\u20131900"],"downwards":[]}
{"upwards":["June 8","Deaths","1901\u2013present"],"downwards":[]}
{"upwards":["June 8","Holidays and observances"],"downwards":[]}
{"upwards":["June 8","References"],"downwards":[]}
{"upwards":["June 8","External links"],"downwards":[]}
{"upwards":["Verbascum"],"downwards":["Description","Cultivation","Other uses","Select species"]}
{"upwards":["Verbascum","Description"],"downwards":[]}
{"upwards":["Verbascum","Cultivation"],"downwards":[]}
{"upwards":["Verbascum","Other uses"],"downwards":[]}
{"upwards":["Verbascum","Select species"],"downwards":[]}
{"upwards":["Tenthredo scrophulariae"],"downwards":["Distribution and habitat","Description","Biology","Gallery","References","External links"]}
{"upwards":["Tenthredo scrophulariae","Distribution and habitat"],"downwards":[]}
{"upwards":["Tenthredo scrophulariae","Description"],"downwards":[]}
{"upwards":["Tenthredo scrophulariae","Biology"],"downwards":[]}
{"upwards":["Tenthredo scrophulariae","Gallery"],"downwards":[]}
{"upwards":["Tenthredo scrophulariae","References"],"downwards":[]}
{"upwards":["Tenthredo scrophulariae","External links"],"downwards":[]}
{"upwards":["June 6"],"downwards":["Events","Births","Deaths","Holidays and observances","References","External links"]}
{"upwards":["June 6","Events"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 6","Events","Pre-1600"],"downwards":[]}
{"upwards":["June 6","Events","1601\u20131900"],"downwards":[]}
{"upwards":["June 6","Events","1901\u2013present"],"downwards":[]}
{"upwards":["June 6","Births"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 6","Births","Pre-1600"],"downwards":[]}
{"upwards":["June 6","Births","1601\u20131900"],"downwards":[]}
{"upwards":["June 6","Births","1901\u2013present"],"downwards":[]}
{"upwards":["June 6","Deaths"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 6","Deaths","Pre-1600"],"downwards":[]}
{"upwards":["June 6","Deaths","1601\u20131900"],"downwards":[]}
{"upwards":["June 6","Deaths","1901\u2013present"],"downwards":[]}
{"upwards":["June 6","Holidays and observances"],"downwards":[]}
{"upwards":["June 6","References"],"downwards":[]}
{"upwards":["June 6","External links"],"downwards":[]}
{"upwards":["Deductive reasoning"],"downwards":["Definition","Conceptions of deduction","Rules of inference","Validity and soundness","Difference from ampliative reasoning","In various fields","History","Related concepts and theories"]}
{"upwards":["Deductive reasoning","Definition"],"downwards":[]}
{"upwards":["Deductive reasoning","Conceptions of deduction"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference"],"downwards":["Prominent rules of inference","Fallacies","Definitory and strategic rules"]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference"],"downwards":["Modus ponens","Modus tollens","Hypothetical syllogism"]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference","Modus ponens"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference","Modus tollens"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference","Hypothetical syllogism"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Fallacies"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Definitory and strategic rules"],"downwards":[]}
{"upwards":["Deductive reasoning","Validity and soundness"],"downwards":[]}
{"upwards":["Deductive reasoning","Difference from ampliative reasoning"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields"],"downwards":["Cognitive psychology","Epistemology","Probability logic"]}
{"upwards":["Deductive reasoning","In various fields","Cognitive psychology"],"downwards":["Psychological theories of deductive reasoning","Intelligence"]}
{"upwards":["Deductive reasoning","In various fields","Cognitive psychology","Psychological theories of deductive reasoning"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields","Cognitive psychology","Intelligence"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields","Epistemology"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields","Probability logic"],"downwards":[]}
{"upwards":["Deductive reasoning","History"],"downwards":[]}
{"upwards":["Deductive reasoning","Related concepts and theories"],"downwards":["Deductivism","Natural deduction","Geometrical method"]}
{"upwards":["Deductive reasoning","Related concepts and theories","Deductivism"],"downwards":[]}
{"upwards":["Deductive reasoning","Related concepts and theories","Natural deduction"],"downwards":[]}
{"upwards":["Deductive reasoning","Related concepts and theories","Geometrical method"],"downwards":[]}
{"upwards":["Wikimedia Foundation"],"downwards":["Mission","History","Projects and initiatives","Technology","Finances","Staff","Independent contractors","Disputes","References","External links"]}
{"upwards":["Wikimedia Foundation","Mission"],"downwards":[]}
{"upwards":["Wikimedia Foundation","History"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Projects and initiatives"],"downwards":["Wikimedia projects","Affiliates","Wikimania"]}
{"upwards":["Wikimedia Foundation","Projects and initiatives","Wikimedia projects"],"downwards":[]}
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{"upwards":["Wikimedia Foundation","Projects and initiatives","Wikimania"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Technology"],"downwards":["Hardware","Software"]}
{"upwards":["Wikimedia Foundation","Technology","Hardware"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Technology","Software"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Finances"],"downwards":["Wikimedia Endowment","Financial development","Expenses","Grants"]}
{"upwards":["Wikimedia Foundation","Finances","Wikimedia Endowment"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Finances","Financial development"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Finances","Expenses"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Finances","Grants"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Staff"],"downwards":["History","Present department structure","Board of trustees"]}
{"upwards":["Wikimedia Foundation","Staff","History"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Staff","Present department structure"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Staff","Board of trustees"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Independent contractors"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Disputes"],"downwards":["Excessive spending and fundraising","Knowledge Engine project"]}
{"upwards":["Wikimedia Foundation","Disputes","Excessive spending and fundraising"],"downwards":[]}
{"upwards":["Wikimedia Foundation","Disputes","Knowledge Engine project"],"downwards":[]}
{"upwards":["Wikimedia Foundation","References"],"downwards":[]}
{"upwards":["Wikimedia Foundation","External links"],"downwards":["Organization","Financials","Charity status","Community"]}
{"upwards":["Wikimedia Foundation","External links","Organization"],"downwards":[]}
{"upwards":["Wikimedia Foundation","External links","Financials"],"downwards":[]}
{"upwards":["Wikimedia Foundation","External links","Charity status"],"downwards":[]}
{"upwards":["Wikimedia Foundation","External links","Community"],"downwards":[]}
{"upwards":["Concept"],"downwards":["Ontology of concepts","Notable theories on the structure of concepts","Ideasthesia","Etymology"]}
{"upwards":["Concept","Ontology of concepts"],"downwards":["Concepts as mental representations","Concepts as abstract objects"]}
{"upwards":["Concept","Ontology of concepts","Concepts as mental representations"],"downwards":["The psychological view of concepts","The physicalist view of concepts"]}
{"upwards":["Concept","Ontology of concepts","Concepts as mental representations","The psychological view of concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as mental representations","The physicalist view of concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects"],"downwards":["A priori concepts","Embodied content","Realist universal concepts","Sense and reference","Concepts in calculus"]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","A priori concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Embodied content"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Realist universal concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Sense and reference"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Concepts in calculus"],"downwards":[]}
{"upwards":["Concept","Notable theories on the structure of concepts"],"downwards":["Classical theory","Prototype theory","Theory-theory"]}
{"upwards":["Concept","Notable theories on the structure of concepts","Classical theory"],"downwards":["Arguments against the classical theory"]}
{"upwards":["Concept","Notable theories on the structure of concepts","Classical theory","Arguments against the classical theory"],"downwards":[]}
{"upwards":["Concept","Notable theories on the structure of concepts","Prototype theory"],"downwards":[]}
{"upwards":["Concept","Notable theories on the structure of concepts","Theory-theory"],"downwards":[]}
{"upwards":["Concept","Ideasthesia"],"downwards":[]}
{"upwards":["Concept","Etymology"],"downwards":[]}
{"upwards":["Ontology"],"downwards":["Etymology","Overview","Types of ontologies","History","Other ontological topics"]}
{"upwards":["Ontology","Etymology"],"downwards":[]}
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{"upwards":["Ontology","Overview","Particulars and universals"],"downwards":[]}
{"upwards":["Ontology","Overview","Abstract and concrete"],"downwards":[]}
{"upwards":["Ontology","Overview","Ontological dependence"],"downwards":[]}
{"upwards":["Ontology","Overview","Identity"],"downwards":[]}
{"upwards":["Ontology","Overview","Modality"],"downwards":[]}
{"upwards":["Ontology","Overview","Substances"],"downwards":[]}
{"upwards":["Ontology","Overview","Properties and relations"],"downwards":[]}
{"upwards":["Ontology","Overview","States of affairs and events"],"downwards":[]}
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{"upwards":["Ontology","Types of ontologies"],"downwards":["Flat vs polycategorical vs hierarchical","Thing ontologies vs fact ontologies","Constituent ontologies vs blob theories","Information science and natural sciences"]}
{"upwards":["Ontology","Types of ontologies","Flat vs polycategorical vs hierarchical"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies","Thing ontologies vs fact ontologies"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies","Constituent ontologies vs blob theories"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies","Information science and natural sciences"],"downwards":[]}
{"upwards":["Ontology","History"],"downwards":["Ancient Greek","Hindu philosophy","Medieval","Modern","20th century"]}
{"upwards":["Ontology","History","Ancient Greek"],"downwards":[]}
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{"upwards":["Ontology","History","Modern"],"downwards":[]}
{"upwards":["Ontology","History","20th century"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics"],"downwards":["Ontological formations","Ontology of fictional characters","Ontological and epistemological certainty","Body and environment, questioning the meaning of being","Ontology and language","Ontology and human geography","Anthropology","Reality and actuality","Microcosmic ontology","Ontological argument","Hintikka's locution for existence"]}
{"upwards":["Ontology","Other ontological topics","Ontological formations"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontology of fictional characters"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontological and epistemological certainty"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Body and environment, questioning the meaning of being"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontology and language"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontology and human geography"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Anthropology"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Reality and actuality"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Microcosmic ontology"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontological argument"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Hintikka's locution for existence"],"downwards":[]}
{"upwards":["Number"],"downwards":["History","Main classification","Subclasses of the integers","Subclasses of the complex numbers","Extensions of the concept"]}
{"upwards":["Number","History"],"downwards":["Numerals","First use of numbers","Zero","Negative numbers","Rational numbers","Irrational numbers","Transcendental numbers and reals","Infinity and infinitesimals","Complex numbers","Prime numbers"]}
{"upwards":["Number","History","Numerals"],"downwards":[]}
{"upwards":["Number","History","First use of numbers"],"downwards":[]}
{"upwards":["Number","History","Zero"],"downwards":[]}
{"upwards":["Number","History","Negative numbers"],"downwards":[]}
{"upwards":["Number","History","Rational numbers"],"downwards":[]}
{"upwards":["Number","History","Irrational numbers"],"downwards":[]}
{"upwards":["Number","History","Transcendental numbers and reals"],"downwards":[]}
{"upwards":["Number","History","Infinity and infinitesimals"],"downwards":[]}
{"upwards":["Number","History","Complex numbers"],"downwards":[]}
{"upwards":["Number","History","Prime numbers"],"downwards":[]}
{"upwards":["Number","Main classification"],"downwards":["Natural numbers","Integers","Rational numbers","Real numbers","Complex numbers"]}
{"upwards":["Number","Main classification","Natural numbers"],"downwards":[]}
{"upwards":["Number","Main classification","Integers"],"downwards":[]}
{"upwards":["Number","Main classification","Rational numbers"],"downwards":[]}
{"upwards":["Number","Main classification","Real numbers"],"downwards":[]}
{"upwards":["Number","Main classification","Complex numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the integers"],"downwards":["Even and odd numbers","Prime numbers","Other classes of integers"]}
{"upwards":["Number","Subclasses of the integers","Even and odd numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the integers","Prime numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the integers","Other classes of integers"],"downwards":[]}
{"upwards":["Number","Subclasses of the complex numbers"],"downwards":["Algebraic, irrational and transcendental numbers","Constructible numbers","Computable numbers"]}
{"upwards":["Number","Subclasses of the complex numbers","Algebraic, irrational and transcendental numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the complex numbers","Constructible numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the complex numbers","Computable numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept"],"downwards":["p-adic numbers","Hypercomplex numbers","Transfinite numbers","Nonstandard numbers"]}
{"upwards":["Number","Extensions of the concept","p-adic numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept","Hypercomplex numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept","Transfinite numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept","Nonstandard numbers"],"downwards":[]}
{"upwards":["Transformation (function)"],"downwards":["Partial transformations","Algebraic structures","Combinatorics"]}
{"upwards":["Transformation (function)","Partial transformations"],"downwards":[]}
{"upwards":["Transformation (function)","Algebraic structures"],"downwards":[]}
{"upwards":["Transformation (function)","Combinatorics"],"downwards":[]}
{"upwards":["Space (mathematics)"],"downwards":["History","Taxonomy of spaces","Types of spaces","Spaces and structure","List of mathematical spaces"]}
{"upwards":["Space (mathematics)","History"],"downwards":["Before the golden age of geometry","The golden age of geometry and afterwards"]}
{"upwards":["Space (mathematics)","History","Before the golden age of geometry"],"downwards":[]}
{"upwards":["Space (mathematics)","History","The golden age of geometry and afterwards"],"downwards":[]}
{"upwards":["Space (mathematics)","Taxonomy of spaces"],"downwards":["Three taxonomic ranks","Relations between species of spaces"]}
{"upwards":["Space (mathematics)","Taxonomy of spaces","Three taxonomic ranks"],"downwards":[]}
{"upwards":["Space (mathematics)","Taxonomy of spaces","Relations between species of spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces"],"downwards":["Linear and topological spaces","Affine and projective spaces","Metric and uniform spaces","Normed, Banach, inner product, and Hilbert spaces","Smooth and Riemannian manifolds","Measurable, measure, and probability spaces","Non-commutative geometry","Schemes","Topoi"]}
{"upwards":["Space (mathematics)","Types of spaces","Linear and topological spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Affine and projective spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Metric and uniform spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Normed, Banach, inner product, and Hilbert spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Smooth and Riemannian manifolds"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Measurable, measure, and probability spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Non-commutative geometry"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Schemes"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Topoi"],"downwards":[]}
{"upwards":["Space (mathematics)","Spaces and structure"],"downwards":[]}
{"upwards":["Space (mathematics)","List of mathematical spaces"],"downwards":[]}
{"upwards":["Number theory"],"downwards":["History","Main subdivisions","Other subfields","Applications","Prizes"]}
{"upwards":["Number theory","History"],"downwards":["Origins","Early modern number theory","Maturity and division into subfields"]}
{"upwards":["Number theory","History","Origins"],"downwards":["Dawn of arithmetic","Classical Greece and the early Hellenistic period","Diophantus","\u0100ryabha\u1e6da, Brahmagupta, Bh\u0101skara","Arithmetic in the Islamic golden age","Western Europe in the Middle Ages"]}
{"upwards":["Number theory","History","Origins","Dawn of arithmetic"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Classical Greece and the early Hellenistic period"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Diophantus"],"downwards":[]}
{"upwards":["Number theory","History","Origins","\u0100ryabha\u1e6da, Brahmagupta, Bh\u0101skara"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Arithmetic in the Islamic golden age"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Western Europe in the Middle Ages"],"downwards":[]}
{"upwards":["Number theory","History","Early modern number theory"],"downwards":["Fermat","Euler","Lagrange, Legendre, and Gauss"]}
{"upwards":["Number theory","History","Early modern number theory","Fermat"],"downwards":[]}
{"upwards":["Number theory","History","Early modern number theory","Euler"],"downwards":[]}
{"upwards":["Number theory","History","Early modern number theory","Lagrange, Legendre, and Gauss"],"downwards":[]}
{"upwards":["Number theory","History","Maturity and division into subfields"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions"],"downwards":["Elementary number theory","Analytic number theory","Algebraic number theory","Diophantine geometry"]}
{"upwards":["Number theory","Main subdivisions","Elementary number theory"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions","Analytic number theory"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions","Algebraic number theory"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions","Diophantine geometry"],"downwards":[]}
{"upwards":["Number theory","Other subfields"],"downwards":["Probabilistic number theory","Arithmetic combinatorics","Computational number theory"]}
{"upwards":["Number theory","Other subfields","Probabilistic number theory"],"downwards":[]}
{"upwards":["Number theory","Other subfields","Arithmetic combinatorics"],"downwards":[]}
{"upwards":["Number theory","Other subfields","Computational number theory"],"downwards":[]}
{"upwards":["Number theory","Applications"],"downwards":[]}
{"upwards":["Number theory","Prizes"],"downwards":[]}
{"upwards":["Combinatorics"],"downwards":["Definition","History","Approaches and subfields of combinatorics","Related fields"]}
{"upwards":["Combinatorics","Definition"],"downwards":[]}
{"upwards":["Combinatorics","History"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics"],"downwards":["Enumerative combinatorics","Analytic combinatorics","Partition theory","Graph theory","Design theory","Finite geometry","Order theory","Matroid theory","Extremal combinatorics","Probabilistic combinatorics","Algebraic combinatorics","Combinatorics on words","Geometric combinatorics","Topological combinatorics","Arithmetic combinatorics","Infinitary combinatorics"]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Enumerative combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Analytic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Partition theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Graph theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Design theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Finite geometry"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Order theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Matroid theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Extremal combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Probabilistic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Algebraic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Combinatorics on words"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Geometric combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Topological combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Arithmetic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Infinitary combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Related fields"],"downwards":["Combinatorial optimization","Coding theory","Discrete and computational geometry","Combinatorics and dynamical systems","Combinatorics and physics"]}
{"upwards":["Combinatorics","Related fields","Combinatorial optimization"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Coding theory"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Discrete and computational geometry"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Combinatorics and dynamical systems"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Combinatorics and physics"],"downwards":[]}
{"upwards":["Point (geometry)"],"downwards":["Points in Euclidean geometry","Dimension of a point","Geometry without points","Point masses and the Dirac delta function"]}
{"upwards":["Point (geometry)","Points in Euclidean geometry"],"downwards":[]}
{"upwards":["Point (geometry)","Dimension of a point"],"downwards":["Vector space dimension","Topological dimension","Hausdorff dimension"]}
{"upwards":["Point (geometry)","Dimension of a point","Vector space dimension"],"downwards":[]}
{"upwards":["Point (geometry)","Dimension of a point","Topological dimension"],"downwards":[]}
{"upwards":["Point (geometry)","Dimension of a point","Hausdorff dimension"],"downwards":[]}
{"upwards":["Point (geometry)","Geometry without points"],"downwards":[]}
{"upwards":["Point (geometry)","Point masses and the Dirac delta function"],"downwards":[]}
{"upwards":["Polygon"],"downwards":["Etymology","Classification","Properties and formulas","Generalizations","Naming","History","In nature","Computer graphics"]}
{"upwards":["Polygon","Etymology"],"downwards":[]}
{"upwards":["Polygon","Classification"],"downwards":["Number of sides","Convexity and intersection","Equality and symmetry","Miscellaneous"]}
{"upwards":["Polygon","Classification","Number of sides"],"downwards":[]}
{"upwards":["Polygon","Classification","Convexity and intersection"],"downwards":[]}
{"upwards":["Polygon","Classification","Equality and symmetry"],"downwards":[]}
{"upwards":["Polygon","Classification","Miscellaneous"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas"],"downwards":["Angles","Area","Centroid"]}
{"upwards":["Polygon","Properties and formulas","Angles"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Area"],"downwards":["Simple polygons","Regular polygons","Self-intersecting"]}
{"upwards":["Polygon","Properties and formulas","Area","Simple polygons"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Area","Regular polygons"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Area","Self-intersecting"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Centroid"],"downwards":[]}
{"upwards":["Polygon","Generalizations"],"downwards":[]}
{"upwards":["Polygon","Naming"],"downwards":[]}
{"upwards":["Polygon","History"],"downwards":[]}
{"upwards":["Polygon","In nature"],"downwards":[]}
{"upwards":["Polygon","Computer graphics"],"downwards":[]}
{"upwards":["Line (geometry)"],"downwards":["Properties","Definition","Other representations","Generalizations of the Euclidean line","Extensions"]}
{"upwards":["Line (geometry)","Properties"],"downwards":["In higher dimensions","Types","In axiomatic systems"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions"],"downwards":["Collinear points"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions","Collinear points"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","Types"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","In axiomatic systems"],"downwards":[]}
{"upwards":["Line (geometry)","Definition"],"downwards":["Linear equation","Parametric equation","Hesse normal form"]}
{"upwards":["Line (geometry)","Definition","Linear equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Parametric equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Hesse normal form"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations"],"downwards":["Vectors","Polar coordinates"]}
{"upwards":["Line (geometry)","Other representations","Vectors"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations","Polar coordinates"],"downwards":[]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line"],"downwards":["Projective geometry"]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line","Projective geometry"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions"],"downwards":["Ray","Line segment","Number line"]}
{"upwards":["Line (geometry)","Extensions","Ray"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Line segment"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Number line"],"downwards":[]}
{"upwards":["Derangement"],"downwards":["Example","Counting derangements","Growth of number of derangements as n approaches \u221e","Generalizations","Computational complexity","References","External links"]}
{"upwards":["Derangement","Example"],"downwards":[]}
{"upwards":["Derangement","Counting derangements"],"downwards":["Derivation by inclusion\u2013exclusion principle"]}
{"upwards":["Derangement","Counting derangements","Derivation by inclusion\u2013exclusion principle"],"downwards":[]}
{"upwards":["Derangement","Growth of number of derangements as n approaches \u221e"],"downwards":["Asymptotic expansion in terms of Bell numbers"]}
{"upwards":["Derangement","Growth of number of derangements as n approaches \u221e","Asymptotic expansion in terms of Bell numbers"],"downwards":[]}
{"upwards":["Derangement","Generalizations"],"downwards":[]}
{"upwards":["Derangement","Computational complexity"],"downwards":[]}
{"upwards":["Derangement","References"],"downwards":[]}
{"upwards":["Derangement","External links"],"downwards":[]}
{"upwards":["Triangle"],"downwards":["Types of triangle","Basic facts","Existence of a triangle","Points, lines, and circles associated with a triangle","Computing the sides and angles","Area","Further formulas for general Euclidean triangles","Figures inscribed in a triangle","Figures circumscribed about a triangle","Specifying the location of a point in a triangle","Non-planar triangles","Triangles in construction"]}
{"upwards":["Triangle","Types of triangle"],"downwards":["By lengths of sides","By internal angles"]}
{"upwards":["Triangle","Types of triangle","By lengths of sides"],"downwards":[]}
{"upwards":["Triangle","Types of triangle","By internal angles"],"downwards":[]}
{"upwards":["Triangle","Basic facts"],"downwards":["Similarity and congruence","Right triangles"]}
{"upwards":["Triangle","Basic facts","Similarity and congruence"],"downwards":[]}
{"upwards":["Triangle","Basic facts","Right triangles"],"downwards":[]}
{"upwards":["Triangle","Existence of a triangle"],"downwards":["Condition on the sides","Conditions on the angles"]}
{"upwards":["Triangle","Existence of a triangle","Condition on the sides"],"downwards":[]}
{"upwards":["Triangle","Existence of a triangle","Conditions on the angles"],"downwards":["Trigonometric conditions"]}
{"upwards":["Triangle","Existence of a triangle","Conditions on the angles","Trigonometric conditions"],"downwards":[]}
{"upwards":["Triangle","Points, lines, and circles associated with a triangle"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles"],"downwards":["Trigonometric ratios in right triangles","Sine, cosine and tangent rules","Solution of triangles"]}
{"upwards":["Triangle","Computing the sides and angles","Trigonometric ratios in right triangles"],"downwards":["Sine, cosine and tangent","Inverse functions"]}
{"upwards":["Triangle","Computing the sides and angles","Trigonometric ratios in right triangles","Sine, cosine and tangent"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles","Trigonometric ratios in right triangles","Inverse functions"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles","Sine, cosine and tangent rules"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles","Solution of triangles"],"downwards":[]}
{"upwards":["Triangle","Area"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles"],"downwards":["Medians, angle bisectors, perpendicular side bisectors, and altitudes","Circumradius and inradius","Adjacent triangles","Centroid","Circumcenter, incenter, and orthocenter","Angles","Morley's trisector theorem"]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Medians, angle bisectors, perpendicular side bisectors, and altitudes"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Circumradius and inradius"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Adjacent triangles"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Centroid"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Circumcenter, incenter, and orthocenter"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Angles"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Morley's trisector theorem"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle"],"downwards":["Conics","Convex polygon","Hexagon","Squares","Triangles"]}
{"upwards":["Triangle","Figures inscribed in a triangle","Conics"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Convex polygon"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Hexagon"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Squares"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Triangles"],"downwards":[]}
{"upwards":["Triangle","Figures circumscribed about a triangle"],"downwards":[]}
{"upwards":["Triangle","Specifying the location of a point in a triangle"],"downwards":[]}
{"upwards":["Triangle","Non-planar triangles"],"downwards":[]}
{"upwards":["Triangle","Triangles in construction"],"downwards":[]}
{"upwards":["Combination"],"downwards":["Number of k-combinations","Number of combinations with repetition","Number of k-combinations for all k","Probability: sampling a random combination","Number of ways to put objects into bins"]}
{"upwards":["Combination","Number of k-combinations"],"downwards":["Example of counting combinations","Enumerating k-combinations"]}
{"upwards":["Combination","Number of k-combinations","Example of counting combinations"],"downwards":[]}
{"upwards":["Combination","Number of k-combinations","Enumerating k-combinations"],"downwards":[]}
{"upwards":["Combination","Number of combinations with repetition"],"downwards":["Example of counting multisubsets"]}
{"upwards":["Combination","Number of combinations with repetition","Example of counting multisubsets"],"downwards":[]}
{"upwards":["Combination","Number of k-combinations for all k"],"downwards":[]}
{"upwards":["Combination","Probability: sampling a random combination"],"downwards":[]}
{"upwards":["Combination","Number of ways to put objects into bins"],"downwards":[]}
{"upwards":["Square"],"downwards":["Characterizations","Properties","Coordinates and equations","Construction","Symmetry","Squares inscribed in triangles","Squaring the circle","Non-Euclidean geometry","Crossed square","Graphs"]}
{"upwards":["Square","Characterizations"],"downwards":[]}
{"upwards":["Square","Properties"],"downwards":["Perimeter and area","Other facts"]}
{"upwards":["Square","Properties","Perimeter and area"],"downwards":[]}
{"upwards":["Square","Properties","Other facts"],"downwards":[]}
{"upwards":["Square","Coordinates and equations"],"downwards":[]}
{"upwards":["Square","Construction"],"downwards":[]}
{"upwards":["Square","Symmetry"],"downwards":[]}
{"upwards":["Square","Squares inscribed in triangles"],"downwards":[]}
{"upwards":["Square","Squaring the circle"],"downwards":[]}
{"upwards":["Square","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Square","Crossed square"],"downwards":[]}
{"upwards":["Square","Graphs"],"downwards":[]}
{"upwards":["Pentagon"],"downwards":["Regular pentagons","Equilateral pentagons","Cyclic pentagons","General convex pentagons","Pentagons in tiling","Pentagons in polyhedra","Pentagons in nature","Other examples"]}
{"upwards":["Pentagon","Regular pentagons"],"downwards":["Derivation of the area formula","Inradius","Chords from the circumscribed circle to the vertices","Point in plane","Geometrical constructions","Physical construction methods","Symmetry","Regular pentagram"]}
{"upwards":["Pentagon","Regular pentagons","Derivation of the area formula"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Inradius"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Chords from the circumscribed circle to the vertices"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Point in plane"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions"],"downwards":["Richmond's method","Carlyle circles","Euclid's method"]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions","Richmond's method"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions","Carlyle circles"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions","Euclid's method"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Physical construction methods"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Symmetry"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Regular pentagram"],"downwards":[]}
{"upwards":["Pentagon","Equilateral pentagons"],"downwards":[]}
{"upwards":["Pentagon","Cyclic pentagons"],"downwards":[]}
{"upwards":["Pentagon","General convex pentagons"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in tiling"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in polyhedra"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in nature"],"downwards":["Plants","Animals","Minerals"]}
{"upwards":["Pentagon","Pentagons in nature","Plants"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in nature","Animals"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in nature","Minerals"],"downwards":[]}
{"upwards":["Pentagon","Other examples"],"downwards":[]}
{"upwards":["Mathematical proof"],"downwards":["History and etymology","Nature and purpose","Methods of proof","Undecidable statements","Heuristic mathematics and experimental mathematics","Related concepts","Ending a proof"]}
{"upwards":["Mathematical proof","History and etymology"],"downwards":[]}
{"upwards":["Mathematical proof","Nature and purpose"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof"],"downwards":["Direct proof","Proof by mathematical induction","Proof by contraposition","Proof by contradiction","Proof by construction","Proof by exhaustion","Probabilistic proof","Combinatorial proof","Nonconstructive proof","Statistical proofs in pure mathematics","Computer-assisted proofs"]}
{"upwards":["Mathematical proof","Methods of proof","Direct proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by mathematical induction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contraposition"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contradiction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by construction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by exhaustion"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Probabilistic proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Combinatorial proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Nonconstructive proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Statistical proofs in pure mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Computer-assisted proofs"],"downwards":[]}
{"upwards":["Mathematical proof","Undecidable statements"],"downwards":[]}
{"upwards":["Mathematical proof","Heuristic mathematics and experimental mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts"],"downwards":["Visual proof","Elementary proof","Two-column proof","Colloquial use of \"mathematical proof\"","Statistical proof using data","Inductive logic proofs and Bayesian analysis","Proofs as mental objects","Influence of mathematical proof methods outside mathematics"]}
{"upwards":["Mathematical proof","Related concepts","Visual proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Elementary proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Two-column proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Colloquial use of \"mathematical proof\""],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Statistical proof using data"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Inductive logic proofs and Bayesian analysis"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Proofs as mental objects"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Influence of mathematical proof methods outside mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Ending a proof"],"downwards":[]}
{"upwards":["Mathematical proof"],"downwards":["History and etymology","Nature and purpose","Methods of proof","Undecidable statements","Heuristic mathematics and experimental mathematics","Related concepts","Ending a proof"]}
{"upwards":["Mathematical proof","History and etymology"],"downwards":[]}
{"upwards":["Mathematical proof","Nature and purpose"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof"],"downwards":["Direct proof","Proof by mathematical induction","Proof by contraposition","Proof by contradiction","Proof by construction","Proof by exhaustion","Probabilistic proof","Combinatorial proof","Nonconstructive proof","Statistical proofs in pure mathematics","Computer-assisted proofs"]}
{"upwards":["Mathematical proof","Methods of proof","Direct proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by mathematical induction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contraposition"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contradiction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by construction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by exhaustion"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Probabilistic proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Combinatorial proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Nonconstructive proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Statistical proofs in pure mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Computer-assisted proofs"],"downwards":[]}
{"upwards":["Mathematical proof","Undecidable statements"],"downwards":[]}
{"upwards":["Mathematical proof","Heuristic mathematics and experimental mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts"],"downwards":["Visual proof","Elementary proof","Two-column proof","Colloquial use of \"mathematical proof\"","Statistical proof using data","Inductive logic proofs and Bayesian analysis","Proofs as mental objects","Influence of mathematical proof methods outside mathematics"]}
{"upwards":["Mathematical proof","Related concepts","Visual proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Elementary proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Two-column proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Colloquial use of \"mathematical proof\""],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Statistical proof using data"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Inductive logic proofs and Bayesian analysis"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Proofs as mental objects"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Influence of mathematical proof methods outside mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Ending a proof"],"downwards":[]}
{"upwards":["Geometry"],"downwards":["History","Main concepts","Contemporary geometry","Applications"]}
{"upwards":["Geometry","History"],"downwards":[]}
{"upwards":["Geometry","Main concepts"],"downwards":["Axioms","Objects","Length, area, and volume","Congruence and similarity","Compass and straightedge constructions","Dimension","Symmetry"]}
{"upwards":["Geometry","Main concepts","Axioms"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects"],"downwards":["Points","Lines","Planes","Angles","Curves","Surfaces","Manifolds"]}
{"upwards":["Geometry","Main concepts","Objects","Points"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Lines"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Planes"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Angles"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Curves"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Surfaces"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Manifolds"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Length, area, and volume"],"downwards":["Metrics and measures"]}
{"upwards":["Geometry","Main concepts","Length, area, and volume","Metrics and measures"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Congruence and similarity"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Compass and straightedge constructions"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Dimension"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Symmetry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry"],"downwards":["Euclidean geometry","Differential geometry","Topology","Algebraic geometry","Complex geometry","Discrete geometry","Computational geometry","Geometric group theory","Convex geometry"]}
{"upwards":["Geometry","Contemporary geometry","Euclidean geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Differential geometry"],"downwards":["Non-Euclidean geometry"]}
{"upwards":["Geometry","Contemporary geometry","Differential geometry","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Topology"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Algebraic geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Complex geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Discrete geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Computational geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Geometric group theory"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Convex geometry"],"downwards":[]}
{"upwards":["Geometry","Applications"],"downwards":["Art","Architecture","Physics","Other fields of mathematics"]}
{"upwards":["Geometry","Applications","Art"],"downwards":[]}
{"upwards":["Geometry","Applications","Architecture"],"downwards":[]}
{"upwards":["Geometry","Applications","Physics"],"downwards":[]}
{"upwards":["Geometry","Applications","Other fields of mathematics"],"downwards":[]}
{"upwards":["Geometry"],"downwards":["History","Main concepts","Contemporary geometry","Applications"]}
{"upwards":["Geometry","History"],"downwards":[]}
{"upwards":["Geometry","Main concepts"],"downwards":["Axioms","Objects","Length, area, and volume","Congruence and similarity","Compass and straightedge constructions","Dimension","Symmetry"]}
{"upwards":["Geometry","Main concepts","Axioms"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects"],"downwards":["Points","Lines","Planes","Angles","Curves","Surfaces","Manifolds"]}
{"upwards":["Geometry","Main concepts","Objects","Points"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Lines"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Planes"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Angles"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Curves"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Surfaces"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Manifolds"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Length, area, and volume"],"downwards":["Metrics and measures"]}
{"upwards":["Geometry","Main concepts","Length, area, and volume","Metrics and measures"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Congruence and similarity"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Compass and straightedge constructions"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Dimension"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Symmetry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry"],"downwards":["Euclidean geometry","Differential geometry","Topology","Algebraic geometry","Complex geometry","Discrete geometry","Computational geometry","Geometric group theory","Convex geometry"]}
{"upwards":["Geometry","Contemporary geometry","Euclidean geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Differential geometry"],"downwards":["Non-Euclidean geometry"]}
{"upwards":["Geometry","Contemporary geometry","Differential geometry","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Topology"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Algebraic geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Complex geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Discrete geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Computational geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Geometric group theory"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Convex geometry"],"downwards":[]}
{"upwards":["Geometry","Applications"],"downwards":["Art","Architecture","Physics","Other fields of mathematics"]}
{"upwards":["Geometry","Applications","Art"],"downwards":[]}
{"upwards":["Geometry","Applications","Architecture"],"downwards":[]}
{"upwards":["Geometry","Applications","Physics"],"downwards":[]}
{"upwards":["Geometry","Applications","Other fields of mathematics"],"downwards":[]}
{"upwards":["Hexagon"],"downwards":["Regular hexagon","Parameters","Point in plane","Symmetry","Dissection","Related polygons and tilings","Hexagonal structures","Tesselations by hexagons","Hexagon inscribed in a conic section","Hexagon tangential to a conic section","Equilateral triangles on the sides of an arbitrary hexagon","Skew hexagon","Convex equilateral hexagon","Gallery of natural and artificial hexagons"]}
{"upwards":["Hexagon","Regular hexagon"],"downwards":[]}
{"upwards":["Hexagon","Parameters"],"downwards":[]}
{"upwards":["Hexagon","Point in plane"],"downwards":[]}
{"upwards":["Hexagon","Symmetry"],"downwards":["A2 and G2 groups"]}
{"upwards":["Hexagon","Symmetry","A2 and G2 groups"],"downwards":[]}
{"upwards":["Hexagon","Dissection"],"downwards":[]}
{"upwards":["Hexagon","Related polygons and tilings"],"downwards":["Self-crossing hexagons"]}
{"upwards":["Hexagon","Related polygons and tilings","Self-crossing hexagons"],"downwards":[]}
{"upwards":["Hexagon","Hexagonal structures"],"downwards":[]}
{"upwards":["Hexagon","Tesselations by hexagons"],"downwards":[]}
{"upwards":["Hexagon","Hexagon inscribed in a conic section"],"downwards":["Cyclic hexagon"]}
{"upwards":["Hexagon","Hexagon inscribed in a conic section","Cyclic hexagon"],"downwards":[]}
{"upwards":["Hexagon","Hexagon tangential to a conic section"],"downwards":[]}
{"upwards":["Hexagon","Equilateral triangles on the sides of an arbitrary hexagon"],"downwards":[]}
{"upwards":["Hexagon","Skew hexagon"],"downwards":["Petrie polygons"]}
{"upwards":["Hexagon","Skew hexagon","Petrie polygons"],"downwards":[]}
{"upwards":["Hexagon","Convex equilateral hexagon"],"downwards":["Polyhedra with hexagons"]}
{"upwards":["Hexagon","Convex equilateral hexagon","Polyhedra with hexagons"],"downwards":[]}
{"upwards":["Hexagon","Gallery of natural and artificial hexagons"],"downwards":[]}
{"upwards":["Parabola"],"downwards":["History","Definition as a locus of points","In a Cartesian coordinate system","As a graph of a function","Similarity to the unit parabola","As a special conic section","In polar coordinates","Conic section and quadratic form","Proof of the reflective property","Pin and string construction","Properties related to Pascal's theorem","Steiner generation","Inscribed angles and the 3-point form","Pole\u2013polar relation","Tangent properties","Facts related to chords and arcs","A geometrical construction to find a sector area","Focal length and radius of curvature at the vertex","As the affine image of the unit parabola","As quadratic B\u00e9zier curve","Numerical integration","As plane section of quadric","As trisectrix","Generalizations","In the physical world"]}
{"upwards":["Parabola","History"],"downwards":[]}
{"upwards":["Parabola","Definition as a locus of points"],"downwards":[]}
{"upwards":["Parabola","In a Cartesian coordinate system"],"downwards":["Axis of symmetry parallel to the y axis","General position"]}
{"upwards":["Parabola","In a Cartesian coordinate system","Axis of symmetry parallel to the y axis"],"downwards":[]}
{"upwards":["Parabola","In a Cartesian coordinate system","General position"],"downwards":[]}
{"upwards":["Parabola","As a graph of a function"],"downwards":[]}
{"upwards":["Parabola","Similarity to the unit parabola"],"downwards":[]}
{"upwards":["Parabola","As a special conic section"],"downwards":[]}
{"upwards":["Parabola","In polar coordinates"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form"],"downwards":["Diagram, description, and definitions","Derivation of quadratic equation","Focal length","Position of the focus","Alternative proof with Dandelin spheres"]}
{"upwards":["Parabola","Conic section and quadratic form","Diagram, description, and definitions"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Derivation of quadratic equation"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Focal length"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Position of the focus"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Alternative proof with Dandelin spheres"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property"],"downwards":["Construction and definitions","Deductions","Other consequences","Alternative proofs"]}
{"upwards":["Parabola","Proof of the reflective property","Construction and definitions"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Deductions"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences"],"downwards":["Tangent bisection property","Intersection of a tangent and perpendicular from focus","Reflection of light striking the convex side"]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences","Tangent bisection property"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences","Intersection of a tangent and perpendicular from focus"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences","Reflection of light striking the convex side"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Alternative proofs"],"downwards":[]}
{"upwards":["Parabola","Pin and string construction"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem"],"downwards":["4-points property","3-points\u20131-tangent property","2-points\u20132-tangents property","Axis direction"]}
{"upwards":["Parabola","Properties related to Pascal's theorem","4-points property"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem","3-points\u20131-tangent property"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem","2-points\u20132-tangents property"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem","Axis direction"],"downwards":[]}
{"upwards":["Parabola","Steiner generation"],"downwards":["Parabola","Dual parabola"]}
{"upwards":["Parabola","Steiner generation","Parabola"],"downwards":[]}
{"upwards":["Parabola","Steiner generation","Dual parabola"],"downwards":[]}
{"upwards":["Parabola","Inscribed angles and the 3-point form"],"downwards":[]}
{"upwards":["Parabola","Pole\u2013polar relation"],"downwards":[]}
{"upwards":["Parabola","Tangent properties"],"downwards":["Two tangent properties related to the latus rectum","Orthoptic property","Lambert's theorem"]}
{"upwards":["Parabola","Tangent properties","Two tangent properties related to the latus rectum"],"downwards":[]}
{"upwards":["Parabola","Tangent properties","Orthoptic property"],"downwards":[]}
{"upwards":["Parabola","Tangent properties","Lambert's theorem"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs"],"downwards":["Focal length calculated from parameters of a chord","Area enclosed between a parabola and a chord","Corollary concerning midpoints and endpoints of chords","Arc length"]}
{"upwards":["Parabola","Facts related to chords and arcs","Focal length calculated from parameters of a chord"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs","Area enclosed between a parabola and a chord"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs","Corollary concerning midpoints and endpoints of chords"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs","Arc length"],"downwards":[]}
{"upwards":["Parabola","A geometrical construction to find a sector area"],"downwards":[]}
{"upwards":["Parabola","Focal length and radius of curvature at the vertex"],"downwards":[]}
{"upwards":["Parabola","As the affine image of the unit parabola"],"downwards":[]}
{"upwards":["Parabola","As quadratic B\u00e9zier curve"],"downwards":[]}
{"upwards":["Parabola","Numerical integration"],"downwards":[]}
{"upwards":["Parabola","As plane section of quadric"],"downwards":[]}
{"upwards":["Parabola","As trisectrix"],"downwards":[]}
{"upwards":["Parabola","Generalizations"],"downwards":[]}
{"upwards":["Parabola","In the physical world"],"downwards":["Gallery"]}
{"upwards":["Parabola","In the physical world","Gallery"],"downwards":[]}
{"upwards":["Octahedron"],"downwards":["Regular octahedron","Irregular octahedra","Octahedra in the physical world","Related polyhedra"]}
{"upwards":["Octahedron","Regular octahedron"],"downwards":["Dimensions","Orthogonal projections","Spherical tiling","Cartesian coordinates","Area and volume","Geometric relations","Characteristic orthoscheme","Topology","Nets","Faceting","Uniform colorings and symmetry"]}
{"upwards":["Octahedron","Regular octahedron","Dimensions"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Orthogonal projections"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Spherical tiling"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Cartesian coordinates"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Area and volume"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations"],"downwards":["Dual","Stellation","Snub octahedron","Tessellations"]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Dual"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Stellation"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Snub octahedron"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Tessellations"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Characteristic orthoscheme"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Topology"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Nets"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Faceting"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Uniform colorings and symmetry"],"downwards":[]}
{"upwards":["Octahedron","Irregular octahedra"],"downwards":[]}
{"upwards":["Octahedron","Octahedra in the physical world"],"downwards":["Octahedra in nature","Octahedra in art and culture","Tetrahedral octet truss"]}
{"upwards":["Octahedron","Octahedra in the physical world","Octahedra in nature"],"downwards":[]}
{"upwards":["Octahedron","Octahedra in the physical world","Octahedra in art and culture"],"downwards":[]}
{"upwards":["Octahedron","Octahedra in the physical world","Tetrahedral octet truss"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra"],"downwards":["Tetratetrahedron","Trigonal antiprism","Square bipyramid","Other related polyhedra"]}
{"upwards":["Octahedron","Related polyhedra","Tetratetrahedron"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra","Trigonal antiprism"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra","Square bipyramid"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra","Other related polyhedra"],"downwards":[]}
{"upwards":["Ellipse"],"downwards":["Definition as locus of points","In Cartesian coordinates","Parametric representation","Polar forms","Eccentricity and the directrix property","Focus-to-focus reflection property","Conjugate diameters","Orthogonal tangents","Drawing ellipses","Inscribed angles and three-point form","Pole-polar relation","Metric properties","In triangle geometry","As plane sections of quadrics","Applications"]}
{"upwards":["Ellipse","Definition as locus of points"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates"],"downwards":["Standard equation","Parameters","Tangent","Shifted ellipse","General ellipse"]}
{"upwards":["Ellipse","In Cartesian coordinates","Standard equation"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters"],"downwards":["Principal axes","Linear eccentricity","Eccentricity","Semi-latus rectum"]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Principal axes"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Linear eccentricity"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Eccentricity"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Semi-latus rectum"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Tangent"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Shifted ellipse"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","General ellipse"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation"],"downwards":["Standard parametric representation","Rational representation","Tangent slope as parameter","General ellipse"]}
{"upwards":["Ellipse","Parametric representation","Standard parametric representation"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation","Rational representation"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation","Tangent slope as parameter"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation","General ellipse"],"downwards":[]}
{"upwards":["Ellipse","Polar forms"],"downwards":["Polar form relative to center","Polar form relative to focus"]}
{"upwards":["Ellipse","Polar forms","Polar form relative to center"],"downwards":[]}
{"upwards":["Ellipse","Polar forms","Polar form relative to focus"],"downwards":[]}
{"upwards":["Ellipse","Eccentricity and the directrix property"],"downwards":[]}
{"upwards":["Ellipse","Focus-to-focus reflection property"],"downwards":[]}
{"upwards":["Ellipse","Conjugate diameters"],"downwards":["Definition of conjugate diameters","Theorem of Apollonios on conjugate diameters"]}
{"upwards":["Ellipse","Conjugate diameters","Definition of conjugate diameters"],"downwards":[]}
{"upwards":["Ellipse","Conjugate diameters","Theorem of Apollonios on conjugate diameters"],"downwards":[]}
{"upwards":["Ellipse","Orthogonal tangents"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses"],"downwards":["de La Hire's point construction","Pins-and-string method","Paper strip methods","Approximation by osculating circles","Steiner generation","As hypotrochoid"]}
{"upwards":["Ellipse","Drawing ellipses","de La Hire's point construction"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Pins-and-string method"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Paper strip methods"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Approximation by osculating circles"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Steiner generation"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","As hypotrochoid"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form"],"downwards":["Circles","Ellipses"]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Circles"],"downwards":["Inscribed angle theorem for circles","Three-point form of circle equation"]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Circles","Inscribed angle theorem for circles"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Circles","Three-point form of circle equation"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Ellipses"],"downwards":["Inscribed angle theorem for ellipses","Three-point form of ellipse equation"]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Ellipses","Inscribed angle theorem for ellipses"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Ellipses","Three-point form of ellipse equation"],"downwards":[]}
{"upwards":["Ellipse","Pole-polar relation"],"downwards":[]}
{"upwards":["Ellipse","Metric properties"],"downwards":["Area","Circumference","Arc length","Curvature"]}
{"upwards":["Ellipse","Metric properties","Area"],"downwards":[]}
{"upwards":["Ellipse","Metric properties","Circumference"],"downwards":[]}
{"upwards":["Ellipse","Metric properties","Arc length"],"downwards":[]}
{"upwards":["Ellipse","Metric properties","Curvature"],"downwards":[]}
{"upwards":["Ellipse","In triangle geometry"],"downwards":[]}
{"upwards":["Ellipse","As plane sections of quadrics"],"downwards":[]}
{"upwards":["Ellipse","Applications"],"downwards":["Physics","Statistics and finance","Computer graphics","Optimization theory"]}
{"upwards":["Ellipse","Applications","Physics"],"downwards":["Elliptical reflectors and acoustics","Planetary orbits","Harmonic oscillators","Phase visualization","Elliptical gears","Optics"]}
{"upwards":["Ellipse","Applications","Physics","Elliptical reflectors and acoustics"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Planetary orbits"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Harmonic oscillators"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Phase visualization"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Elliptical gears"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Optics"],"downwards":[]}
{"upwards":["Ellipse","Applications","Statistics and finance"],"downwards":[]}
{"upwards":["Ellipse","Applications","Computer graphics"],"downwards":[]}
{"upwards":["Ellipse","Applications","Optimization theory"],"downwards":[]}
{"upwards":["Dodecahedron"],"downwards":["Regular dodecahedron","Other pentagonal dodecahedra","Rhombic dodecahedron","Other dodecahedra","Practical usage"]}
{"upwards":["Dodecahedron","Regular dodecahedron"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra"],"downwards":["Pyritohedron","Tetartoid","Dual of triangular gyrobianticupola"]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron"],"downwards":["Crystal pyrite","Cartesian coordinates","Geometric freedom"]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron","Crystal pyrite"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron","Cartesian coordinates"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron","Geometric freedom"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Tetartoid"],"downwards":["Cartesian coordinates","Geometric freedom"]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Tetartoid","Cartesian coordinates"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Tetartoid","Geometric freedom"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Dual of triangular gyrobianticupola"],"downwards":[]}
{"upwards":["Dodecahedron","Rhombic dodecahedron"],"downwards":[]}
{"upwards":["Dodecahedron","Other dodecahedra"],"downwards":[]}
{"upwards":["Dodecahedron","Practical usage"],"downwards":[]}
{"upwards":["Polyhedron"],"downwards":["Definition","Characteristics","Convex polyhedra","Symmetries","Other important families of polyhedra","Generalisations of polyhedra","Alternative usages","History","In nature"]}
{"upwards":["Polyhedron","Definition"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics"],"downwards":["Number of faces","Topological classification","Duality","Vertex figures","Surface area and distances","Volume","Dehn invariant"]}
{"upwards":["Polyhedron","Characteristics","Number of faces"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Topological classification"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Duality"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Vertex figures"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Surface area and distances"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Volume"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Dehn invariant"],"downwards":[]}
{"upwards":["Polyhedron","Convex polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries"],"downwards":["Regular polyhedra","Uniform polyhedra and their duals","Isohedra","Symmetry groups"]}
{"upwards":["Polyhedron","Symmetries","Regular polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries","Uniform polyhedra and their duals"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries","Isohedra"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries","Symmetry groups"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra"],"downwards":["Polyhedra with regular faces","Pyramids","Stellations and facettings","Zonohedra","Space-filling polyhedra","Lattice polyhedra","Flexible polyhedra","Compounds","Orthogonal polyhedra","Embedded regular maps with planar faces","Canonical polyhedra"]}
{"upwards":["Polyhedron","Other important families of polyhedra","Polyhedra with regular faces"],"downwards":["Equal regular faces","Johnson solids"]}
{"upwards":["Polyhedron","Other important families of polyhedra","Polyhedra with regular faces","Equal regular faces"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Polyhedra with regular faces","Johnson solids"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Pyramids"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Stellations and facettings"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Zonohedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Space-filling polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Lattice polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Flexible polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Compounds"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Orthogonal polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Embedded regular maps with planar faces"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Canonical polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra"],"downwards":["Apeirohedra","Complex polyhedra","Curved polyhedra","Ideal polyhedra","Skeletons and polyhedra as graphs"]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Apeirohedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Complex polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Curved polyhedra"],"downwards":["Spherical polyhedra","Curved spacefilling polyhedra"]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Curved polyhedra","Spherical polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Curved polyhedra","Curved spacefilling polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Ideal polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Skeletons and polyhedra as graphs"],"downwards":[]}
{"upwards":["Polyhedron","Alternative usages"],"downwards":["Higher-dimensional polyhedra","Topological polyhedra","Abstract polyhedra"]}
{"upwards":["Polyhedron","Alternative usages","Higher-dimensional polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Alternative usages","Topological polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Alternative usages","Abstract polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","History"],"downwards":["Before the Greeks","Ancient Greece","Ancient China","Medieval Islam","Renaissance","17th\u201319th centuries","20th\u201321st centuries"]}
{"upwards":["Polyhedron","History","Before the Greeks"],"downwards":[]}
{"upwards":["Polyhedron","History","Ancient Greece"],"downwards":[]}
{"upwards":["Polyhedron","History","Ancient China"],"downwards":[]}
{"upwards":["Polyhedron","History","Medieval Islam"],"downwards":[]}
{"upwards":["Polyhedron","History","Renaissance"],"downwards":[]}
{"upwards":["Polyhedron","History","17th\u201319th centuries"],"downwards":[]}
{"upwards":["Polyhedron","History","20th\u201321st centuries"],"downwards":[]}
{"upwards":["Polyhedron","In nature"],"downwards":[]}
{"upwards":["Sphere"],"downwards":["Basic terminology","Equations","Properties","Treatment by area of mathematics","Curves on a sphere","Generalizations","History","Gallery","Regions"]}
{"upwards":["Sphere","Basic terminology"],"downwards":[]}
{"upwards":["Sphere","Equations"],"downwards":["Parametric"]}
{"upwards":["Sphere","Equations","Parametric"],"downwards":[]}
{"upwards":["Sphere","Properties"],"downwards":["Enclosed volume","Surface area","Other geometric properties","Properties of the sphere"]}
{"upwards":["Sphere","Properties","Enclosed volume"],"downwards":[]}
{"upwards":["Sphere","Properties","Surface area"],"downwards":[]}
{"upwards":["Sphere","Properties","Other geometric properties"],"downwards":["Pencil of spheres"]}
{"upwards":["Sphere","Properties","Other geometric properties","Pencil of spheres"],"downwards":[]}
{"upwards":["Sphere","Properties","Properties of the sphere"],"downwards":[]}
{"upwards":["Sphere","Treatment by area of mathematics"],"downwards":["Spherical geometry","Differential geometry","Topology"]}
{"upwards":["Sphere","Treatment by area of mathematics","Spherical geometry"],"downwards":[]}
{"upwards":["Sphere","Treatment by area of mathematics","Differential geometry"],"downwards":[]}
{"upwards":["Sphere","Treatment by area of mathematics","Topology"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere"],"downwards":["Circles","Loxodrome","Clelia curves","Spherical conics","Intersection of a sphere with a more general surface"]}
{"upwards":["Sphere","Curves on a sphere","Circles"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Loxodrome"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Clelia curves"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Spherical conics"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Intersection of a sphere with a more general surface"],"downwards":[]}
{"upwards":["Sphere","Generalizations"],"downwards":["Ellipsoids","Dimensionality","Metric spaces"]}
{"upwards":["Sphere","Generalizations","Ellipsoids"],"downwards":[]}
{"upwards":["Sphere","Generalizations","Dimensionality"],"downwards":[]}
{"upwards":["Sphere","Generalizations","Metric spaces"],"downwards":[]}
{"upwards":["Sphere","History"],"downwards":[]}
{"upwards":["Sphere","Gallery"],"downwards":[]}
{"upwards":["Sphere","Regions"],"downwards":[]}
{"upwards":["Circle"],"downwards":["Euclid's definition","Topological definition","Terminology","History","Analytic results","Properties","Compass and straightedge constructions","Circle of Apollonius","Inscription in or circumscription about other figures","Limiting case of other figures","In other p-norms","Locus of constant sum","Squaring the circle","Significance in art and symbolism","Unicode"]}
{"upwards":["Circle","Euclid's definition"],"downwards":[]}
{"upwards":["Circle","Topological definition"],"downwards":[]}
{"upwards":["Circle","Terminology"],"downwards":[]}
{"upwards":["Circle","History"],"downwards":[]}
{"upwards":["Circle","Analytic results"],"downwards":["Circumference","Area enclosed","Equations","Tangent lines"]}
{"upwards":["Circle","Analytic results","Circumference"],"downwards":[]}
{"upwards":["Circle","Analytic results","Area enclosed"],"downwards":[]}
{"upwards":["Circle","Analytic results","Equations"],"downwards":["Cartesian coordinates","Polar coordinates","Complex plane"]}
{"upwards":["Circle","Analytic results","Equations","Cartesian coordinates"],"downwards":[]}
{"upwards":["Circle","Analytic results","Equations","Polar coordinates"],"downwards":[]}
{"upwards":["Circle","Analytic results","Equations","Complex plane"],"downwards":[]}
{"upwards":["Circle","Analytic results","Tangent lines"],"downwards":[]}
{"upwards":["Circle","Properties"],"downwards":["Chord","Tangent","Theorems","Inscribed angles","Sagitta"]}
{"upwards":["Circle","Properties","Chord"],"downwards":[]}
{"upwards":["Circle","Properties","Tangent"],"downwards":[]}
{"upwards":["Circle","Properties","Theorems"],"downwards":[]}
{"upwards":["Circle","Properties","Inscribed angles"],"downwards":[]}
{"upwards":["Circle","Properties","Sagitta"],"downwards":[]}
{"upwards":["Circle","Compass and straightedge constructions"],"downwards":["Construction with given diameter","Construction through three noncollinear points"]}
{"upwards":["Circle","Compass and straightedge constructions","Construction with given diameter"],"downwards":[]}
{"upwards":["Circle","Compass and straightedge constructions","Construction through three noncollinear points"],"downwards":[]}
{"upwards":["Circle","Circle of Apollonius"],"downwards":["Cross-ratios","Generalised circles"]}
{"upwards":["Circle","Circle of Apollonius","Cross-ratios"],"downwards":[]}
{"upwards":["Circle","Circle of Apollonius","Generalised circles"],"downwards":[]}
{"upwards":["Circle","Inscription in or circumscription about other figures"],"downwards":[]}
{"upwards":["Circle","Limiting case of other figures"],"downwards":[]}
{"upwards":["Circle","In other p-norms"],"downwards":[]}
{"upwards":["Circle","Locus of constant sum"],"downwards":[]}
{"upwards":["Circle","Squaring the circle"],"downwards":[]}
{"upwards":["Circle","Significance in art and symbolism"],"downwards":[]}
{"upwards":["Circle","Unicode"],"downwards":[]}
{"upwards":["Cube"],"downwards":["Orthogonal projections","Spherical tiling","Cartesian coordinates","Equation in three dimensional space","Formulas","Doubling the cube","Uniform colorings and symmetry","Geometric relations","In theology","Other dimensions","Related polyhedra","Cubical graph"]}
{"upwards":["Cube","Orthogonal projections"],"downwards":[]}
{"upwards":["Cube","Spherical tiling"],"downwards":[]}
{"upwards":["Cube","Cartesian coordinates"],"downwards":[]}
{"upwards":["Cube","Equation in three dimensional space"],"downwards":[]}
{"upwards":["Cube","Formulas"],"downwards":["Point in space"]}
{"upwards":["Cube","Formulas","Point in space"],"downwards":[]}
{"upwards":["Cube","Doubling the cube"],"downwards":[]}
{"upwards":["Cube","Uniform colorings and symmetry"],"downwards":[]}
{"upwards":["Cube","Geometric relations"],"downwards":[]}
{"upwards":["Cube","In theology"],"downwards":[]}
{"upwards":["Cube","Other dimensions"],"downwards":[]}
{"upwards":["Cube","Related polyhedra"],"downwards":["In uniform honeycombs and polychora"]}
{"upwards":["Cube","Related polyhedra","In uniform honeycombs and polychora"],"downwards":[]}
{"upwards":["Cube","Cubical graph"],"downwards":[]}
{"upwards":["Ellipsoid"],"downwards":["Standard equation","Parameterization","Volume","Surface area","Plane sections","Pins-and-string construction","In general position","Applications","In higher dimensions"]}
{"upwards":["Ellipsoid","Standard equation"],"downwards":[]}
{"upwards":["Ellipsoid","Parameterization"],"downwards":[]}
{"upwards":["Ellipsoid","Volume"],"downwards":[]}
{"upwards":["Ellipsoid","Surface area"],"downwards":["Approximate formula"]}
{"upwards":["Ellipsoid","Surface area","Approximate formula"],"downwards":[]}
{"upwards":["Ellipsoid","Plane sections"],"downwards":["Determining the ellipse of a plane section"]}
{"upwards":["Ellipsoid","Plane sections","Determining the ellipse of a plane section"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction"],"downwards":["Steps of the construction","Semi-axes","Converse","Confocal ellipsoids","Limit case, ellipsoid of revolution","Properties of the focal hyperbola","Property of the focal ellipse"]}
{"upwards":["Ellipsoid","Pins-and-string construction","Steps of the construction"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Semi-axes"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Converse"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Confocal ellipsoids"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Limit case, ellipsoid of revolution"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Properties of the focal hyperbola"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Property of the focal ellipse"],"downwards":[]}
{"upwards":["Ellipsoid","In general position"],"downwards":["As a quadric","Parametric representation"]}
{"upwards":["Ellipsoid","In general position","As a quadric"],"downwards":[]}
{"upwards":["Ellipsoid","In general position","Parametric representation"],"downwards":[]}
{"upwards":["Ellipsoid","Applications"],"downwards":["Dynamical properties","Fluid dynamics","In probability and statistics"]}
{"upwards":["Ellipsoid","Applications","Dynamical properties"],"downwards":[]}
{"upwards":["Ellipsoid","Applications","Fluid dynamics"],"downwards":[]}
{"upwards":["Ellipsoid","Applications","In probability and statistics"],"downwards":[]}
{"upwards":["Ellipsoid","In higher dimensions"],"downwards":[]}
{"upwards":["Icosahedron"],"downwards":["Regular icosahedra","Stellated icosahedra","Pyritohedral symmetry","Other icosahedra"]}
{"upwards":["Icosahedron","Regular icosahedra"],"downwards":["Convex regular icosahedron","Great icosahedron"]}
{"upwards":["Icosahedron","Regular icosahedra","Convex regular icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Regular icosahedra","Great icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Stellated icosahedra"],"downwards":[]}
{"upwards":["Icosahedron","Pyritohedral symmetry"],"downwards":["Cartesian coordinates","Jessen's icosahedron","Cuboctahedron"]}
{"upwards":["Icosahedron","Pyritohedral symmetry","Cartesian coordinates"],"downwards":[]}
{"upwards":["Icosahedron","Pyritohedral symmetry","Jessen's icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Pyritohedral symmetry","Cuboctahedron"],"downwards":[]}
{"upwards":["Icosahedron","Other icosahedra"],"downwards":["Rhombic icosahedron","Pyramid and prism symmetries","Johnson solids"]}
{"upwards":["Icosahedron","Other icosahedra","Rhombic icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Other icosahedra","Pyramid and prism symmetries"],"downwards":[]}
{"upwards":["Icosahedron","Other icosahedra","Johnson solids"],"downwards":[]}
{"upwards":["Hyperbola"],"downwards":["Etymology and history","Definitions","In Cartesian coordinates","In polar coordinates","Parametric equations","Hyperbolic functions","Properties","Arc length","Derived curves","Elliptic coordinates","Conic section analysis of the hyperbolic appearance of circles","Applications","Hyperbolas as plane sections of quadrics"]}
{"upwards":["Hyperbola","Etymology and history"],"downwards":[]}
{"upwards":["Hyperbola","Definitions"],"downwards":["As locus of points","Hyperbola with equation y = A\/x","By the directrix property","As plane section of a cone","Pin and string construction","Steiner generation of a hyperbola","Inscribed angles for hyperbolas y = a\/(x \u2212 b) + c and the 3-point-form","As an affine image of the unit hyperbola x\u00b2 \u2212 y\u00b2 = 1","As an affine image of the hyperbola y = 1\/x","Reciprocation of a circle","Quadratic equation"]}
{"upwards":["Hyperbola","Definitions","As locus of points"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Hyperbola with equation y = A\/x"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","By the directrix property"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As plane section of a cone"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Pin and string construction"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Steiner generation of a hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Inscribed angles for hyperbolas y = a\/(x \u2212 b) + c and the 3-point-form"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the unit hyperbola x\u00b2 \u2212 y\u00b2 = 1"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x"],"downwards":["Tangent construction","Point construction","Tangent\u2013asymptotes triangle"]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x","Tangent construction"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x","Point construction"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x","Tangent\u2013asymptotes triangle"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Reciprocation of a circle"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Quadratic equation"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates"],"downwards":["Equation","Asymptotes","Semi-latus rectum","Tangent","Rectangular hyperbola","Parametric representation with hyperbolic sine\/cosine","Conjugate hyperbola"]}
{"upwards":["Hyperbola","In Cartesian coordinates","Equation"],"downwards":["Eccentricity"]}
{"upwards":["Hyperbola","In Cartesian coordinates","Equation","Eccentricity"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Asymptotes"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Semi-latus rectum"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Tangent"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Rectangular hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Parametric representation with hyperbolic sine\/cosine"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Conjugate hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","In polar coordinates"],"downwards":[]}
{"upwards":["Hyperbola","Parametric equations"],"downwards":[]}
{"upwards":["Hyperbola","Hyperbolic functions"],"downwards":[]}
{"upwards":["Hyperbola","Properties"],"downwards":["The tangent bisects the angle between the lines to the foci","Midpoints of parallel chords","Orthogonal tangents \u2013 orthoptic","Pole-polar relation for a hyperbola","Other properties"]}
{"upwards":["Hyperbola","Properties","The tangent bisects the angle between the lines to the foci"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Midpoints of parallel chords"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Orthogonal tangents \u2013 orthoptic"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Pole-polar relation for a hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Other properties"],"downwards":[]}
{"upwards":["Hyperbola","Arc length"],"downwards":[]}
{"upwards":["Hyperbola","Derived curves"],"downwards":[]}
{"upwards":["Hyperbola","Elliptic coordinates"],"downwards":[]}
{"upwards":["Hyperbola","Conic section analysis of the hyperbolic appearance of circles"],"downwards":[]}
{"upwards":["Hyperbola","Applications"],"downwards":["Sundials","Multilateration","Path followed by a particle","Korteweg\u2013de Vries equation","Angle trisection","Efficient portfolio frontier","Biochemistry"]}
{"upwards":["Hyperbola","Applications","Sundials"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Multilateration"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Path followed by a particle"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Korteweg\u2013de Vries equation"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Angle trisection"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Efficient portfolio frontier"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Biochemistry"],"downwards":[]}
{"upwards":["Hyperbola","Hyperbolas as plane sections of quadrics"],"downwards":[]}
{"upwards":["Tetrahedron"],"downwards":["Regular tetrahedron","Irregular tetrahedra","General properties","Integer tetrahedra","Related polyhedra and compounds","Applications","Tetrahedral graph"]}
{"upwards":["Tetrahedron","Regular tetrahedron"],"downwards":["Coordinates for a regular tetrahedron","Angles and distances","Isometries of the regular tetrahedron","Orthogonal projections of the regular tetrahedron","Cross section of regular tetrahedron","Spherical tiling","Helical stacking"]}
{"upwards":["Tetrahedron","Regular tetrahedron","Coordinates for a regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Angles and distances"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Isometries of the regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Orthogonal projections of the regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Cross section of regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Spherical tiling"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Helical stacking"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra"],"downwards":["Trirectangular tetrahedron","Disphenoid","Orthoschemes","Space-filling tetrahedra","Fundamental domains","Isometries of irregular tetrahedra"]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Trirectangular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Disphenoid"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Orthoschemes"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Space-filling tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Fundamental domains"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Isometries of irregular tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","General properties"],"downwards":["Volume","Distance between the edges","Properties analogous to those of a triangle","Geometric relations","A law of sines for tetrahedra and the space of all shapes of tetrahedra","Law of cosines for tetrahedra","Interior point","Inradius","Circumradius","Circumcenter","Centroid","Faces"]}
{"upwards":["Tetrahedron","General properties","Volume"],"downwards":["Heron-type formula for the volume of a tetrahedron","Volume divider","Non-Euclidean volume"]}
{"upwards":["Tetrahedron","General properties","Volume","Heron-type formula for the volume of a tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Volume","Volume divider"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Volume","Non-Euclidean volume"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Distance between the edges"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Properties analogous to those of a triangle"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Geometric relations"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","A law of sines for tetrahedra and the space of all shapes of tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Law of cosines for tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Interior point"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Inradius"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Circumradius"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Circumcenter"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Centroid"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Faces"],"downwards":[]}
{"upwards":["Tetrahedron","Integer tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","Related polyhedra and compounds"],"downwards":[]}
{"upwards":["Tetrahedron","Applications"],"downwards":["Numerical analysis","Structural engineering","Aviation","Chemistry","Electricity and electronics","Color space","Games","Geology","Popular culture"]}
{"upwards":["Tetrahedron","Applications","Numerical analysis"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Structural engineering"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Aviation"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Chemistry"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Electricity and electronics"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Color space"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Games"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Geology"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Popular culture"],"downwards":[]}
{"upwards":["Tetrahedron","Tetrahedral graph"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan"],"downwards":["Early life and education","Early political career","Premiership","Presidency","Ideology and public image","Personal life","Electoral history","Honours and accolades","Bibliography"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Early life and education"],"downwards":["Education"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Early life and education","Education"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Early political career"],"downwards":["Mayor of Istanbul (1994\u20131998)","Imprisonment","Justice and Development Party"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Early political career","Mayor of Istanbul (1994\u20131998)"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Early political career","Imprisonment"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Early political career","Justice and Development Party"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership"],"downwards":["General elections","Referendums","Domestic policy","Foreign policy","Protests"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","General elections"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Referendums"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy"],"downwards":["Kurdish issue","Armenian genocide","Human rights","Economy","Education","Infrastructure","Justice","Civil\u2013military relations","Health care"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Kurdish issue"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Armenian genocide"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Human rights"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Economy"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Education"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Infrastructure"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Justice"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Civil\u2013military relations"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Domestic policy","Health care"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy"],"downwards":["European Union","Greece and Cyprus dispute","Armenia","Russia","United States","Iraq","Israel","Syria","Saudi Arabia","Egypt","Somalia"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","European Union"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Greece and Cyprus dispute"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Armenia"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Russia"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","United States"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Iraq"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Israel"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Syria"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Saudi Arabia"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Egypt"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Foreign policy","Somalia"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Premiership","Protests"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency"],"downwards":["Presidential elections","Referendum","Domestic policy","Foreign policy","Events"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Presidential elections"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Referendum"],"downwards":["Local elections"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Referendum","Local elections"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Domestic policy"],"downwards":["Presidential palace","The media","State of emergency and purges","Economic policy"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Domestic policy","Presidential palace"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Domestic policy","The media"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Domestic policy","State of emergency and purges"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Domestic policy","Economic policy"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy"],"downwards":["Europe","Greece","The Balkans","United Kingdom","Israel","Syrian Civil War","China","Japan","Qatar blockade","Myanmar","United States","Venezuela","Ukraine and Russian invasion of Ukraine"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Europe"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Greece"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","The Balkans"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","United Kingdom"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Israel"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Syrian Civil War"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","China"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Japan"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Qatar blockade"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Myanmar"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","United States"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Venezuela"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Foreign policy","Ukraine and Russian invasion of Ukraine"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Events"],"downwards":["Coup d'\u00e9tat attempt","2023 earthquake"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Events","Coup d'\u00e9tat attempt"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Presidency","Events","2023 earthquake"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image"],"downwards":["Ziya G\u00f6kalp","Ottomanism","Authoritarianism","Erdo\u011fanism","Views on minorities"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Ziya G\u00f6kalp"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Ottomanism"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Authoritarianism"],"downwards":["Suppression of dissent"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Authoritarianism","Suppression of dissent"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Erdo\u011fanism"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Views on minorities"],"downwards":["LGBT","Jews"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Views on minorities","LGBT"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Ideology and public image","Views on minorities","Jews"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Personal life"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Electoral history"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Honours and accolades"],"downwards":["Foreign honours","Other awards"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Honours and accolades","Foreign honours"],"downwards":["Supranational"]}
{"upwards":["Recep Tayyip Erdo\u011fan","Honours and accolades","Foreign honours","Supranational"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Honours and accolades","Other awards"],"downwards":[]}
{"upwards":["Recep Tayyip Erdo\u011fan","Bibliography"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine"],"downwards":["Background","Prelude","Invasion","Reactions","Foreign involvement","Casualties","Impacts","Peace efforts"]}
{"upwards":["Russian invasion of Ukraine","Background"],"downwards":["Ukrainian unrest, Russian activity in Crimea and Donbas"]}
{"upwards":["Russian invasion of Ukraine","Background","Ukrainian unrest, Russian activity in Crimea and Donbas"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Prelude"],"downwards":["Putin's invasion  announcement"]}
{"upwards":["Russian invasion of Ukraine","Prelude","Putin's invasion  announcement"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion"],"downwards":["Initial invasion of Ukraine (24 February \u2013 7 April)","Southeastern front (8 April \u2013 5 September)","Russian annexations and Ukrainian counterattacks (6 September\u00a0\u2013 11 November 2022)","Winter stalemate, attrition campaign and military surge (12 November 2022\u00a0\u2013 present)","Events in Crimea","Missile attacks and aerial warfare","Naval blockade and engagements","Command","Nuclear threats","Ukrainian resistance"]}
{"upwards":["Russian invasion of Ukraine","Invasion","Initial invasion of Ukraine (24 February \u2013 7 April)"],"downwards":["Kyiv and northern front","North-eastern front","Southern front","Eastern front"]}
{"upwards":["Russian invasion of Ukraine","Invasion","Initial invasion of Ukraine (24 February \u2013 7 April)","Kyiv and northern front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Initial invasion of Ukraine (24 February \u2013 7 April)","North-eastern front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Initial invasion of Ukraine (24 February \u2013 7 April)","Southern front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Initial invasion of Ukraine (24 February \u2013 7 April)","Eastern front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Southeastern front (8 April \u2013 5 September)"],"downwards":["Fall of Mariupol","Fall of Sievierodonetsk and Lysychansk","Kharkiv front","Kherson-Mykolaiv front","Zaporizhzhia front"]}
{"upwards":["Russian invasion of Ukraine","Invasion","Southeastern front (8 April \u2013 5 September)","Fall of Mariupol"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Southeastern front (8 April \u2013 5 September)","Fall of Sievierodonetsk and Lysychansk"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Southeastern front (8 April \u2013 5 September)","Kharkiv front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Southeastern front (8 April \u2013 5 September)","Kherson-Mykolaiv front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Southeastern front (8 April \u2013 5 September)","Zaporizhzhia front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Russian annexations and Ukrainian counterattacks (6 September\u00a0\u2013 11 November 2022)"],"downwards":["Russian annexation of Donetsk, Kherson, Luhansk and Zaporizhzhia oblasts","Zaporizhzhia front","Kherson counteroffensive","Kharkiv counteroffensive"]}
{"upwards":["Russian invasion of Ukraine","Invasion","Russian annexations and Ukrainian counterattacks (6 September\u00a0\u2013 11 November 2022)","Russian annexation of Donetsk, Kherson, Luhansk and Zaporizhzhia oblasts"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Russian annexations and Ukrainian counterattacks (6 September\u00a0\u2013 11 November 2022)","Zaporizhzhia front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Russian annexations and Ukrainian counterattacks (6 September\u00a0\u2013 11 November 2022)","Kherson counteroffensive"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Russian annexations and Ukrainian counterattacks (6 September\u00a0\u2013 11 November 2022)","Kharkiv counteroffensive"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Winter stalemate, attrition campaign and military surge (12 November 2022\u00a0\u2013 present)"],"downwards":["Battle of Bakhmut","Southern front","Diplomatic developments","Continued infrastructure attacks"]}
{"upwards":["Russian invasion of Ukraine","Invasion","Winter stalemate, attrition campaign and military surge (12 November 2022\u00a0\u2013 present)","Battle of Bakhmut"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Winter stalemate, attrition campaign and military surge (12 November 2022\u00a0\u2013 present)","Southern front"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Winter stalemate, attrition campaign and military surge (12 November 2022\u00a0\u2013 present)","Diplomatic developments"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Winter stalemate, attrition campaign and military surge (12 November 2022\u00a0\u2013 present)","Continued infrastructure attacks"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Events in Crimea"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Missile attacks and aerial warfare"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Naval blockade and engagements"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Command"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Nuclear threats"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Invasion","Ukrainian resistance"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Reactions"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Foreign involvement"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Casualties"],"downwards":["Prisoners of war"]}
{"upwards":["Russian invasion of Ukraine","Casualties","Prisoners of war"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Impacts"],"downwards":["Humanitarian impact","Refugee crisis","Environmental impact"]}
{"upwards":["Russian invasion of Ukraine","Impacts","Humanitarian impact"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Impacts","Refugee crisis"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Impacts","Environmental impact"],"downwards":[]}
{"upwards":["Russian invasion of Ukraine","Peace efforts"],"downwards":[]}
{"upwards":["Interpretation (model theory)"],"downwards":["Definition","Bi-interpretability","Example","References"]}
{"upwards":["Interpretation (model theory)","Definition"],"downwards":[]}
{"upwards":["Interpretation (model theory)","Bi-interpretability"],"downwards":[]}
{"upwards":["Interpretation (model theory)","Example"],"downwards":[]}
{"upwards":["Interpretation (model theory)","References"],"downwards":[]}
{"upwards":["Elementary equivalence"],"downwards":["Elementarily equivalent structures","Elementary substructures and elementary extensions","Tarski\u2013Vaught test","Elementary embeddings","References"]}
{"upwards":["Elementary equivalence","Elementarily equivalent structures"],"downwards":[]}
{"upwards":["Elementary equivalence","Elementary substructures and elementary extensions"],"downwards":[]}
{"upwards":["Elementary equivalence","Tarski\u2013Vaught test"],"downwards":[]}
{"upwards":["Elementary equivalence","Elementary embeddings"],"downwards":[]}
{"upwards":["Elementary equivalence","References"],"downwards":[]}
{"upwards":["Tree (graph theory)"],"downwards":["Definitions","Properties","Enumeration","Types of trees"]}
{"upwards":["Tree (graph theory)","Definitions"],"downwards":["Tree","Forest","Polytree","Polyforest","Rooted tree","Ordered tree"]}
{"upwards":["Tree (graph theory)","Definitions","Tree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Forest"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Polytree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Polyforest"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Rooted tree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Ordered tree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Properties"],"downwards":[]}
{"upwards":["Tree (graph theory)","Enumeration"],"downwards":["Labeled trees","Unlabeled trees"]}
{"upwards":["Tree (graph theory)","Enumeration","Labeled trees"],"downwards":[]}
{"upwards":["Tree (graph theory)","Enumeration","Unlabeled trees"],"downwards":[]}
{"upwards":["Tree (graph theory)","Types of trees"],"downwards":[]}
{"upwards":["Module (mathematics)"],"downwards":["Introduction and definition","Examples","Submodules and homomorphisms","Types of modules","Further notions"]}
{"upwards":["Module (mathematics)","Introduction and definition"],"downwards":["Motivation","Formal definition"]}
{"upwards":["Module (mathematics)","Introduction and definition","Motivation"],"downwards":[]}
{"upwards":["Module (mathematics)","Introduction and definition","Formal definition"],"downwards":[]}
{"upwards":["Module (mathematics)","Examples"],"downwards":[]}
{"upwards":["Module (mathematics)","Submodules and homomorphisms"],"downwards":[]}
{"upwards":["Module (mathematics)","Types of modules"],"downwards":[]}
{"upwards":["Module (mathematics)","Further notions"],"downwards":["Relation to representation theory","Generalizations"]}
{"upwards":["Module (mathematics)","Further notions","Relation to representation theory"],"downwards":[]}
{"upwards":["Module (mathematics)","Further notions","Generalizations"],"downwards":[]}
{"upwards":["Topological space"],"downwards":["History","Definitions","Comparison of topologies","Continuous functions","Examples of topological spaces","Topological constructions","Classification of topological spaces","Topological spaces with algebraic structure","Topological spaces with order structure"]}
{"upwards":["Topological space","History"],"downwards":[]}
{"upwards":["Topological space","Definitions"],"downwards":["Definition via neighbourhoods","Definition via open sets","Definition via closed sets","Other definitions"]}
{"upwards":["Topological space","Definitions","Definition via neighbourhoods"],"downwards":[]}
{"upwards":["Topological space","Definitions","Definition via open sets"],"downwards":["Examples of topologies"]}
{"upwards":["Topological space","Definitions","Definition via open sets","Examples of topologies"],"downwards":[]}
{"upwards":["Topological space","Definitions","Definition via closed sets"],"downwards":[]}
{"upwards":["Topological space","Definitions","Other definitions"],"downwards":[]}
{"upwards":["Topological space","Comparison of topologies"],"downwards":[]}
{"upwards":["Topological space","Continuous functions"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces"],"downwards":["Metric spaces","Proximity spaces","Uniform spaces","Function spaces","Cauchy spaces","Convergence spaces","Grothendieck sites","Other spaces"]}
{"upwards":["Topological space","Examples of topological spaces","Metric spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Proximity spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Uniform spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Function spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Cauchy spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Convergence spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Grothendieck sites"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Other spaces"],"downwards":[]}
{"upwards":["Topological space","Topological constructions"],"downwards":[]}
{"upwards":["Topological space","Classification of topological spaces"],"downwards":[]}
{"upwards":["Topological space","Topological spaces with algebraic structure"],"downwards":[]}
{"upwards":["Topological space","Topological spaces with order structure"],"downwards":[]}
{"upwards":["Lattice (order)"],"downwards":["Definition","Bounded lattice","Connection to other algebraic structures","Examples","Examples of non-lattices","Morphisms of lattices","Sublattices","Properties of lattices","Free lattices","Important lattice-theoretic notions"]}
{"upwards":["Lattice (order)","Definition"],"downwards":["As partially ordered set","As algebraic structure","Connection between the two definitions"]}
{"upwards":["Lattice (order)","Definition","As partially ordered set"],"downwards":[]}
{"upwards":["Lattice (order)","Definition","As algebraic structure"],"downwards":[]}
{"upwards":["Lattice (order)","Definition","Connection between the two definitions"],"downwards":[]}
{"upwards":["Lattice (order)","Bounded lattice"],"downwards":[]}
{"upwards":["Lattice (order)","Connection to other algebraic structures"],"downwards":[]}
{"upwards":["Lattice (order)","Examples"],"downwards":[]}
{"upwards":["Lattice (order)","Examples of non-lattices"],"downwards":[]}
{"upwards":["Lattice (order)","Morphisms of lattices"],"downwards":[]}
{"upwards":["Lattice (order)","Sublattices"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices"],"downwards":["Completeness","Conditional completeness","Distributivity","Modularity","Semimodularity","Continuity and algebraicity","Complements and pseudo-complements","Jordan\u2013Dedekind chain condition","Graded\/ranked"]}
{"upwards":["Lattice (order)","Properties of lattices","Completeness"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Conditional completeness"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Distributivity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Modularity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Semimodularity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Continuity and algebraicity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Complements and pseudo-complements"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Jordan\u2013Dedekind chain condition"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Graded\/ranked"],"downwards":[]}
{"upwards":["Lattice (order)","Free lattices"],"downwards":[]}
{"upwards":["Lattice (order)","Important lattice-theoretic notions"],"downwards":[]}
{"upwards":["Hyperboloid"],"downwards":["Parametric representations","Properties","In more than three dimensions","Hyperboloid structures","Relation to the sphere"]}
{"upwards":["Hyperboloid","Parametric representations"],"downwards":["Generalised equations"]}
{"upwards":["Hyperboloid","Parametric representations","Generalised equations"],"downwards":[]}
{"upwards":["Hyperboloid","Properties"],"downwards":["Hyperboloid of one sheet","Hyperboloid of two sheets","Other properties"]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of one sheet"],"downwards":["Lines on the surface","Plane sections"]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of one sheet","Lines on the surface"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of one sheet","Plane sections"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of two sheets"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Other properties"],"downwards":["Symmetries","Curvature"]}
{"upwards":["Hyperboloid","Properties","Other properties","Symmetries"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Other properties","Curvature"],"downwards":[]}
{"upwards":["Hyperboloid","In more than three dimensions"],"downwards":[]}
{"upwards":["Hyperboloid","Hyperboloid structures"],"downwards":[]}
{"upwards":["Hyperboloid","Relation to the sphere"],"downwards":[]}
{"upwards":["Cone"],"downwards":["Further terminology","Measurements and equations","Projective geometry","Generalizations"]}
{"upwards":["Cone","Further terminology"],"downwards":[]}
{"upwards":["Cone","Measurements and equations"],"downwards":["Volume","Center of mass","Right circular cone","Elliptic cone"]}
{"upwards":["Cone","Measurements and equations","Volume"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Center of mass"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Right circular cone"],"downwards":["Volume","Slant height","Surface area","Circular sector","Equation form"]}
{"upwards":["Cone","Measurements and equations","Right circular cone","Volume"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Right circular cone","Slant height"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Right circular cone","Surface area"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Right circular cone","Circular sector"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Right circular cone","Equation form"],"downwards":[]}
{"upwards":["Cone","Measurements and equations","Elliptic cone"],"downwards":[]}
{"upwards":["Cone","Projective geometry"],"downwards":[]}
{"upwards":["Cone","Generalizations"],"downwards":[]}
{"upwards":["Matrix (mathematics)"],"downwards":["Definition","Notation","Basic operations","Linear equations","Linear transformations","Square matrix","Computational aspects","Decomposition","Abstract algebraic aspects and generalizations","Applications","History"]}
{"upwards":["Matrix (mathematics)","Definition"],"downwards":["Size"]}
{"upwards":["Matrix (mathematics)","Definition","Size"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Notation"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations"],"downwards":["Addition, scalar multiplication, and transposition","Matrix multiplication","Row operations","Submatrix"]}
{"upwards":["Matrix (mathematics)","Basic operations","Addition, scalar multiplication, and transposition"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations","Matrix multiplication"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations","Row operations"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations","Submatrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Linear equations"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Linear transformations"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix"],"downwards":["Main types","Main operations"]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types"],"downwards":["Diagonal and triangular matrix","Identity matrix","Symmetric or skew-symmetric matrix","Invertible matrix and its inverse","Definite matrix","Orthogonal matrix"]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Diagonal and triangular matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Identity matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Symmetric or skew-symmetric matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Invertible matrix and its inverse"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Definite matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Orthogonal matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations"],"downwards":["Trace","Determinant","Eigenvalues and eigenvectors"]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations","Trace"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations","Determinant"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations","Eigenvalues and eigenvectors"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Computational aspects"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Decomposition"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations"],"downwards":["Matrices with more general entries","Relationship to linear maps","Matrix groups","Infinite matrices","Empty matrix"]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Matrices with more general entries"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Relationship to linear maps"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Matrix groups"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Infinite matrices"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Empty matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications"],"downwards":["Graph theory","Analysis and geometry","Probability theory and statistics","Symmetries and transformations in physics","Linear combinations of quantum states","Normal modes","Geometrical optics","Electronics"]}
{"upwards":["Matrix (mathematics)","Applications","Graph theory"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Analysis and geometry"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Probability theory and statistics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Symmetries and transformations in physics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Linear combinations of quantum states"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Normal modes"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Geometrical optics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Electronics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","History"],"downwards":["Other historical usages of the word \"matrix\" in mathematics"]}
{"upwards":["Matrix (mathematics)","History","Other historical usages of the word \"matrix\" in mathematics"],"downwards":[]}
{"upwards":["Topology"],"downwards":["Motivation","History","Concepts","Topics","Applications"]}
{"upwards":["Topology","Motivation"],"downwards":[]}
{"upwards":["Topology","History"],"downwards":[]}
{"upwards":["Topology","Concepts"],"downwards":["Topologies on sets","Continuous functions and homeomorphisms","Manifolds"]}
{"upwards":["Topology","Concepts","Topologies on sets"],"downwards":[]}
{"upwards":["Topology","Concepts","Continuous functions and homeomorphisms"],"downwards":[]}
{"upwards":["Topology","Concepts","Manifolds"],"downwards":[]}
{"upwards":["Topology","Topics"],"downwards":["General topology","Algebraic topology","Differential topology","Geometric topology","Generalizations"]}
{"upwards":["Topology","Topics","General topology"],"downwards":[]}
{"upwards":["Topology","Topics","Algebraic topology"],"downwards":[]}
{"upwards":["Topology","Topics","Differential topology"],"downwards":[]}
{"upwards":["Topology","Topics","Geometric topology"],"downwards":[]}
{"upwards":["Topology","Topics","Generalizations"],"downwards":[]}
{"upwards":["Topology","Applications"],"downwards":["Biology","Computer science","Physics","Robotics","Games and puzzles","Fiber art"]}
{"upwards":["Topology","Applications","Biology"],"downwards":[]}
{"upwards":["Topology","Applications","Computer science"],"downwards":[]}
{"upwards":["Topology","Applications","Physics"],"downwards":[]}
{"upwards":["Topology","Applications","Robotics"],"downwards":[]}
{"upwards":["Topology","Applications","Games and puzzles"],"downwards":[]}
{"upwards":["Topology","Applications","Fiber art"],"downwards":[]}
{"upwards":["Tensor"],"downwards":["Definition","Examples","Properties","Notation","Operations","Applications","Generalizations","History"]}
{"upwards":["Tensor","Definition"],"downwards":["As multidimensional arrays","As multilinear maps","Using tensor products","Tensors in infinite dimensions","Tensor fields"]}
{"upwards":["Tensor","Definition","As multidimensional arrays"],"downwards":[]}
{"upwards":["Tensor","Definition","As multilinear maps"],"downwards":[]}
{"upwards":["Tensor","Definition","Using tensor products"],"downwards":[]}
{"upwards":["Tensor","Definition","Tensors in infinite dimensions"],"downwards":[]}
{"upwards":["Tensor","Definition","Tensor fields"],"downwards":[]}
{"upwards":["Tensor","Examples"],"downwards":[]}
{"upwards":["Tensor","Properties"],"downwards":[]}
{"upwards":["Tensor","Notation"],"downwards":["Ricci calculus","Einstein summation convention","Penrose graphical notation","Abstract index notation","Component-free notation"]}
{"upwards":["Tensor","Notation","Ricci calculus"],"downwards":[]}
{"upwards":["Tensor","Notation","Einstein summation convention"],"downwards":[]}
{"upwards":["Tensor","Notation","Penrose graphical notation"],"downwards":[]}
{"upwards":["Tensor","Notation","Abstract index notation"],"downwards":[]}
{"upwards":["Tensor","Notation","Component-free notation"],"downwards":[]}
{"upwards":["Tensor","Operations"],"downwards":["Tensor product","Contraction","Raising or lowering an index"]}
{"upwards":["Tensor","Operations","Tensor product"],"downwards":[]}
{"upwards":["Tensor","Operations","Contraction"],"downwards":[]}
{"upwards":["Tensor","Operations","Raising or lowering an index"],"downwards":[]}
{"upwards":["Tensor","Applications"],"downwards":["Continuum mechanics","Other examples from physics","Computer vision and optics","Machine learning"]}
{"upwards":["Tensor","Applications","Continuum mechanics"],"downwards":[]}
{"upwards":["Tensor","Applications","Other examples from physics"],"downwards":[]}
{"upwards":["Tensor","Applications","Computer vision and optics"],"downwards":[]}
{"upwards":["Tensor","Applications","Machine learning"],"downwards":[]}
{"upwards":["Tensor","Generalizations"],"downwards":["Tensor products of vector spaces","Tensors in infinite dimensions","Tensor densities","Geometric objects","Spinors"]}
{"upwards":["Tensor","Generalizations","Tensor products of vector spaces"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Tensors in infinite dimensions"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Tensor densities"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Geometric objects"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Spinors"],"downwards":[]}
{"upwards":["Tensor","History"],"downwards":[]}
{"upwards":["Oxford University Press"],"downwards":["History","Museum","Scholarly journals","Series and titles","Clarendon Scholarships","Controversies"]}
{"upwards":["Oxford University Press","History"],"downwards":[]}
{"upwards":["Oxford University Press","Museum"],"downwards":["Clarendon Press"]}
{"upwards":["Oxford University Press","Museum","Clarendon Press"],"downwards":[]}
{"upwards":["Oxford University Press","Scholarly journals"],"downwards":[]}
{"upwards":["Oxford University Press","Series and titles"],"downwards":[]}
{"upwards":["Oxford University Press","Clarendon Scholarships"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies"],"downwards":["Tehran Book Fair Controversy","Malcolm vs Oxford University 1986-1992","Closure of Poetry List","Tax Exemption Controversy","East African Bribery Scandal"]}
{"upwards":["Oxford University Press","Controversies","Tehran Book Fair Controversy"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","Malcolm vs Oxford University 1986-1992"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","Closure of Poetry List"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","Tax Exemption Controversy"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","East African Bribery Scandal"],"downwards":[]}
{"upwards":["Cylinder"],"downwards":["Types","Properties","Cylindrical surfaces","Projective geometry","Prisms"]}
{"upwards":["Cylinder","Types"],"downwards":["Right circular cylinders"]}
{"upwards":["Cylinder","Types","Right circular cylinders"],"downwards":[]}
{"upwards":["Cylinder","Properties"],"downwards":["Cylindric sections","Volume","Surface area","Right circular hollow cylinder (cylindrical shell)","On the Sphere and Cylinder"]}
{"upwards":["Cylinder","Properties","Cylindric sections"],"downwards":[]}
{"upwards":["Cylinder","Properties","Volume"],"downwards":[]}
{"upwards":["Cylinder","Properties","Surface area"],"downwards":[]}
{"upwards":["Cylinder","Properties","Right circular hollow cylinder (cylindrical shell)"],"downwards":[]}
{"upwards":["Cylinder","Properties","On the Sphere and Cylinder"],"downwards":[]}
{"upwards":["Cylinder","Cylindrical surfaces"],"downwards":["Elliptic cylinder","Hyperbolic cylinder","Parabolic cylinder"]}
{"upwards":["Cylinder","Cylindrical surfaces","Elliptic cylinder"],"downwards":[]}
{"upwards":["Cylinder","Cylindrical surfaces","Hyperbolic cylinder"],"downwards":[]}
{"upwards":["Cylinder","Cylindrical surfaces","Parabolic cylinder"],"downwards":[]}
{"upwards":["Cylinder","Projective geometry"],"downwards":[]}
{"upwards":["Cylinder","Prisms"],"downwards":[]}
{"upwards":["Graph theory"],"downwards":["Definitions","Applications","History","Representation","Problems"]}
{"upwards":["Graph theory","Definitions"],"downwards":["Graph","Directed graph"]}
{"upwards":["Graph theory","Definitions","Graph"],"downwards":[]}
{"upwards":["Graph theory","Definitions","Directed graph"],"downwards":[]}
{"upwards":["Graph theory","Applications"],"downwards":["Computer science","Linguistics","Physics and chemistry","Social sciences","Biology","Mathematics","Other topics"]}
{"upwards":["Graph theory","Applications","Computer science"],"downwards":[]}
{"upwards":["Graph theory","Applications","Linguistics"],"downwards":[]}
{"upwards":["Graph theory","Applications","Physics and chemistry"],"downwards":[]}
{"upwards":["Graph theory","Applications","Social sciences"],"downwards":[]}
{"upwards":["Graph theory","Applications","Biology"],"downwards":[]}
{"upwards":["Graph theory","Applications","Mathematics"],"downwards":[]}
{"upwards":["Graph theory","Applications","Other topics"],"downwards":[]}
{"upwards":["Graph theory","History"],"downwards":[]}
{"upwards":["Graph theory","Representation"],"downwards":["Visual: Graph drawing","Tabular: Graph data structures"]}
{"upwards":["Graph theory","Representation","Visual: Graph drawing"],"downwards":[]}
{"upwards":["Graph theory","Representation","Tabular: Graph data structures"],"downwards":[]}
{"upwards":["Graph theory","Problems"],"downwards":["Enumeration","Subgraphs, induced subgraphs, and minors","Graph coloring","Subsumption and unification","Route problems","Network flow","Visibility problems","Covering problems","Decomposition problems","Graph classes"]}
{"upwards":["Graph theory","Problems","Enumeration"],"downwards":[]}
{"upwards":["Graph theory","Problems","Subgraphs, induced subgraphs, and minors"],"downwards":[]}
{"upwards":["Graph theory","Problems","Graph coloring"],"downwards":[]}
{"upwards":["Graph theory","Problems","Subsumption and unification"],"downwards":[]}
{"upwards":["Graph theory","Problems","Route problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Network flow"],"downwards":[]}
{"upwards":["Graph theory","Problems","Visibility problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Covering problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Decomposition problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Graph classes"],"downwards":[]}
{"upwards":["Manifold"],"downwards":["Motivating examples","Mathematical definition","Charts, atlases, and transition maps","Manifold with boundary","Construction","History","Additional structure","Classification and invariants","Surfaces","Maps of manifolds","Generalizations of manifolds"]}
{"upwards":["Manifold","Motivating examples"],"downwards":["Circle","Sphere","Other curves"]}
{"upwards":["Manifold","Motivating examples","Circle"],"downwards":[]}
{"upwards":["Manifold","Motivating examples","Sphere"],"downwards":[]}
{"upwards":["Manifold","Motivating examples","Other curves"],"downwards":[]}
{"upwards":["Manifold","Mathematical definition"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps"],"downwards":["Charts","Atlases","Transition maps","Additional structure"]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Charts"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Atlases"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Transition maps"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Additional structure"],"downwards":[]}
{"upwards":["Manifold","Manifold with boundary"],"downwards":["Boundary and interior"]}
{"upwards":["Manifold","Manifold with boundary","Boundary and interior"],"downwards":[]}
{"upwards":["Manifold","Construction"],"downwards":["Charts","Patchwork","Identifying points of a manifold","Gluing along boundaries","Cartesian products"]}
{"upwards":["Manifold","Construction","Charts"],"downwards":["Sphere with charts"]}
{"upwards":["Manifold","Construction","Charts","Sphere with charts"],"downwards":[]}
{"upwards":["Manifold","Construction","Patchwork"],"downwards":["Intrinsic and extrinsic view","n-Sphere as a patchwork"]}
{"upwards":["Manifold","Construction","Patchwork","Intrinsic and extrinsic view"],"downwards":[]}
{"upwards":["Manifold","Construction","Patchwork","n-Sphere as a patchwork"],"downwards":[]}
{"upwards":["Manifold","Construction","Identifying points of a manifold"],"downwards":[]}
{"upwards":["Manifold","Construction","Gluing along boundaries"],"downwards":[]}
{"upwards":["Manifold","Construction","Cartesian products"],"downwards":[]}
{"upwards":["Manifold","History"],"downwards":["Early development","Synthesis","Poincar\u00e9's definition","Topology of manifolds: highlights"]}
{"upwards":["Manifold","History","Early development"],"downwards":[]}
{"upwards":["Manifold","History","Synthesis"],"downwards":[]}
{"upwards":["Manifold","History","Poincar\u00e9's definition"],"downwards":[]}
{"upwards":["Manifold","History","Topology of manifolds: highlights"],"downwards":[]}
{"upwards":["Manifold","Additional structure"],"downwards":["Topological manifolds","Differentiable manifolds","Riemannian manifolds","Finsler manifolds","Lie groups","Other types of manifolds"]}
{"upwards":["Manifold","Additional structure","Topological manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Differentiable manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Riemannian manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Finsler manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Lie groups"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Other types of manifolds"],"downwards":[]}
{"upwards":["Manifold","Classification and invariants"],"downwards":[]}
{"upwards":["Manifold","Surfaces"],"downwards":["Orientability","Genus and the Euler characteristic"]}
{"upwards":["Manifold","Surfaces","Orientability"],"downwards":["M\u00f6bius strip","Klein bottle","Real projective plane"]}
{"upwards":["Manifold","Surfaces","Orientability","M\u00f6bius strip"],"downwards":[]}
{"upwards":["Manifold","Surfaces","Orientability","Klein bottle"],"downwards":[]}
{"upwards":["Manifold","Surfaces","Orientability","Real projective plane"],"downwards":[]}
{"upwards":["Manifold","Surfaces","Genus and the Euler characteristic"],"downwards":[]}
{"upwards":["Manifold","Maps of manifolds"],"downwards":["Scalar-valued functions"]}
{"upwards":["Manifold","Maps of manifolds","Scalar-valued functions"],"downwards":[]}
{"upwards":["Manifold","Generalizations of manifolds"],"downwards":[]}
{"upwards":["Scalar (mathematics)"],"downwards":["Etymology","Definitions and properties"]}
{"upwards":["Scalar (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties"],"downwards":["Scalars of vector spaces","Scalars as vector components","Scalars in normed vector spaces","Scalars in modules","Scaling transformation"]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars of vector spaces"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars as vector components"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars in normed vector spaces"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars in modules"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scaling transformation"],"downwards":[]}
{"upwards":["Vector space"],"downwards":["Definition and basic properties","Related concepts and properties","History","Examples","Linear maps and matrices","Basic constructions","Vector spaces with additional structure","Related structures","Related concepts"]}
{"upwards":["Vector space","Definition and basic properties"],"downwards":[]}
{"upwards":["Vector space","Related concepts and properties"],"downwards":[]}
{"upwards":["Vector space","History"],"downwards":[]}
{"upwards":["Vector space","Examples"],"downwards":["Arrows in the plane","Second example: ordered pairs of numbers","Coordinate space","Complex numbers and other field extensions","Function spaces","Linear equations"]}
{"upwards":["Vector space","Examples","Arrows in the plane"],"downwards":[]}
{"upwards":["Vector space","Examples","Second example: ordered pairs of numbers"],"downwards":[]}
{"upwards":["Vector space","Examples","Coordinate space"],"downwards":[]}
{"upwards":["Vector space","Examples","Complex numbers and other field extensions"],"downwards":[]}
{"upwards":["Vector space","Examples","Function spaces"],"downwards":[]}
{"upwards":["Vector space","Examples","Linear equations"],"downwards":[]}
{"upwards":["Vector space","Linear maps and matrices"],"downwards":["Matrices","Eigenvalues and eigenvectors"]}
{"upwards":["Vector space","Linear maps and matrices","Matrices"],"downwards":[]}
{"upwards":["Vector space","Linear maps and matrices","Eigenvalues and eigenvectors"],"downwards":[]}
{"upwards":["Vector space","Basic constructions"],"downwards":["Subspaces and quotient spaces","Direct product and direct sum","Tensor product"]}
{"upwards":["Vector space","Basic constructions","Subspaces and quotient spaces"],"downwards":[]}
{"upwards":["Vector space","Basic constructions","Direct product and direct sum"],"downwards":[]}
{"upwards":["Vector space","Basic constructions","Tensor product"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure"],"downwards":["Normed vector spaces and inner product spaces","Topological vector spaces","Algebras over fields"]}
{"upwards":["Vector space","Vector spaces with additional structure","Normed vector spaces and inner product spaces"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure","Topological vector spaces"],"downwards":["Banach spaces","Hilbert spaces"]}
{"upwards":["Vector space","Vector spaces with additional structure","Topological vector spaces","Banach spaces"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure","Topological vector spaces","Hilbert spaces"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure","Algebras over fields"],"downwards":[]}
{"upwards":["Vector space","Related structures"],"downwards":["Vector bundles","Modules","Affine and projective spaces"]}
{"upwards":["Vector space","Related structures","Vector bundles"],"downwards":[]}
{"upwards":["Vector space","Related structures","Modules"],"downwards":[]}
{"upwards":["Vector space","Related structures","Affine and projective spaces"],"downwards":[]}
{"upwards":["Vector space","Related concepts"],"downwards":[]}
{"upwards":["Linear algebra"],"downwards":["History","Vector spaces","Matrices","Linear systems","Endomorphisms and square matrices","Duality","Relationship with geometry","Usage and applications","Extensions and generalizations"]}
{"upwards":["Linear algebra","History"],"downwards":[]}
{"upwards":["Linear algebra","Vector spaces"],"downwards":["Linear maps","Subspaces, span, and basis"]}
{"upwards":["Linear algebra","Vector spaces","Linear maps"],"downwards":[]}
{"upwards":["Linear algebra","Vector spaces","Subspaces, span, and basis"],"downwards":[]}
{"upwards":["Linear algebra","Matrices"],"downwards":[]}
{"upwards":["Linear algebra","Linear systems"],"downwards":[]}
{"upwards":["Linear algebra","Endomorphisms and square matrices"],"downwards":["Determinant","Eigenvalues and eigenvectors"]}
{"upwards":["Linear algebra","Endomorphisms and square matrices","Determinant"],"downwards":[]}
{"upwards":["Linear algebra","Endomorphisms and square matrices","Eigenvalues and eigenvectors"],"downwards":[]}
{"upwards":["Linear algebra","Duality"],"downwards":["Dual map","Inner-product spaces"]}
{"upwards":["Linear algebra","Duality","Dual map"],"downwards":[]}
{"upwards":["Linear algebra","Duality","Inner-product spaces"],"downwards":[]}
{"upwards":["Linear algebra","Relationship with geometry"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications"],"downwards":["Geometry of ambient space","Functional analysis","Study of complex systems","Scientific computation"]}
{"upwards":["Linear algebra","Usage and applications","Geometry of ambient space"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications","Functional analysis"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications","Study of complex systems"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications","Scientific computation"],"downwards":[]}
{"upwards":["Linear algebra","Extensions and generalizations"],"downwards":["Module theory","Multilinear algebra and tensors","Topological vector spaces"]}
{"upwards":["Linear algebra","Extensions and generalizations","Module theory"],"downwards":[]}
{"upwards":["Linear algebra","Extensions and generalizations","Multilinear algebra and tensors"],"downwards":[]}
{"upwards":["Linear algebra","Extensions and generalizations","Topological vector spaces"],"downwards":[]}
{"upwards":["Glossary of graph theory"],"downwards":["Symbols","A","B","C","D","E","F","G","H","I","K","L","M","N","O","P","Q","R","S","T","U","V","W"]}
{"upwards":["Glossary of graph theory","Symbols"],"downwards":[]}
{"upwards":["Glossary of graph theory","A"],"downwards":[]}
{"upwards":["Glossary of graph theory","B"],"downwards":[]}
{"upwards":["Glossary of graph theory","C"],"downwards":[]}
{"upwards":["Glossary of graph theory","D"],"downwards":[]}
{"upwards":["Glossary of graph theory","E"],"downwards":[]}
{"upwards":["Glossary of graph theory","F"],"downwards":[]}
{"upwards":["Glossary of graph theory","G"],"downwards":[]}
{"upwards":["Glossary of graph theory","H"],"downwards":[]}
{"upwards":["Glossary of graph theory","I"],"downwards":[]}
{"upwards":["Glossary of graph theory","K"],"downwards":[]}
{"upwards":["Glossary of graph theory","L"],"downwards":[]}
{"upwards":["Glossary of graph theory","M"],"downwards":[]}
{"upwards":["Glossary of graph theory","N"],"downwards":[]}
{"upwards":["Glossary of graph theory","O"],"downwards":[]}
{"upwards":["Glossary of graph theory","P"],"downwards":[]}
{"upwards":["Glossary of graph theory","Q"],"downwards":[]}
{"upwards":["Glossary of graph theory","R"],"downwards":[]}
{"upwards":["Glossary of graph theory","S"],"downwards":[]}
{"upwards":["Glossary of graph theory","T"],"downwards":[]}
{"upwards":["Glossary of graph theory","U"],"downwards":[]}
{"upwards":["Glossary of graph theory","V"],"downwards":[]}
{"upwards":["Glossary of graph theory","W"],"downwards":[]}
{"upwards":["Paraboloid"],"downwards":["Properties and applications","Cylinder between pencils of elliptic and hyperbolic paraboloids","Curvature","Geometric representation of multiplication table","Dimensions of a paraboloidal dish"]}
{"upwards":["Paraboloid","Properties and applications"],"downwards":["Elliptic paraboloid","Hyperbolic paraboloid"]}
{"upwards":["Paraboloid","Properties and applications","Elliptic paraboloid"],"downwards":["Parabolic reflector"]}
{"upwards":["Paraboloid","Properties and applications","Elliptic paraboloid","Parabolic reflector"],"downwards":[]}
{"upwards":["Paraboloid","Properties and applications","Hyperbolic paraboloid"],"downwards":["Examples in architecture"]}
{"upwards":["Paraboloid","Properties and applications","Hyperbolic paraboloid","Examples in architecture"],"downwards":[]}
{"upwards":["Paraboloid","Cylinder between pencils of elliptic and hyperbolic paraboloids"],"downwards":[]}
{"upwards":["Paraboloid","Curvature"],"downwards":[]}
{"upwards":["Paraboloid","Geometric representation of multiplication table"],"downwards":[]}
{"upwards":["Paraboloid","Dimensions of a paraboloidal dish"],"downwards":[]}
{"upwards":["Lattice (group)"],"downwards":["Symmetry considerations and examples","Dividing space according to a lattice","Lattice points in convex sets","Computational lattice problems","Lattices in two dimensions: detailed discussion","Lattices in three dimensions","Lattices in complex space","In Lie groups","Lattices in general vector spaces","Related notions"]}
{"upwards":["Lattice (group)","Symmetry considerations and examples"],"downwards":[]}
{"upwards":["Lattice (group)","Dividing space according to a lattice"],"downwards":[]}
{"upwards":["Lattice (group)","Lattice points in convex sets"],"downwards":[]}
{"upwards":["Lattice (group)","Computational lattice problems"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in two dimensions: detailed discussion"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in three dimensions"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in complex space"],"downwards":[]}
{"upwards":["Lattice (group)","In Lie groups"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in general vector spaces"],"downwards":[]}
{"upwards":["Lattice (group)","Related notions"],"downwards":[]}
{"upwards":["Abstract and concrete"],"downwards":["In philosophy","Concrete and abstract thought in psychology"]}
{"upwards":["Abstract and concrete","In philosophy"],"downwards":["Abstract objects and causality","Quasi-abstract entities"]}
{"upwards":["Abstract and concrete","In philosophy","Abstract objects and causality"],"downwards":[]}
{"upwards":["Abstract and concrete","In philosophy","Quasi-abstract entities"],"downwards":[]}
{"upwards":["Abstract and concrete","Concrete and abstract thought in psychology"],"downwards":[]}
{"upwards":["Philip J. Davis"],"downwards":["Publications (Books Only)","Notes","References","External links"]}
{"upwards":["Philip J. Davis","Publications (Books Only)"],"downwards":[]}
{"upwards":["Philip J. Davis","Notes"],"downwards":[]}
{"upwards":["Philip J. Davis","References"],"downwards":[]}
{"upwards":["Philip J. Davis","External links"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)"],"downwards":["Vectors in Euclidean geometry","Vector spaces","Vectors in algebra","Data represented by vectors"]}
{"upwards":["Vector (mathematics and physics)","Vectors in Euclidean geometry"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vector spaces"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vectors in algebra"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Data represented by vectors"],"downwards":[]}
{"upwards":["John P. Burgess"],"downwards":["Education and career","Selected publications","References","External links"]}
{"upwards":["John P. Burgess","Education and career"],"downwards":[]}
{"upwards":["John P. Burgess","Selected publications"],"downwards":[]}
{"upwards":["John P. Burgess","References"],"downwards":[]}
{"upwards":["John P. Burgess","External links"],"downwards":[]}
{"upwards":["Reuben Hersh"],"downwards":["Education","Academic career","Books"]}
{"upwards":["Reuben Hersh","Education"],"downwards":[]}
{"upwards":["Reuben Hersh","Academic career"],"downwards":[]}
{"upwards":["Reuben Hersh","Books"],"downwards":[]}
{"upwards":["Vertex (graph theory)"],"downwards":["Types of vertices"]}
{"upwards":["Vertex (graph theory)","Types of vertices"],"downwards":[]}
{"upwards":["Bonnie Gold"],"downwards":["Education and career","Contributions","Recognition","References","External links"]}
{"upwards":["Bonnie Gold","Education and career"],"downwards":[]}
{"upwards":["Bonnie Gold","Contributions"],"downwards":[]}
{"upwards":["Bonnie Gold","Recognition"],"downwards":[]}
{"upwards":["Bonnie Gold","References"],"downwards":[]}
{"upwards":["Bonnie Gold","External links"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)"],"downwards":["Definitions","Types of graphs","Properties of graphs","Examples","Graph operations","Generalizations"]}
{"upwards":["Graph (discrete mathematics)","Definitions"],"downwards":["Graph","Directed graph","Mixed graph","Weighted graph"]}
{"upwards":["Graph (discrete mathematics)","Definitions","Graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Definitions","Directed graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Definitions","Mixed graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Definitions","Weighted graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs"],"downwards":["Oriented graph","Regular graph","Complete graph","Finite graph","Connected graph","Bipartite graph","Path graph","Planar graph","Cycle graph","Tree","Polytree","Advanced classes"]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Oriented graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Regular graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Complete graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Finite graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Connected graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Bipartite graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Path graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Planar graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Cycle graph"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Tree"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Polytree"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Types of graphs","Advanced classes"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Properties of graphs"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Examples"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Graph operations"],"downwards":[]}
{"upwards":["Graph (discrete mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Mathematical structure"],"downwards":["History","Example: the real numbers"]}
{"upwards":["Mathematical structure","History"],"downwards":[]}
{"upwards":["Mathematical structure","Example: the real numbers"],"downwards":[]}
{"upwards":["Binary relation"],"downwards":["Definition","Operations","Examples","Special types of binary relations","Sets versus classes","Homogeneous relation","Heterogeneous relation","Calculus of relations","Induced concept lattice","Particular relations","Preorder R\\R","Fringe of a relation","Mathematical heaps"]}
{"upwards":["Binary relation","Definition"],"downwards":[]}
{"upwards":["Binary relation","Operations"],"downwards":["Union","Intersection","Composition","Converse","Complement","Restriction","Matrix representation"]}
{"upwards":["Binary relation","Operations","Union"],"downwards":[]}
{"upwards":["Binary relation","Operations","Intersection"],"downwards":[]}
{"upwards":["Binary relation","Operations","Composition"],"downwards":[]}
{"upwards":["Binary relation","Operations","Converse"],"downwards":[]}
{"upwards":["Binary relation","Operations","Complement"],"downwards":[]}
{"upwards":["Binary relation","Operations","Restriction"],"downwards":[]}
{"upwards":["Binary relation","Operations","Matrix representation"],"downwards":[]}
{"upwards":["Binary relation","Examples"],"downwards":[]}
{"upwards":["Binary relation","Special types of binary relations"],"downwards":[]}
{"upwards":["Binary relation","Sets versus classes"],"downwards":[]}
{"upwards":["Binary relation","Homogeneous relation"],"downwards":[]}
{"upwards":["Binary relation","Heterogeneous relation"],"downwards":[]}
{"upwards":["Binary relation","Calculus of relations"],"downwards":[]}
{"upwards":["Binary relation","Induced concept lattice"],"downwards":[]}
{"upwards":["Binary relation","Particular relations"],"downwards":["Difunctional","Ferrers type","Contact"]}
{"upwards":["Binary relation","Particular relations","Difunctional"],"downwards":[]}
{"upwards":["Binary relation","Particular relations","Ferrers type"],"downwards":[]}
{"upwards":["Binary relation","Particular relations","Contact"],"downwards":[]}
{"upwards":["Binary relation","Preorder R\\R"],"downwards":[]}
{"upwards":["Binary relation","Fringe of a relation"],"downwards":[]}
{"upwards":["Binary relation","Mathematical heaps"],"downwards":[]}
{"upwards":["Recursive definition"],"downwards":["Form of recursive definitions","Examples of recursive definitions"]}
{"upwards":["Recursive definition","Form of recursive definitions"],"downwards":["Principle of recursive definition"]}
{"upwards":["Recursive definition","Form of recursive definitions","Principle of recursive definition"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions"],"downwards":["Elementary functions","Prime numbers","Non-negative even numbers","Well formed formulas"]}
{"upwards":["Recursive definition","Examples of recursive definitions","Elementary functions"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions","Prime numbers"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions","Non-negative even numbers"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions","Well formed formulas"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)"],"downwards":["Vectors in Euclidean geometry","Vector spaces","Vectors in algebra","Data represented by vectors"]}
{"upwards":["Vector (mathematics and physics)","Vectors in Euclidean geometry"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vector spaces"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vectors in algebra"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Data represented by vectors"],"downwards":[]}
{"upwards":["Terminology"],"downwards":["Overview","Types of terminology","Terminology as a discipline","Terminological theories"]}
{"upwards":["Terminology","Overview"],"downwards":[]}
{"upwards":["Terminology","Types of terminology"],"downwards":[]}
{"upwards":["Terminology","Terminology as a discipline"],"downwards":["Science"]}
{"upwards":["Terminology","Terminology as a discipline","Science"],"downwards":[]}
{"upwards":["Terminology","Terminological theories"],"downwards":[]}
{"upwards":["Stanford Encyclopedia of Philosophy"],"downwards":["Approach and history"]}
{"upwards":["Stanford Encyclopedia of Philosophy","Approach and history"],"downwards":[]}
{"upwards":["Analytic geometry"],"downwards":["History","Coordinates","Equations and curves","Distance and angle","Transformations","Finding intersections of geometric objects","Geometric axis","Spherical and nonlinear planes and their tangents"]}
{"upwards":["Analytic geometry","History"],"downwards":["Ancient Greece","Persia","Western Europe"]}
{"upwards":["Analytic geometry","History","Ancient Greece"],"downwards":[]}
{"upwards":["Analytic geometry","History","Persia"],"downwards":[]}
{"upwards":["Analytic geometry","History","Western Europe"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates"],"downwards":["Cartesian coordinates (in a plane or space)","Polar coordinates (in a plane)","Cylindrical coordinates (in a space)","Spherical coordinates (in a space)"]}
{"upwards":["Analytic geometry","Coordinates","Cartesian coordinates (in a plane or space)"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates","Polar coordinates (in a plane)"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates","Cylindrical coordinates (in a space)"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates","Spherical coordinates (in a space)"],"downwards":[]}
{"upwards":["Analytic geometry","Equations and curves"],"downwards":["Lines and planes","Conic sections","Quadric surfaces"]}
{"upwards":["Analytic geometry","Equations and curves","Lines and planes"],"downwards":[]}
{"upwards":["Analytic geometry","Equations and curves","Conic sections"],"downwards":[]}
{"upwards":["Analytic geometry","Equations and curves","Quadric surfaces"],"downwards":[]}
{"upwards":["Analytic geometry","Distance and angle"],"downwards":[]}
{"upwards":["Analytic geometry","Transformations"],"downwards":[]}
{"upwards":["Analytic geometry","Finding intersections of geometric objects"],"downwards":["Finding intercepts"]}
{"upwards":["Analytic geometry","Finding intersections of geometric objects","Finding intercepts"],"downwards":[]}
{"upwards":["Analytic geometry","Geometric axis"],"downwards":[]}
{"upwards":["Analytic geometry","Spherical and nonlinear planes and their tangents"],"downwards":["Tangent lines and planes"]}
{"upwards":["Analytic geometry","Spherical and nonlinear planes and their tangents","Tangent lines and planes"],"downwards":[]}
{"upwards":["Euclidean plane"],"downwards":["History","In geometry","In linear algebra","In calculus","In topology","In graph theory"]}
{"upwards":["Euclidean plane","History"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry"],"downwards":["Coordinate systems","Embedding in three-dimensional space","Polytopes","Circle","Other shapes"]}
{"upwards":["Euclidean plane","In geometry","Coordinate systems"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry","Embedding in three-dimensional space"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry","Polytopes"],"downwards":["Convex","Degenerate (spherical)","Non-convex"]}
{"upwards":["Euclidean plane","In geometry","Polytopes","Convex"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry","Polytopes","Degenerate (spherical)"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry","Polytopes","Non-convex"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry","Circle"],"downwards":[]}
{"upwards":["Euclidean plane","In geometry","Other shapes"],"downwards":[]}
{"upwards":["Euclidean plane","In linear algebra"],"downwards":["Dot product, angle, and length"]}
{"upwards":["Euclidean plane","In linear algebra","Dot product, angle, and length"],"downwards":[]}
{"upwards":["Euclidean plane","In calculus"],"downwards":["Gradient","Line integrals and double integrals","Fundamental theorem of line integrals","Green's theorem"]}
{"upwards":["Euclidean plane","In calculus","Gradient"],"downwards":[]}
{"upwards":["Euclidean plane","In calculus","Line integrals and double integrals"],"downwards":[]}
{"upwards":["Euclidean plane","In calculus","Fundamental theorem of line integrals"],"downwards":[]}
{"upwards":["Euclidean plane","In calculus","Green's theorem"],"downwards":[]}
{"upwards":["Euclidean plane","In topology"],"downwards":[]}
{"upwards":["Euclidean plane","In graph theory"],"downwards":[]}
{"upwards":["Union (set theory)"],"downwards":["Union of two sets","Algebraic properties","Finite unions","Arbitrary unions","Notation encoding"]}
{"upwards":["Union (set theory)","Union of two sets"],"downwards":[]}
{"upwards":["Union (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Union (set theory)","Finite unions"],"downwards":[]}
{"upwards":["Union (set theory)","Arbitrary unions"],"downwards":["Notations"]}
{"upwards":["Union (set theory)","Arbitrary unions","Notations"],"downwards":[]}
{"upwards":["Union (set theory)","Notation encoding"],"downwards":[]}
{"upwards":["Intersection (set theory)"],"downwards":["Notation and terminology","Definition","Algebraic properties","Arbitrary intersections","Nullary intersection"]}
{"upwards":["Intersection (set theory)","Notation and terminology"],"downwards":[]}
{"upwards":["Intersection (set theory)","Definition"],"downwards":["Intersecting and disjoint sets"]}
{"upwards":["Intersection (set theory)","Definition","Intersecting and disjoint sets"],"downwards":[]}
{"upwards":["Intersection (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Intersection (set theory)","Arbitrary intersections"],"downwards":[]}
{"upwards":["Intersection (set theory)","Nullary intersection"],"downwards":[]}
{"upwards":["Conjunct"],"downwards":["Semantic functions"]}
{"upwards":["Conjunct","Semantic functions"],"downwards":[]}
{"upwards":["If and only if"],"downwards":["Definition","Usage","Distinction from \"if\" and \"only if\"","In terms of Euler diagrams","More general usage"]}
{"upwards":["If and only if","Definition"],"downwards":[]}
{"upwards":["If and only if","Usage"],"downwards":["Notation","Proofs","Origin of iff and pronunciation","Usage in definitions"]}
{"upwards":["If and only if","Usage","Notation"],"downwards":[]}
{"upwards":["If and only if","Usage","Proofs"],"downwards":[]}
{"upwards":["If and only if","Usage","Origin of iff and pronunciation"],"downwards":[]}
{"upwards":["If and only if","Usage","Usage in definitions"],"downwards":[]}
{"upwards":["If and only if","Distinction from \"if\" and \"only if\""],"downwards":[]}
{"upwards":["If and only if","In terms of Euler diagrams"],"downwards":[]}
{"upwards":["If and only if","More general usage"],"downwards":[]}
{"upwards":["Image (mathematics)"],"downwards":["Definition","Inverse image","Notation for image and inverse image","Examples","Properties"]}
{"upwards":["Image (mathematics)","Definition"],"downwards":["Image of an element","Image of a subset","Image of a function","Generalization to binary relations"]}
{"upwards":["Image (mathematics)","Definition","Image of an element"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a subset"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a function"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Generalization to binary relations"],"downwards":[]}
{"upwards":["Image (mathematics)","Inverse image"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image"],"downwards":["Arrow notation","Star notation","Other terminology"]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Arrow notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Star notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Other terminology"],"downwards":[]}
{"upwards":["Image (mathematics)","Examples"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties"],"downwards":["General","Multiple functions","Multiple subsets of domain or codomain"]}
{"upwards":["Image (mathematics)","Properties","General"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple functions"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple subsets of domain or codomain"],"downwards":[]}
{"upwards":["Image (mathematics)"],"downwards":["Definition","Inverse image","Notation for image and inverse image","Examples","Properties"]}
{"upwards":["Image (mathematics)","Definition"],"downwards":["Image of an element","Image of a subset","Image of a function","Generalization to binary relations"]}
{"upwards":["Image (mathematics)","Definition","Image of an element"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a subset"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a function"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Generalization to binary relations"],"downwards":[]}
{"upwards":["Image (mathematics)","Inverse image"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image"],"downwards":["Arrow notation","Star notation","Other terminology"]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Arrow notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Star notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Other terminology"],"downwards":[]}
{"upwards":["Image (mathematics)","Examples"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties"],"downwards":["General","Multiple functions","Multiple subsets of domain or codomain"]}
{"upwards":["Image (mathematics)","Properties","General"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple functions"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple subsets of domain or codomain"],"downwards":[]}
{"upwards":["Norbert Wiener"],"downwards":["Biography","Work","In popular culture","Publications"]}
{"upwards":["Norbert Wiener","Biography"],"downwards":["Youth","Harvard and World War I","After the war","During and after World War II","Personal life","Awards and honors","Doctoral students"]}
{"upwards":["Norbert Wiener","Biography","Youth"],"downwards":[]}
{"upwards":["Norbert Wiener","Biography","Harvard and World War I"],"downwards":[]}
{"upwards":["Norbert Wiener","Biography","After the war"],"downwards":[]}
{"upwards":["Norbert Wiener","Biography","During and after World War II"],"downwards":[]}
{"upwards":["Norbert Wiener","Biography","Personal life"],"downwards":[]}
{"upwards":["Norbert Wiener","Biography","Awards and honors"],"downwards":[]}
{"upwards":["Norbert Wiener","Biography","Doctoral students"],"downwards":[]}
{"upwards":["Norbert Wiener","Work"],"downwards":["Wiener equation","Wiener filter","In mathematics"]}
{"upwards":["Norbert Wiener","Work","Wiener equation"],"downwards":[]}
{"upwards":["Norbert Wiener","Work","Wiener filter"],"downwards":[]}
{"upwards":["Norbert Wiener","Work","In mathematics"],"downwards":[]}
{"upwards":["Norbert Wiener","In popular culture"],"downwards":[]}
{"upwards":["Norbert Wiener","Publications"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Nicolas Bourbaki"],"downwards":["Background","The Bourbaki collective","Working method","Works","Membership","Influence and criticism"]}
{"upwards":["Nicolas Bourbaki","Background"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective"],"downwards":["Founding","World War II","Postwar until the present"]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective","Founding"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective","World War II"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective","Postwar until the present"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Working method"],"downwards":["Humor"]}
{"upwards":["Nicolas Bourbaki","Working method","Humor"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works"],"downwards":["\u00c9l\u00e9ments de math\u00e9matique","S\u00e9minaire Bourbaki","Articles","La Tribu"]}
{"upwards":["Nicolas Bourbaki","Works","\u00c9l\u00e9ments de math\u00e9matique"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works","S\u00e9minaire Bourbaki"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works","Articles"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works","La Tribu"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Membership"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Influence and criticism"],"downwards":["Influence","Praise","Criticism"]}
{"upwards":["Nicolas Bourbaki","Influence and criticism","Influence"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Influence and criticism","Praise"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Influence and criticism","Criticism"],"downwards":[]}
{"upwards":["Commutative diagram"],"downwards":["Description","Examples","Diagram chasing","In higher category theory","Diagrams as functors"]}
{"upwards":["Commutative diagram","Description"],"downwards":["Arrow symbols","Verifying commutativity"]}
{"upwards":["Commutative diagram","Description","Arrow symbols"],"downwards":[]}
{"upwards":["Commutative diagram","Description","Verifying commutativity"],"downwards":[]}
{"upwards":["Commutative diagram","Examples"],"downwards":["Example 1","Example 2"]}
{"upwards":["Commutative diagram","Examples","Example 1"],"downwards":[]}
{"upwards":["Commutative diagram","Examples","Example 2"],"downwards":[]}
{"upwards":["Commutative diagram","Diagram chasing"],"downwards":[]}
{"upwards":["Commutative diagram","In higher category theory"],"downwards":[]}
{"upwards":["Commutative diagram","Diagrams as functors"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski"],"downwards":["Biography and studies","Legacy","Research","Publications"]}
{"upwards":["Kazimierz Kuratowski","Biography and studies"],"downwards":["Doctoral thesis","Academic career until World War II","During and after the war"]}
{"upwards":["Kazimierz Kuratowski","Biography and studies","Doctoral thesis"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Biography and studies","Academic career until World War II"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Biography and studies","During and after the war"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Legacy"],"downwards":["Kazimierz Kuratowski Prize"]}
{"upwards":["Kazimierz Kuratowski","Legacy","Kazimierz Kuratowski Prize"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Research"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Publications"],"downwards":[]}
{"upwards":["Number line"],"downwards":["History","Drawing the number line","Comparing numbers","Portions of the number line","Extensions of the concept","Advanced concepts","In real algebras"]}
{"upwards":["Number line","History"],"downwards":[]}
{"upwards":["Number line","Drawing the number line"],"downwards":[]}
{"upwards":["Number line","Comparing numbers"],"downwards":[]}
{"upwards":["Number line","Portions of the number line"],"downwards":[]}
{"upwards":["Number line","Extensions of the concept"],"downwards":["Logarithmic scale","Combining number lines"]}
{"upwards":["Number line","Extensions of the concept","Logarithmic scale"],"downwards":[]}
{"upwards":["Number line","Extensions of the concept","Combining number lines"],"downwards":[]}
{"upwards":["Number line","Advanced concepts"],"downwards":["As a linear continuum","As a metric space","As a topological space","As a vector space","As a measure space"]}
{"upwards":["Number line","Advanced concepts","As a linear continuum"],"downwards":[]}
{"upwards":["Number line","Advanced concepts","As a metric space"],"downwards":[]}
{"upwards":["Number line","Advanced concepts","As a topological space"],"downwards":[]}
{"upwards":["Number line","Advanced concepts","As a vector space"],"downwards":[]}
{"upwards":["Number line","Advanced concepts","As a measure space"],"downwards":[]}
{"upwards":["Number line","In real algebras"],"downwards":[]}
{"upwards":["J. Barkley Rosser"],"downwards":["Selected publications","References","External links"]}
{"upwards":["J. Barkley Rosser","Selected publications"],"downwards":[]}
{"upwards":["J. Barkley Rosser","References"],"downwards":[]}
{"upwards":["J. Barkley Rosser","External links"],"downwards":[]}
{"upwards":["Interval (mathematics)"],"downwards":["Terminology","Notations for intervals","Classification of intervals","Properties of intervals","Dyadic intervals","Generalizations","Topological algebra"]}
{"upwards":["Interval (mathematics)","Terminology"],"downwards":["Note on conflicting terminology"]}
{"upwards":["Interval (mathematics)","Terminology","Note on conflicting terminology"],"downwards":[]}
{"upwards":["Interval (mathematics)","Notations for intervals"],"downwards":["Including or excluding endpoints","Infinite endpoints","Integer intervals"]}
{"upwards":["Interval (mathematics)","Notations for intervals","Including or excluding endpoints"],"downwards":[]}
{"upwards":["Interval (mathematics)","Notations for intervals","Infinite endpoints"],"downwards":[]}
{"upwards":["Interval (mathematics)","Notations for intervals","Integer intervals"],"downwards":[]}
{"upwards":["Interval (mathematics)","Classification of intervals"],"downwards":[]}
{"upwards":["Interval (mathematics)","Properties of intervals"],"downwards":[]}
{"upwards":["Interval (mathematics)","Dyadic intervals"],"downwards":[]}
{"upwards":["Interval (mathematics)","Generalizations"],"downwards":["Multi-dimensional intervals","Complex intervals"]}
{"upwards":["Interval (mathematics)","Generalizations","Multi-dimensional intervals"],"downwards":[]}
{"upwards":["Interval (mathematics)","Generalizations","Complex intervals"],"downwards":[]}
{"upwards":["Interval (mathematics)","Topological algebra"],"downwards":[]}
{"upwards":["Felix Hausdorff"],"downwards":["Life","Work and reception","Hausdorff as name-giver","Writings","Collected works","References"]}
{"upwards":["Felix Hausdorff","Life"],"downwards":["Childhood and youth","Degree, doctorate and Habilitation","Lecturer in Leipzig","First professorship","Under the Nazi dictatorship and suicide"]}
{"upwards":["Felix Hausdorff","Life","Childhood and youth"],"downwards":[]}
{"upwards":["Felix Hausdorff","Life","Degree, doctorate and Habilitation"],"downwards":[]}
{"upwards":["Felix Hausdorff","Life","Lecturer in Leipzig"],"downwards":[]}
{"upwards":["Felix Hausdorff","Life","First professorship"],"downwards":[]}
{"upwards":["Felix Hausdorff","Life","Under the Nazi dictatorship and suicide"],"downwards":[]}
{"upwards":["Felix Hausdorff","Work and reception"],"downwards":["Hausdorff as philosopher and writer (Paul Mongr\u00e9)","Theory of ordered sets","The \"Magnum Opus\": \"Principles of set theory\"","Descriptive set theory, measure theory and analysis","His last works"]}
{"upwards":["Felix Hausdorff","Work and reception","Hausdorff as philosopher and writer (Paul Mongr\u00e9)"],"downwards":[]}
{"upwards":["Felix Hausdorff","Work and reception","Theory of ordered sets"],"downwards":[]}
{"upwards":["Felix Hausdorff","Work and reception","The \"Magnum Opus\": \"Principles of set theory\""],"downwards":[]}
{"upwards":["Felix Hausdorff","Work and reception","Descriptive set theory, measure theory and analysis"],"downwards":[]}
{"upwards":["Felix Hausdorff","Work and reception","His last works"],"downwards":[]}
{"upwards":["Felix Hausdorff","Hausdorff as name-giver"],"downwards":[]}
{"upwards":["Felix Hausdorff","Writings"],"downwards":["As Paul Mongr\u00e9","As Felix Hausdorff"]}
{"upwards":["Felix Hausdorff","Writings","As Paul Mongr\u00e9"],"downwards":[]}
{"upwards":["Felix Hausdorff","Writings","As Felix Hausdorff"],"downwards":[]}
{"upwards":["Felix Hausdorff","Collected works"],"downwards":[]}
{"upwards":["Felix Hausdorff","References"],"downwards":[]}
{"upwards":["Anthony Morse"],"downwards":["Career","References","External links"]}
{"upwards":["Anthony Morse","Career"],"downwards":[]}
{"upwards":["Anthony Morse","References"],"downwards":[]}
{"upwards":["Anthony Morse","External links"],"downwards":[]}
{"upwards":["Bracket"],"downwards":["History","Typography","Round brackets or parentheses","Square brackets or brackets","Curly brackets or braces","Angle brackets or chevrons","Other brackets","Unicode"]}
{"upwards":["Bracket","History"],"downwards":[]}
{"upwards":["Bracket","Typography"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses"],"downwards":["Uses of ( )"]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )"],"downwards":["Enumerations","Accounting","Parentheses in mathematics","Parentheses in programming languages","Taxonomy","Chemistry and physics","Pop songs"]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Enumerations"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Accounting"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Parentheses in mathematics"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Parentheses in programming languages"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Taxonomy"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Chemistry and physics"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Pop songs"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets"],"downwards":["Uses of [ ]"]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]"],"downwards":["Proofreading","Law","Square brackets in mathematics","Chemistry","Square brackets in programming languages"]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Proofreading"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Law"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Square brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Chemistry"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Square brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces"],"downwards":["Uses of { }"]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }"],"downwards":["Music","Curly brackets in programming languages","Curly brackets in mathematics"]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Music"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Curly brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Curly brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons"],"downwards":["Shape","Uses of \u27e8 \u27e9"]}
{"upwards":["Bracket","Angle brackets or chevrons","Shape"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9"],"downwards":["Angle brackets in mathematics","Physics and mechanics","Angle brackets in programming languages"]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Angle brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Physics and mechanics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Angle brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Other brackets"],"downwards":["Lenticular brackets\u3010\u3011","Floor \u230a \u230b and ceiling \u2308 \u2309 corner brackets","Quine corners \u231c\u231d and half brackets \u2e24 \u2e25 or \u2e22 \u2e23","Double brackets \u27e6 \u27e7","Brackets with quills \u2045 \u2046"]}
{"upwards":["Bracket","Other brackets","Lenticular brackets\u3010\u3011"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Floor \u230a \u230b and ceiling \u2308 \u2309 corner brackets"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Quine corners \u231c\u231d and half brackets \u2e24 \u2e25 or \u2e22 \u2e23"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Double brackets \u27e6 \u27e7"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Brackets with quills \u2045 \u2046"],"downwards":[]}
{"upwards":["Bracket","Unicode"],"downwards":[]}
{"upwards":["Unordered pair"],"downwards":["Notes","References"]}
{"upwards":["Unordered pair","Notes"],"downwards":[]}
{"upwards":["Unordered pair","References"],"downwards":[]}
{"upwards":["Iterated binary operation"],"downwards":["Definition","Non-associative binary operation","Notation"]}
{"upwards":["Iterated binary operation","Definition"],"downwards":[]}
{"upwards":["Iterated binary operation","Non-associative binary operation"],"downwards":[]}
{"upwards":["Iterated binary operation","Notation"],"downwards":[]}
{"upwards":["Stewart Shapiro"],"downwards":["Education and career","Publications"]}
{"upwards":["Stewart Shapiro","Education and career"],"downwards":[]}
{"upwards":["Stewart Shapiro","Publications"],"downwards":["Books","Editorships"]}
{"upwards":["Stewart Shapiro","Publications","Books"],"downwards":[]}
{"upwards":["Stewart Shapiro","Publications","Editorships"],"downwards":[]}
{"upwards":["Tuple"],"downwards":["Etymology","Properties","Definitions","n-tuples of m-sets","Type theory"]}
{"upwards":["Tuple","Etymology"],"downwards":["Names for tuples of specific lengths"]}
{"upwards":["Tuple","Etymology","Names for tuples of specific lengths"],"downwards":[]}
{"upwards":["Tuple","Properties"],"downwards":[]}
{"upwards":["Tuple","Definitions"],"downwards":["Tuples as functions","Tuples as nested ordered pairs","Tuples as nested sets"]}
{"upwards":["Tuple","Definitions","Tuples as functions"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested ordered pairs"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested sets"],"downwards":[]}
{"upwards":["Tuple","n-tuples of m-sets"],"downwards":[]}
{"upwards":["Tuple","Type theory"],"downwards":[]}
{"upwards":["Ordinal number"],"downwards":["Ordinals extend the natural numbers","Definitions","Transfinite sequence","Transfinite induction","Arithmetic of ordinals","Ordinals and cardinals","Some \"large\" countable ordinals","Topology and ordinals","History"]}
{"upwards":["Ordinal number","Ordinals extend the natural numbers"],"downwards":[]}
{"upwards":["Ordinal number","Definitions"],"downwards":["Well-ordered sets","Definition of an ordinal as an equivalence class","Von Neumann definition of ordinals","Other definitions"]}
{"upwards":["Ordinal number","Definitions","Well-ordered sets"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Definition of an ordinal as an equivalence class"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Von Neumann definition of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Other definitions"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite sequence"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction"],"downwards":["Transfinite recursion","Successor and limit ordinals","Indexing classes of ordinals","Closed unbounded sets and classes"]}
{"upwards":["Ordinal number","Transfinite induction","Transfinite recursion"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Successor and limit ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Indexing classes of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Closed unbounded sets and classes"],"downwards":[]}
{"upwards":["Ordinal number","Arithmetic of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals"],"downwards":["Initial ordinal of a cardinal","Cofinality"]}
{"upwards":["Ordinal number","Ordinals and cardinals","Initial ordinal of a cardinal"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals","Cofinality"],"downwards":[]}
{"upwards":["Ordinal number","Some \"large\" countable ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Topology and ordinals"],"downwards":[]}
{"upwards":["Ordinal number","History"],"downwards":[]}
{"upwards":["Willard Van Orman Quine"],"downwards":["Biography","Work","In popular culture","Bibliography","Filmography"]}
{"upwards":["Willard Van Orman Quine","Biography"],"downwards":["Education","World War II","Personal","Harvard"]}
{"upwards":["Willard Van Orman Quine","Biography","Education"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","World War II"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","Personal"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","Harvard"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work"],"downwards":["Logic","Set theory","Metaphysics","Epistemology"]}
{"upwards":["Willard Van Orman Quine","Work","Logic"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Set theory"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics"],"downwards":["Rejection of the analytic\u2013synthetic distinction","Confirmation holism and ontological relativity","Existence and its contrary","Ontological commitment","Indispensability argument for mathematical realism"]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Rejection of the analytic\u2013synthetic distinction"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Confirmation holism and ontological relativity"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Existence and its contrary"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Ontological commitment"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Indispensability argument for mathematical realism"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Epistemology"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","In popular culture"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Bibliography"],"downwards":["Selected books","Important articles"]}
{"upwards":["Willard Van Orman Quine","Bibliography","Selected books"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Bibliography","Important articles"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Filmography"],"downwards":[]}
{"upwards":["Product (category theory)"],"downwards":["Definition","Examples","Discussion","Distributivity"]}
{"upwards":["Product (category theory)","Definition"],"downwards":["Product of two objects","Product of an arbitrary family","Equational definition","As a limit","Universal property"]}
{"upwards":["Product (category theory)","Definition","Product of two objects"],"downwards":[]}
{"upwards":["Product (category theory)","Definition","Product of an arbitrary family"],"downwards":[]}
{"upwards":["Product (category theory)","Definition","Equational definition"],"downwards":[]}
{"upwards":["Product (category theory)","Definition","As a limit"],"downwards":[]}
{"upwards":["Product (category theory)","Definition","Universal property"],"downwards":[]}
{"upwards":["Product (category theory)","Examples"],"downwards":[]}
{"upwards":["Product (category theory)","Discussion"],"downwards":[]}
{"upwards":["Product (category theory)","Distributivity"],"downwards":[]}
{"upwards":["Wikipedia, the free encyclopedia"],"downwards":[]}
{"upwards":["Singleton (mathematics)"],"downwards":["Properties","In category theory","Definition by indicator functions","Definition in Principia Mathematica"]}
{"upwards":["Singleton (mathematics)","Properties"],"downwards":[]}
{"upwards":["Singleton (mathematics)","In category theory"],"downwards":[]}
{"upwards":["Singleton (mathematics)","Definition by indicator functions"],"downwards":[]}
{"upwards":["Singleton (mathematics)","Definition in Principia Mathematica"],"downwards":[]}
{"upwards":["Paradoxes of the Infinite"],"downwards":["References"]}
{"upwards":["Paradoxes of the Infinite","References"],"downwards":[]}
{"upwards":["Set (mathematics)"],"downwards":["History","How sets are defined and set notation","Membership","The empty set","Singleton sets","Subsets","Euler and Venn diagrams","Special sets of numbers in mathematics","Functions","Cardinality","Power sets","Partitions","Basic operations","Applications","Principle of inclusion and exclusion"]}
{"upwards":["Set (mathematics)","History"],"downwards":["Naive set theory","Axiomatic set theory"]}
{"upwards":["Set (mathematics)","History","Naive set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","History","Axiomatic set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation"],"downwards":["Roster notation","Semantic definition","Set-builder notation","Classifying methods of definition"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation"],"downwards":["Infinite sets in roster notation"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation","Infinite sets in roster notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Semantic definition"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Set-builder notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Classifying methods of definition"],"downwards":[]}
{"upwards":["Set (mathematics)","Membership"],"downwards":[]}
{"upwards":["Set (mathematics)","The empty set"],"downwards":[]}
{"upwards":["Set (mathematics)","Singleton sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Subsets"],"downwards":[]}
{"upwards":["Set (mathematics)","Euler and Venn diagrams"],"downwards":[]}
{"upwards":["Set (mathematics)","Special sets of numbers in mathematics"],"downwards":[]}
{"upwards":["Set (mathematics)","Functions"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality"],"downwards":["Infinite sets and infinite cardinality","The continuum hypothesis"]}
{"upwards":["Set (mathematics)","Cardinality","Infinite sets and infinite cardinality"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality","The continuum hypothesis"],"downwards":[]}
{"upwards":["Set (mathematics)","Power sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Partitions"],"downwards":[]}
{"upwards":["Set (mathematics)","Basic operations"],"downwards":[]}
{"upwards":["Set (mathematics)","Applications"],"downwards":[]}
{"upwards":["Set (mathematics)","Principle of inclusion and exclusion"],"downwards":[]}
{"upwards":["Bernard Bolzano"],"downwards":["Family","Career","Mathematical work","Philosophical work","Philosophical legacy","Writings"]}
{"upwards":["Bernard Bolzano","Family"],"downwards":[]}
{"upwards":["Bernard Bolzano","Career"],"downwards":[]}
{"upwards":["Bernard Bolzano","Mathematical work"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work"],"downwards":["Wissenschaftslehre (Theory of Science)"]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)"],"downwards":["Introduction to Wissenschaftslehre","Metaphysics","Satz an Sich (proposition in itself)","Ideas and objects","Sensation and simple ideas","Logic","Truth","Judgments and cognitions"]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Introduction to Wissenschaftslehre"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Metaphysics"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Satz an Sich (proposition in itself)"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Ideas and objects"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Sensation and simple ideas"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Logic"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Truth"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical work","Wissenschaftslehre (Theory of Science)","Judgments and cognitions"],"downwards":[]}
{"upwards":["Bernard Bolzano","Philosophical legacy"],"downwards":[]}
{"upwards":["Bernard Bolzano","Writings"],"downwards":["Translations and compilations"]}
{"upwards":["Bernard Bolzano","Writings","Translations and compilations"],"downwards":[]}
{"upwards":["Element (mathematics)"],"downwards":["Sets","Notation and terminology","Cardinality of sets","Examples","Formal relation"]}
{"upwards":["Element (mathematics)","Sets"],"downwards":[]}
{"upwards":["Element (mathematics)","Notation and terminology"],"downwards":[]}
{"upwards":["Element (mathematics)","Cardinality of sets"],"downwards":[]}
{"upwards":["Element (mathematics)","Examples"],"downwards":[]}
{"upwards":["Element (mathematics)","Formal relation"],"downwards":[]}
{"upwards":["Naive set theory"],"downwards":["Method","Sets, membership and equality","Specifying sets","Subsets","Universal sets and absolute complements","Unions, intersections, and relative complements","Ordered pairs and Cartesian products","Some important sets","Paradoxes in early set theory"]}
{"upwards":["Naive set theory","Method"],"downwards":["Paradoxes","Cantor's theory","Axiomatic theories","Consistency","Utility"]}
{"upwards":["Naive set theory","Method","Paradoxes"],"downwards":[]}
{"upwards":["Naive set theory","Method","Cantor's theory"],"downwards":[]}
{"upwards":["Naive set theory","Method","Axiomatic theories"],"downwards":[]}
{"upwards":["Naive set theory","Method","Consistency"],"downwards":[]}
{"upwards":["Naive set theory","Method","Utility"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality"],"downwards":["Note on consistency","Membership","Equality","Empty set"]}
{"upwards":["Naive set theory","Sets, membership and equality","Note on consistency"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Membership"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Equality"],"downwards":[]}
{"upwards":["Naive set theory","Sets, membership and equality","Empty set"],"downwards":[]}
{"upwards":["Naive set theory","Specifying sets"],"downwards":[]}
{"upwards":["Naive set theory","Subsets"],"downwards":[]}
{"upwards":["Naive set theory","Universal sets and absolute complements"],"downwards":[]}
{"upwards":["Naive set theory","Unions, intersections, and relative complements"],"downwards":[]}
{"upwards":["Naive set theory","Ordered pairs and Cartesian products"],"downwards":[]}
{"upwards":["Naive set theory","Some important sets"],"downwards":[]}
{"upwards":["Naive set theory","Paradoxes in early set theory"],"downwards":[]}
{"upwards":["Extensional and intensional definitions"],"downwards":["Intensional definition","Extensional definition","History"]}
{"upwards":["Extensional and intensional definitions","Intensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","Extensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","History"],"downwards":[]}
{"upwards":["Enumerative definition"],"downwards":[]}
{"upwards":["Cartesian product"],"downwards":["Set-theoretic definition","Examples","Most common implementation (set theory)","Cartesian products of several sets","Other forms","Definitions outside set theory"]}
{"upwards":["Cartesian product","Set-theoretic definition"],"downwards":[]}
{"upwards":["Cartesian product","Examples"],"downwards":["A deck of cards","A two-dimensional coordinate system"]}
{"upwards":["Cartesian product","Examples","A deck of cards"],"downwards":[]}
{"upwards":["Cartesian product","Examples","A two-dimensional coordinate system"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)"],"downwards":["Non-commutativity and non-associativity","Intersections, unions, and subsets","Cardinality"]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Non-commutativity and non-associativity"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Intersections, unions, and subsets"],"downwards":[]}
{"upwards":["Cartesian product","Most common implementation (set theory)","Cardinality"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets"],"downwards":["n-ary Cartesian product","n-ary Cartesian power","Infinite Cartesian products"]}
{"upwards":["Cartesian product","Cartesian products of several sets","n-ary Cartesian product"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets","n-ary Cartesian power"],"downwards":[]}
{"upwards":["Cartesian product","Cartesian products of several sets","Infinite Cartesian products"],"downwards":[]}
{"upwards":["Cartesian product","Other forms"],"downwards":["Abbreviated form","Cartesian product of functions","Cylinder"]}
{"upwards":["Cartesian product","Other forms","Abbreviated form"],"downwards":[]}
{"upwards":["Cartesian product","Other forms","Cartesian product of functions"],"downwards":[]}
{"upwards":["Cartesian product","Other forms","Cylinder"],"downwards":[]}
{"upwards":["Cartesian product","Definitions outside set theory"],"downwards":["Category theory","Graph theory"]}
{"upwards":["Cartesian product","Definitions outside set theory","Category theory"],"downwards":[]}
{"upwards":["Cartesian product","Definitions outside set theory","Graph theory"],"downwards":[]}
{"upwards":["Empty set"],"downwards":["Notation","Properties","In other areas of mathematics","Questioned existence"]}
{"upwards":["Empty set","Notation"],"downwards":[]}
{"upwards":["Empty set","Properties"],"downwards":["Operations on the empty set"]}
{"upwards":["Empty set","Properties","Operations on the empty set"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics"],"downwards":["Extended real numbers","Topology","Category theory","Set theory"]}
{"upwards":["Empty set","In other areas of mathematics","Extended real numbers"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Topology"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Category theory"],"downwards":[]}
{"upwards":["Empty set","In other areas of mathematics","Set theory"],"downwards":[]}
{"upwards":["Empty set","Questioned existence"],"downwards":["Axiomatic set theory","Philosophical issues"]}
{"upwards":["Empty set","Questioned existence","Axiomatic set theory"],"downwards":[]}
{"upwards":["Empty set","Questioned existence","Philosophical issues"],"downwards":[]}
{"upwards":["Russell's paradox"],"downwards":["Informal presentation","Formal presentation","Philosophical implications","Set-theoretic responses","History","Applied versions","Applications and related topics","Related paradoxes"]}
{"upwards":["Russell's paradox","Informal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Formal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Philosophical implications"],"downwards":[]}
{"upwards":["Russell's paradox","Set-theoretic responses"],"downwards":[]}
{"upwards":["Russell's paradox","History"],"downwards":[]}
{"upwards":["Russell's paradox","Applied versions"],"downwards":[]}
{"upwards":["Russell's paradox","Applications and related topics"],"downwards":["Russell-like paradoxes"]}
{"upwards":["Russell's paradox","Applications and related topics","Russell-like paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox","Related paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox"],"downwards":["Informal presentation","Formal presentation","Philosophical implications","Set-theoretic responses","History","Applied versions","Applications and related topics","Related paradoxes"]}
{"upwards":["Russell's paradox","Informal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Formal presentation"],"downwards":[]}
{"upwards":["Russell's paradox","Philosophical implications"],"downwards":[]}
{"upwards":["Russell's paradox","Set-theoretic responses"],"downwards":[]}
{"upwards":["Russell's paradox","History"],"downwards":[]}
{"upwards":["Russell's paradox","Applied versions"],"downwards":[]}
{"upwards":["Russell's paradox","Applications and related topics"],"downwards":["Russell-like paradoxes"]}
{"upwards":["Russell's paradox","Applications and related topics","Russell-like paradoxes"],"downwards":[]}
{"upwards":["Russell's paradox","Related paradoxes"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Ellipsis"],"downwards":["Background","In writing","In different languages","Usage in computer system menus","In mathematical notation","Computer science","On Internet chat rooms and in text messaging","Computer representations"]}
{"upwards":["Ellipsis","Background"],"downwards":[]}
{"upwards":["Ellipsis","In writing"],"downwards":[]}
{"upwards":["Ellipsis","In different languages"],"downwards":["In English","In Polish","In Russian","In Japanese","In Chinese","In Spanish","In French","In German","In Italian"]}
{"upwards":["Ellipsis","In different languages","In English"],"downwards":["American English","British English"]}
{"upwards":["Ellipsis","In different languages","In English","American English"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In English","British English"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Polish"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Russian"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Japanese"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Chinese"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Spanish"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In French"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In German"],"downwards":[]}
{"upwards":["Ellipsis","In different languages","In Italian"],"downwards":[]}
{"upwards":["Ellipsis","Usage in computer system menus"],"downwards":[]}
{"upwards":["Ellipsis","In mathematical notation"],"downwards":[]}
{"upwards":["Ellipsis","Computer science"],"downwards":["Programming languages","HTML and CSS"]}
{"upwards":["Ellipsis","Computer science","Programming languages"],"downwards":[]}
{"upwards":["Ellipsis","Computer science","HTML and CSS"],"downwards":[]}
{"upwards":["Ellipsis","On Internet chat rooms and in text messaging"],"downwards":[]}
{"upwards":["Ellipsis","Computer representations"],"downwards":["Input"]}
{"upwards":["Ellipsis","Computer representations","Input"],"downwards":[]}
{"upwards":["Euler diagram"],"downwards":["History","Relation between Euler and Venn diagrams","Gallery"]}
{"upwards":["Euler diagram","History"],"downwards":[]}
{"upwards":["Euler diagram","Relation between Euler and Venn diagrams"],"downwards":["Example: Euler- to Venn-diagram and Karnaugh map"]}
{"upwards":["Euler diagram","Relation between Euler and Venn diagrams","Example: Euler- to Venn-diagram and Karnaugh map"],"downwards":[]}
{"upwards":["Euler diagram","Gallery"],"downwards":[]}
{"upwards":["Well-defined expression"],"downwards":["Example","\"Definition\" as anticipation of definition","Independence of representative","Well-defined notation","Other uses of the term"]}
{"upwards":["Well-defined expression","Example"],"downwards":[]}
{"upwards":["Well-defined expression","\"Definition\" as anticipation of definition"],"downwards":[]}
{"upwards":["Well-defined expression","Independence of representative"],"downwards":["Functions with one argument","Operations"]}
{"upwards":["Well-defined expression","Independence of representative","Functions with one argument"],"downwards":[]}
{"upwards":["Well-defined expression","Independence of representative","Operations"],"downwards":[]}
{"upwards":["Well-defined expression","Well-defined notation"],"downwards":[]}
{"upwards":["Well-defined expression","Other uses of the term"],"downwards":[]}
{"upwards":["Sequence"],"downwards":["Examples and notation","Formal definition and basic properties","Limits and convergence","Series","Use in other fields of mathematics"]}
{"upwards":["Sequence","Examples and notation"],"downwards":["Examples","Indexing","Defining a sequence by recursion"]}
{"upwards":["Sequence","Examples and notation","Examples"],"downwards":[]}
{"upwards":["Sequence","Examples and notation","Indexing"],"downwards":[]}
{"upwards":["Sequence","Examples and notation","Defining a sequence by recursion"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties"],"downwards":["Definition","Finite and infinite","Increasing and decreasing","Bounded","Subsequences","Other types of sequences"]}
{"upwards":["Sequence","Formal definition and basic properties","Definition"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Finite and infinite"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Increasing and decreasing"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Bounded"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Subsequences"],"downwards":[]}
{"upwards":["Sequence","Formal definition and basic properties","Other types of sequences"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence"],"downwards":["Formal definition of convergence","Applications and important results","Cauchy sequences","Infinite limits"]}
{"upwards":["Sequence","Limits and convergence","Formal definition of convergence"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Applications and important results"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Cauchy sequences"],"downwards":[]}
{"upwards":["Sequence","Limits and convergence","Infinite limits"],"downwards":[]}
{"upwards":["Sequence","Series"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics"],"downwards":["Topology","Analysis","Linear algebra","Abstract algebra","Set theory","Computing","Streams"]}
{"upwards":["Sequence","Use in other fields of mathematics","Topology"],"downwards":["Product topology"]}
{"upwards":["Sequence","Use in other fields of mathematics","Topology","Product topology"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Analysis"],"downwards":["Sequence spaces"]}
{"upwards":["Sequence","Use in other fields of mathematics","Analysis","Sequence spaces"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Linear algebra"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra"],"downwards":["Free monoid","Exact sequences","Spectral sequences"]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Free monoid"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Exact sequences"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Abstract algebra","Spectral sequences"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Set theory"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Computing"],"downwards":[]}
{"upwards":["Sequence","Use in other fields of mathematics","Streams"],"downwards":[]}
{"upwards":["Extensional and intensional definitions"],"downwards":["Intensional definition","Extensional definition","History"]}
{"upwards":["Extensional and intensional definitions","Intensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","Extensional definition"],"downwards":[]}
{"upwards":["Extensional and intensional definitions","History"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Primitive notion"],"downwards":["Details","Examples","Russell's primitives"]}
{"upwards":["Primitive notion","Details"],"downwards":[]}
{"upwards":["Primitive notion","Examples"],"downwards":[]}
{"upwards":["Primitive notion","Russell's primitives"],"downwards":[]}
{"upwards":["Letter case"],"downwards":["Terminology","Typographical considerations","Bicameral script","Stylistic or specialised usage","Case folding and case conversion","History"]}
{"upwards":["Letter case","Terminology"],"downwards":[]}
{"upwards":["Letter case","Typographical considerations"],"downwards":[]}
{"upwards":["Letter case","Bicameral script"],"downwards":["Capitalisation","Exceptional letters and digraphs","Related features"]}
{"upwards":["Letter case","Bicameral script","Capitalisation"],"downwards":[]}
{"upwards":["Letter case","Bicameral script","Exceptional letters and digraphs"],"downwards":[]}
{"upwards":["Letter case","Bicameral script","Related features"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage"],"downwards":["Headings and publication titles","Multi-word proper nouns","Unit symbols and prefixes in the metric system","Use within programming languages","Studly caps"]}
{"upwards":["Letter case","Stylistic or specialised usage","Headings and publication titles"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Multi-word proper nouns"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Unit symbols and prefixes in the metric system"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages"],"downwards":["Camel case","Snake case","Kebab case"]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages","Camel case"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages","Snake case"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Use within programming languages","Kebab case"],"downwards":[]}
{"upwards":["Letter case","Stylistic or specialised usage","Studly caps"],"downwards":[]}
{"upwards":["Letter case","Case folding and case conversion"],"downwards":["Unicode case folding and script identification","Methods in word processing","Methods in programming"]}
{"upwards":["Letter case","Case folding and case conversion","Unicode case folding and script identification"],"downwards":[]}
{"upwards":["Letter case","Case folding and case conversion","Methods in word processing"],"downwards":[]}
{"upwards":["Letter case","Case folding and case conversion","Methods in programming"],"downwards":[]}
{"upwards":["Letter case","History"],"downwards":["Type cases"]}
{"upwards":["Letter case","History","Type cases"],"downwards":[]}
{"upwards":["Extensionality"],"downwards":["Example","In mathematics"]}
{"upwards":["Extensionality","Example"],"downwards":[]}
{"upwards":["Extensionality","In mathematics"],"downwards":[]}
{"upwards":["Infinite set"],"downwards":["Properties","Examples"]}
{"upwards":["Infinite set","Properties"],"downwards":[]}
{"upwards":["Infinite set","Examples"],"downwards":["Countably infinite sets","Uncountably infinite sets"]}
{"upwards":["Infinite set","Examples","Countably infinite sets"],"downwards":[]}
{"upwards":["Infinite set","Examples","Uncountably infinite sets"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Vertical bar"],"downwards":["Usage","Encoding"]}
{"upwards":["Vertical bar","Usage"],"downwards":["Mathematics","Physics","Computing","Phonetics and orthography","Literature"]}
{"upwards":["Vertical bar","Usage","Mathematics"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Physics"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing"],"downwards":["Pipe","Disjunction","Concatenation","Delimiter","Backus\u2013Naur form","Concurrency operator","APL","List comprehensions","Text markup"]}
{"upwards":["Vertical bar","Usage","Computing","Pipe"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Disjunction"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Concatenation"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Delimiter"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Backus\u2013Naur form"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Concurrency operator"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","APL"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","List comprehensions"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Computing","Text markup"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Phonetics and orthography"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature"],"downwards":["Punctuation","Poetry","Notation","Music scoring"]}
{"upwards":["Vertical bar","Usage","Literature","Punctuation"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature","Poetry"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature","Notation"],"downwards":[]}
{"upwards":["Vertical bar","Usage","Literature","Music scoring"],"downwards":[]}
{"upwards":["Vertical bar","Encoding"],"downwards":["Solid vertical bar vs broken bar","Unicode code points","Code pages and other historical encodings"]}
{"upwards":["Vertical bar","Encoding","Solid vertical bar vs broken bar"],"downwards":[]}
{"upwards":["Vertical bar","Encoding","Unicode code points"],"downwards":[]}
{"upwards":["Vertical bar","Encoding","Code pages and other historical encodings"],"downwards":[]}
{"upwards":["Mathematical model"],"downwards":["Elements of a mathematical model","Classifications","Construction","Significance in the natural sciences","Some applications","Examples"]}
{"upwards":["Mathematical model","Elements of a mathematical model"],"downwards":[]}
{"upwards":["Mathematical model","Classifications"],"downwards":[]}
{"upwards":["Mathematical model","Construction"],"downwards":["A priori information","Complexity","Training, tuning, and fitting","Evaluation and assessment"]}
{"upwards":["Mathematical model","Construction","A priori information"],"downwards":["Subjective information"]}
{"upwards":["Mathematical model","Construction","A priori information","Subjective information"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Complexity"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Training, tuning, and fitting"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment"],"downwards":["Prediction of empirical data","Scope of the model","Philosophical considerations"]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Prediction of empirical data"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Scope of the model"],"downwards":[]}
{"upwards":["Mathematical model","Construction","Evaluation and assessment","Philosophical considerations"],"downwards":[]}
{"upwards":["Mathematical model","Significance in the natural sciences"],"downwards":[]}
{"upwards":["Mathematical model","Some applications"],"downwards":[]}
{"upwards":["Mathematical model","Examples"],"downwards":[]}
{"upwards":["Bracket"],"downwards":["History","Typography","Round brackets or parentheses","Square brackets or brackets","Curly brackets or braces","Angle brackets or chevrons","Other brackets","Unicode"]}
{"upwards":["Bracket","History"],"downwards":[]}
{"upwards":["Bracket","Typography"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses"],"downwards":["Uses of ( )"]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )"],"downwards":["Enumerations","Accounting","Parentheses in mathematics","Parentheses in programming languages","Taxonomy","Chemistry and physics","Pop songs"]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Enumerations"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Accounting"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Parentheses in mathematics"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Parentheses in programming languages"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Taxonomy"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Chemistry and physics"],"downwards":[]}
{"upwards":["Bracket","Round brackets or parentheses","Uses of ( )","Pop songs"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets"],"downwards":["Uses of [ ]"]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]"],"downwards":["Proofreading","Law","Square brackets in mathematics","Chemistry","Square brackets in programming languages"]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Proofreading"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Law"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Square brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Chemistry"],"downwards":[]}
{"upwards":["Bracket","Square brackets or brackets","Uses of [ ]","Square brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces"],"downwards":["Uses of { }"]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }"],"downwards":["Music","Curly brackets in programming languages","Curly brackets in mathematics"]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Music"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Curly brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Curly brackets or braces","Uses of { }","Curly brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons"],"downwards":["Shape","Uses of \u27e8 \u27e9"]}
{"upwards":["Bracket","Angle brackets or chevrons","Shape"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9"],"downwards":["Angle brackets in mathematics","Physics and mechanics","Angle brackets in programming languages"]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Angle brackets in mathematics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Physics and mechanics"],"downwards":[]}
{"upwards":["Bracket","Angle brackets or chevrons","Uses of \u27e8 \u27e9","Angle brackets in programming languages"],"downwards":[]}
{"upwards":["Bracket","Other brackets"],"downwards":["Lenticular brackets\u3010\u3011","Floor \u230a \u230b and ceiling \u2308 \u2309 corner brackets","Quine corners \u231c\u231d and half brackets \u2e24 \u2e25 or \u2e22 \u2e23","Double brackets \u27e6 \u27e7","Brackets with quills \u2045 \u2046"]}
{"upwards":["Bracket","Other brackets","Lenticular brackets\u3010\u3011"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Floor \u230a \u230b and ceiling \u2308 \u2309 corner brackets"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Quine corners \u231c\u231d and half brackets \u2e24 \u2e25 or \u2e22 \u2e23"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Double brackets \u27e6 \u27e7"],"downwards":[]}
{"upwards":["Bracket","Other brackets","Brackets with quills \u2045 \u2046"],"downwards":[]}
{"upwards":["Bracket","Unicode"],"downwards":[]}
{"upwards":["Cantor's paradox"],"downwards":["Statements and proofs","Discussion and consequences","Historical notes","References","External links"]}
{"upwards":["Cantor's paradox","Statements and proofs"],"downwards":[]}
{"upwards":["Cantor's paradox","Discussion and consequences"],"downwards":[]}
{"upwards":["Cantor's paradox","Historical notes"],"downwards":[]}
{"upwards":["Cantor's paradox","References"],"downwards":[]}
{"upwards":["Cantor's paradox","External links"],"downwards":[]}
{"upwards":["Set-builder notation"],"downwards":["Sets defined by enumeration","Sets defined by a predicate","More complex expressions on the left side of the notation","Equivalent predicates yield equal sets","Set existence axiom","In programming languages"]}
{"upwards":["Set-builder notation","Sets defined by enumeration"],"downwards":[]}
{"upwards":["Set-builder notation","Sets defined by a predicate"],"downwards":["Specifying the domain","Examples"]}
{"upwards":["Set-builder notation","Sets defined by a predicate","Specifying the domain"],"downwards":[]}
{"upwards":["Set-builder notation","Sets defined by a predicate","Examples"],"downwards":[]}
{"upwards":["Set-builder notation","More complex expressions on the left side of the notation"],"downwards":[]}
{"upwards":["Set-builder notation","Equivalent predicates yield equal sets"],"downwards":[]}
{"upwards":["Set-builder notation","Set existence axiom"],"downwards":[]}
{"upwards":["Set-builder notation","In programming languages"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory"],"downwards":["History","Axioms","Motivation via the cumulative hierarchy","Metamathematics","Criticisms"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","History"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms"],"downwards":["1. Axiom of extensionality","2. Axiom of regularity (also called the axiom of foundation)","3. Axiom schema of specification (or of separation, or of restricted comprehension)","4. Axiom of pairing","5. Axiom of union","6. Axiom schema of replacement","7. Axiom of infinity","8. Axiom of power set","9. Well-ordering axiom","Zorn's lemma"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","1. Axiom of extensionality"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","2. Axiom of regularity (also called the axiom of foundation)"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","3. Axiom schema of specification (or of separation, or of restricted comprehension)"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","4. Axiom of pairing"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","5. Axiom of union"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","6. Axiom schema of replacement"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","7. Axiom of infinity"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","8. Axiom of power set"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","9. Well-ordering axiom"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Axioms","Zorn's lemma"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Motivation via the cumulative hierarchy"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics"],"downwards":["Virtual classes","Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Consistency","Independence","Proposed additions"]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Virtual classes"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Von Neumann\u2013Bernays\u2013G\u00f6del set theory"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Consistency"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Independence"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Metamathematics","Proposed additions"],"downwards":[]}
{"upwards":["Zermelo\u2013Fraenkel set theory","Criticisms"],"downwards":[]}
{"upwards":["Mathematical object"],"downwards":["List of mathematical objects by branch"]}
{"upwards":["Mathematical object","List of mathematical objects by branch"],"downwards":[]}
{"upwards":["Equality (mathematics)"],"downwards":["Etymology","Basic properties","Equality as predicate","Identities","Equations","Approximate equality","Relation with equivalence, congruence, and isomorphism","Logical definitions","Equality in set theory"]}
{"upwards":["Equality (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Equality (mathematics)","Basic properties"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality as predicate"],"downwards":[]}
{"upwards":["Equality (mathematics)","Identities"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equations"],"downwards":[]}
{"upwards":["Equality (mathematics)","Approximate equality"],"downwards":[]}
{"upwards":["Equality (mathematics)","Relation with equivalence, congruence, and isomorphism"],"downwards":[]}
{"upwards":["Equality (mathematics)","Logical definitions"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality in set theory"],"downwards":["Set equality based on first-order logic with equality","Set equality based on first-order logic without equality"]}
{"upwards":["Equality (mathematics)","Equality in set theory","Set equality based on first-order logic with equality"],"downwards":[]}
{"upwards":["Equality (mathematics)","Equality in set theory","Set equality based on first-order logic without equality"],"downwards":[]}
{"upwards":["Tuple"],"downwards":["Etymology","Properties","Definitions","n-tuples of m-sets","Type theory"]}
{"upwards":["Tuple","Etymology"],"downwards":["Names for tuples of specific lengths"]}
{"upwards":["Tuple","Etymology","Names for tuples of specific lengths"],"downwards":[]}
{"upwards":["Tuple","Properties"],"downwards":[]}
{"upwards":["Tuple","Definitions"],"downwards":["Tuples as functions","Tuples as nested ordered pairs","Tuples as nested sets"]}
{"upwards":["Tuple","Definitions","Tuples as functions"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested ordered pairs"],"downwards":[]}
{"upwards":["Tuple","Definitions","Tuples as nested sets"],"downwards":[]}
{"upwards":["Tuple","n-tuples of m-sets"],"downwards":[]}
{"upwards":["Tuple","Type theory"],"downwards":[]}
{"upwards":["Permutation"],"downwards":["History","Permutations without repetitions","Definition","Notations","Other uses of the term permutation","Properties","Permutations of totally ordered sets","Permutations in computing"]}
{"upwards":["Permutation","History"],"downwards":[]}
{"upwards":["Permutation","Permutations without repetitions"],"downwards":[]}
{"upwards":["Permutation","Definition"],"downwards":[]}
{"upwards":["Permutation","Notations"],"downwards":["Two-line notation","One-line notation","Cycle notation","Canonical cycle notation","Composition of permutations"]}
{"upwards":["Permutation","Notations","Two-line notation"],"downwards":[]}
{"upwards":["Permutation","Notations","One-line notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Cycle notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Canonical cycle notation"],"downwards":[]}
{"upwards":["Permutation","Notations","Composition of permutations"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation"],"downwards":["k-permutations of n","Permutations with repetition","Permutations of multisets","Circular permutations"]}
{"upwards":["Permutation","Other uses of the term permutation","k-permutations of n"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Permutations with repetition"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Permutations of multisets"],"downwards":[]}
{"upwards":["Permutation","Other uses of the term permutation","Circular permutations"],"downwards":[]}
{"upwards":["Permutation","Properties"],"downwards":["Cycle type","Conjugating permutations","Permutation order","Parity of a permutation","Matrix representation"]}
{"upwards":["Permutation","Properties","Cycle type"],"downwards":[]}
{"upwards":["Permutation","Properties","Conjugating permutations"],"downwards":[]}
{"upwards":["Permutation","Properties","Permutation order"],"downwards":[]}
{"upwards":["Permutation","Properties","Parity of a permutation"],"downwards":[]}
{"upwards":["Permutation","Properties","Matrix representation"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets"],"downwards":["Ascents, descents, runs and exceedances","Foata's transition lemma","Inversions"]}
{"upwards":["Permutation","Permutations of totally ordered sets","Ascents, descents, runs and exceedances"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets","Foata's transition lemma"],"downwards":[]}
{"upwards":["Permutation","Permutations of totally ordered sets","Inversions"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing"],"downwards":["Numbering permutations","Algorithms to generate permutations","Applications"]}
{"upwards":["Permutation","Permutations in computing","Numbering permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations"],"downwards":["Random generation of permutations","Generation in lexicographic order","Generation with minimal changes","Meandric permutations"]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Random generation of permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Generation in lexicographic order"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Generation with minimal changes"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Algorithms to generate permutations","Meandric permutations"],"downwards":[]}
{"upwards":["Permutation","Permutations in computing","Applications"],"downwards":[]}
{"upwards":["Philosophy"],"downwards":["Definitions","Historical overview","Branches of philosophy","Methods of philosophy","Applied and professional philosophy"]}
{"upwards":["Philosophy","Definitions"],"downwards":[]}
{"upwards":["Philosophy","Historical overview"],"downwards":["Western philosophy","Middle Eastern philosophy","Eastern philosophy","African philosophy","Indigenous American philosophy"]}
{"upwards":["Philosophy","Historical overview","Western philosophy"],"downwards":["Ancient era","Medieval era","Modern era"]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Ancient era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Medieval era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Western philosophy","Modern era"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy"],"downwards":["Pre-Islamic philosophy","Islamic philosophy"]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy","Pre-Islamic philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Middle Eastern philosophy","Islamic philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy"],"downwards":["Indian philosophy","Buddhist philosophy","East Asian philosophy"]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","Indian philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","Buddhist philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Eastern philosophy","East Asian philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","African philosophy"],"downwards":[]}
{"upwards":["Philosophy","Historical overview","Indigenous American philosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy"],"downwards":["Aesthetics","Ethics","Epistemology","Metaphysics","Logic","Mind and language","Philosophy of science","Political philosophy","Philosophy of religion","Metaphilosophy","Other subdivisions"]}
{"upwards":["Philosophy","Branches of philosophy","Aesthetics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Ethics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Epistemology"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Metaphysics"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Logic"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Mind and language"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Philosophy of science"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Political philosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Philosophy of religion"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Metaphilosophy"],"downwards":[]}
{"upwards":["Philosophy","Branches of philosophy","Other subdivisions"],"downwards":[]}
{"upwards":["Philosophy","Methods of philosophy"],"downwards":["Philosophical progress"]}
{"upwards":["Philosophy","Methods of philosophy","Philosophical progress"],"downwards":[]}
{"upwards":["Philosophy","Applied and professional philosophy"],"downwards":["Women in philosophy"]}
{"upwards":["Philosophy","Applied and professional philosophy","Women in philosophy"],"downwards":[]}
{"upwards":["Independence (mathematical logic)"],"downwards":["Usage note","Independence results in set theory","Applications to physical theory"]}
{"upwards":["Independence (mathematical logic)","Usage note"],"downwards":[]}
{"upwards":["Independence (mathematical logic)","Independence results in set theory"],"downwards":[]}
{"upwards":["Independence (mathematical logic)","Applications to physical theory"],"downwards":[]}
{"upwards":["Paul Cohen"],"downwards":["Early life and education","Career","Death","Selected publications"]}
{"upwards":["Paul Cohen","Early life and education"],"downwards":[]}
{"upwards":["Paul Cohen","Career"],"downwards":["Continuum hypothesis"]}
{"upwards":["Paul Cohen","Career","Continuum hypothesis"],"downwards":[]}
{"upwards":["Paul Cohen","Death"],"downwards":[]}
{"upwards":["Paul Cohen","Selected publications"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Union (set theory)"],"downwards":["Union of two sets","Algebraic properties","Finite unions","Arbitrary unions","Notation encoding"]}
{"upwards":["Union (set theory)","Union of two sets"],"downwards":[]}
{"upwards":["Union (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Union (set theory)","Finite unions"],"downwards":[]}
{"upwards":["Union (set theory)","Arbitrary unions"],"downwards":["Notations"]}
{"upwards":["Union (set theory)","Arbitrary unions","Notations"],"downwards":[]}
{"upwards":["Union (set theory)","Notation encoding"],"downwards":[]}
{"upwards":["Axiom"],"downwards":["Etymology","Historical development","Mathematical logic"]}
{"upwards":["Axiom","Etymology"],"downwards":[]}
{"upwards":["Axiom","Historical development"],"downwards":["Early Greeks","Modern development","Other sciences"]}
{"upwards":["Axiom","Historical development","Early Greeks"],"downwards":[]}
{"upwards":["Axiom","Historical development","Modern development"],"downwards":[]}
{"upwards":["Axiom","Historical development","Other sciences"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic"],"downwards":["Logical axioms","Non-logical axioms","Role in mathematical logic","Further discussion"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic"],"downwards":["Deductive systems and completeness"]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic","Deductive systems and completeness"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Further discussion"],"downwards":[]}
{"upwards":["Theorem"],"downwards":["Theoremhood and truth","Epistemological considerations","Informal account of theorems","Relation with scientific theories","Terminology","Layout","Lore","Theorems in logic"]}
{"upwards":["Theorem","Theoremhood and truth"],"downwards":[]}
{"upwards":["Theorem","Epistemological considerations"],"downwards":[]}
{"upwards":["Theorem","Informal account of theorems"],"downwards":[]}
{"upwards":["Theorem","Relation with scientific theories"],"downwards":[]}
{"upwards":["Theorem","Terminology"],"downwards":[]}
{"upwards":["Theorem","Layout"],"downwards":[]}
{"upwards":["Theorem","Lore"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic"],"downwards":["Syntax and semantics","Interpretation of a formal theorem","Theorems and theories"]}
{"upwards":["Theorem","Theorems in logic","Syntax and semantics"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Interpretation of a formal theorem"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Theorems and theories"],"downwards":[]}
{"upwards":["Italic type"],"downwards":["History","Examples","Usage","Oblique type compared to italics","More complex usage"]}
{"upwards":["Italic type","History"],"downwards":[]}
{"upwards":["Italic type","Examples"],"downwards":[]}
{"upwards":["Italic type","Usage"],"downwards":[]}
{"upwards":["Italic type","Oblique type compared to italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage"],"downwards":["Italics within italics","Left-leaning italics","Upright italics","Parentheses","Substitutes","OpenType","Web pages","Unicode"]}
{"upwards":["Italic type","More complex usage","Italics within italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Left-leaning italics"],"downwards":["Iranic font style"]}
{"upwards":["Italic type","More complex usage","Left-leaning italics","Iranic font style"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Upright italics"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Parentheses"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Substitutes"],"downwards":[]}
{"upwards":["Italic type","More complex usage","OpenType"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Web pages"],"downwards":[]}
{"upwards":["Italic type","More complex usage","Unicode"],"downwards":[]}
{"upwards":["Algebraic number"],"downwards":["Examples","Properties","Field","Related fields","Algebraic integers","Special classes","Notes","References"]}
{"upwards":["Algebraic number","Examples"],"downwards":[]}
{"upwards":["Algebraic number","Properties"],"downwards":[]}
{"upwards":["Algebraic number","Field"],"downwards":[]}
{"upwards":["Algebraic number","Related fields"],"downwards":["Numbers defined by radicals","Closed-form number"]}
{"upwards":["Algebraic number","Related fields","Numbers defined by radicals"],"downwards":[]}
{"upwards":["Algebraic number","Related fields","Closed-form number"],"downwards":[]}
{"upwards":["Algebraic number","Algebraic integers"],"downwards":[]}
{"upwards":["Algebraic number","Special classes"],"downwards":[]}
{"upwards":["Algebraic number","Notes"],"downwards":[]}
{"upwards":["Algebraic number","References"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
{"upwards":["Fraction","Vocabulary"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions"],"downwards":["Simple, common, or vulgar fractions","Proper and improper fractions","Reciprocals and the \"invisible denominator\"","Ratios","Decimal fractions and percentages","Mixed numbers","Historical notions"]}
{"upwards":["Fraction","Forms of fractions","Simple, common, or vulgar fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Proper and improper fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Reciprocals and the \"invisible denominator\""],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Ratios"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Decimal fractions and percentages"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions"],"downwards":["Egyptian fraction","Complex and compound fractions"]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Egyptian fraction"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Complex and compound fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions"],"downwards":["Equivalent fractions","Comparing fractions","Addition","Subtraction","Multiplication","Division","Converting between decimals and fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions"],"downwards":["Simplifying (reducing) fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions","Simplifying (reducing) fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Comparing fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Addition"],"downwards":["Adding unlike quantities"]}
{"upwards":["Fraction","Arithmetic with fractions","Addition","Adding unlike quantities"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Subtraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication"],"downwards":["Multiplying a fraction by another fraction","Multiplying a fraction by a whole number","Multiplying mixed numbers"]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by another fraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by a whole number"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Division"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions"],"downwards":["Converting repeating decimals to fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
{"upwards":["Fraction","Algebraic fractions"],"downwards":[]}
{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Complex number"],"downwards":["Definition","Notation","Visualization","History","Relations and operations","Properties","Formal construction","Complex analysis","Applications","Generalizations and related notions"]}
{"upwards":["Complex number","Definition"],"downwards":[]}
{"upwards":["Complex number","Notation"],"downwards":[]}
{"upwards":["Complex number","Visualization"],"downwards":["Cartesian complex plane","Polar complex plane","Complex graphs"]}
{"upwards":["Complex number","Visualization","Cartesian complex plane"],"downwards":[]}
{"upwards":["Complex number","Visualization","Polar complex plane"],"downwards":["Modulus and argument"]}
{"upwards":["Complex number","Visualization","Polar complex plane","Modulus and argument"],"downwards":[]}
{"upwards":["Complex number","Visualization","Complex graphs"],"downwards":[]}
{"upwards":["Complex number","History"],"downwards":[]}
{"upwards":["Complex number","Relations and operations"],"downwards":["Equality","Ordering","Conjugate","Addition and subtraction","Multiplication and square","Reciprocal and division","Multiplication and division in polar form","Square root","Exponential function","Complex logarithm","Exponentiation"]}
{"upwards":["Complex number","Relations and operations","Equality"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Ordering"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Conjugate"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and square"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Reciprocal and division"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and division in polar form"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Square root"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function"],"downwards":["Functional equation","Euler's formula"]}
{"upwards":["Complex number","Relations and operations","Exponential function","Functional equation"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function","Euler's formula"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Complex logarithm"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponentiation"],"downwards":["Integer and fractional exponents"]}
{"upwards":["Complex number","Relations and operations","Exponentiation","Integer and fractional exponents"],"downwards":[]}
{"upwards":["Complex number","Properties"],"downwards":["Field structure","Solutions of polynomial equations","Algebraic characterization","Characterization as a topological field"]}
{"upwards":["Complex number","Properties","Field structure"],"downwards":[]}
{"upwards":["Complex number","Properties","Solutions of polynomial equations"],"downwards":[]}
{"upwards":["Complex number","Properties","Algebraic characterization"],"downwards":[]}
{"upwards":["Complex number","Properties","Characterization as a topological field"],"downwards":[]}
{"upwards":["Complex number","Formal construction"],"downwards":["Construction as ordered pairs","Construction as a quotient field","Matrix representation of complex numbers"]}
{"upwards":["Complex number","Formal construction","Construction as ordered pairs"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Construction as a quotient field"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Matrix representation of complex numbers"],"downwards":[]}
{"upwards":["Complex number","Complex analysis"],"downwards":["Complex exponential and related functions","Holomorphic functions"]}
{"upwards":["Complex number","Complex analysis","Complex exponential and related functions"],"downwards":[]}
{"upwards":["Complex number","Complex analysis","Holomorphic functions"],"downwards":[]}
{"upwards":["Complex number","Applications"],"downwards":["Geometry","Algebraic number theory","Analytic number theory","Improper integrals","Dynamic equations","Linear algebra","In applied mathematics","In physics"]}
{"upwards":["Complex number","Applications","Geometry"],"downwards":["Shapes","Fractal geometry","Triangles"]}
{"upwards":["Complex number","Applications","Geometry","Shapes"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Fractal geometry"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Triangles"],"downwards":[]}
{"upwards":["Complex number","Applications","Algebraic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Analytic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Improper integrals"],"downwards":[]}
{"upwards":["Complex number","Applications","Dynamic equations"],"downwards":[]}
{"upwards":["Complex number","Applications","Linear algebra"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics"],"downwards":["Control theory","Signal analysis"]}
{"upwards":["Complex number","Applications","In applied mathematics","Control theory"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics","Signal analysis"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics"],"downwards":["Electromagnetism and electrical engineering","Fluid dynamics","Quantum mechanics","Relativity"]}
{"upwards":["Complex number","Applications","In physics","Electromagnetism and electrical engineering"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Fluid dynamics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Quantum mechanics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Relativity"],"downwards":[]}
{"upwards":["Complex number","Generalizations and related notions"],"downwards":[]}
{"upwards":["Integer"],"downwards":["History","Algebraic properties","Order-theoretic properties","Construction","Computer science","Cardinality"]}
{"upwards":["Integer","History"],"downwards":[]}
{"upwards":["Integer","Algebraic properties"],"downwards":[]}
{"upwards":["Integer","Order-theoretic properties"],"downwards":[]}
{"upwards":["Integer","Construction"],"downwards":["Traditional development","Equivalence classes of ordered pairs","Other approaches"]}
{"upwards":["Integer","Construction","Traditional development"],"downwards":[]}
{"upwards":["Integer","Construction","Equivalence classes of ordered pairs"],"downwards":[]}
{"upwards":["Integer","Construction","Other approaches"],"downwards":[]}
{"upwards":["Integer","Computer science"],"downwards":[]}
{"upwards":["Integer","Cardinality"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Function (mathematics)"],"downwards":["Definition","Notation","Other terms","Specifying a function","Representing a function","General properties","Multivariate function","In calculus","Function space","Multi-valued functions","In the foundations of mathematics and set theory","In computer science"]}
{"upwards":["Function (mathematics)","Definition"],"downwards":["Total, univalent relation","Set exponentiation"]}
{"upwards":["Function (mathematics)","Definition","Total, univalent relation"],"downwards":[]}
{"upwards":["Function (mathematics)","Definition","Set exponentiation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation"],"downwards":["Functional notation","Arrow notation","Index notation","Dot notation","Specialized notations"]}
{"upwards":["Function (mathematics)","Notation","Functional notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Arrow notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Index notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Dot notation"],"downwards":[]}
{"upwards":["Function (mathematics)","Notation","Specialized notations"],"downwards":[]}
{"upwards":["Function (mathematics)","Other terms"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function"],"downwards":["By listing function values","By a formula","Inverse and implicit functions","Using differential calculus","By recurrence"]}
{"upwards":["Function (mathematics)","Specifying a function","By listing function values"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","By a formula"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","Inverse and implicit functions"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","Using differential calculus"],"downwards":[]}
{"upwards":["Function (mathematics)","Specifying a function","By recurrence"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function"],"downwards":["Graphs and plots","Tables","Bar chart"]}
{"upwards":["Function (mathematics)","Representing a function","Graphs and plots"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function","Tables"],"downwards":[]}
{"upwards":["Function (mathematics)","Representing a function","Bar chart"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties"],"downwards":["Standard functions","Function composition","Image and preimage","Injective, surjective and bijective functions","Restriction and extension"]}
{"upwards":["Function (mathematics)","General properties","Standard functions"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Function composition"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Image and preimage"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Injective, surjective and bijective functions"],"downwards":[]}
{"upwards":["Function (mathematics)","General properties","Restriction and extension"],"downwards":[]}
{"upwards":["Function (mathematics)","Multivariate function"],"downwards":[]}
{"upwards":["Function (mathematics)","In calculus"],"downwards":["Real function","Vector-valued function"]}
{"upwards":["Function (mathematics)","In calculus","Real function"],"downwards":[]}
{"upwards":["Function (mathematics)","In calculus","Vector-valued function"],"downwards":[]}
{"upwards":["Function (mathematics)","Function space"],"downwards":[]}
{"upwards":["Function (mathematics)","Multi-valued functions"],"downwards":[]}
{"upwards":["Function (mathematics)","In the foundations of mathematics and set theory"],"downwards":[]}
{"upwards":["Function (mathematics)","In computer science"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
{"upwards":["Fraction","Vocabulary"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions"],"downwards":["Simple, common, or vulgar fractions","Proper and improper fractions","Reciprocals and the \"invisible denominator\"","Ratios","Decimal fractions and percentages","Mixed numbers","Historical notions"]}
{"upwards":["Fraction","Forms of fractions","Simple, common, or vulgar fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Proper and improper fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Reciprocals and the \"invisible denominator\""],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Ratios"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Decimal fractions and percentages"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions"],"downwards":["Egyptian fraction","Complex and compound fractions"]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Egyptian fraction"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Complex and compound fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions"],"downwards":["Equivalent fractions","Comparing fractions","Addition","Subtraction","Multiplication","Division","Converting between decimals and fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions"],"downwards":["Simplifying (reducing) fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions","Simplifying (reducing) fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Comparing fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Addition"],"downwards":["Adding unlike quantities"]}
{"upwards":["Fraction","Arithmetic with fractions","Addition","Adding unlike quantities"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Subtraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication"],"downwards":["Multiplying a fraction by another fraction","Multiplying a fraction by a whole number","Multiplying mixed numbers"]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by another fraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by a whole number"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Division"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions"],"downwards":["Converting repeating decimals to fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
{"upwards":["Fraction","Algebraic fractions"],"downwards":[]}
{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Natural number"],"downwards":["History","Notation","Properties","Generalizations","Formal definitions"]}
{"upwards":["Natural number","History"],"downwards":["Ancient roots","Modern definitions"]}
{"upwards":["Natural number","History","Ancient roots"],"downwards":[]}
{"upwards":["Natural number","History","Modern definitions"],"downwards":[]}
{"upwards":["Natural number","Notation"],"downwards":[]}
{"upwards":["Natural number","Properties"],"downwards":["Addition","Multiplication","Relationship between addition and multiplication","Order","Division","Algebraic properties satisfied by the natural numbers"]}
{"upwards":["Natural number","Properties","Addition"],"downwards":[]}
{"upwards":["Natural number","Properties","Multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Relationship between addition and multiplication"],"downwards":[]}
{"upwards":["Natural number","Properties","Order"],"downwards":[]}
{"upwards":["Natural number","Properties","Division"],"downwards":[]}
{"upwards":["Natural number","Properties","Algebraic properties satisfied by the natural numbers"],"downwards":[]}
{"upwards":["Natural number","Generalizations"],"downwards":[]}
{"upwards":["Natural number","Formal definitions"],"downwards":["Peano axioms","Set-theoretic definition"]}
{"upwards":["Natural number","Formal definitions","Peano axioms"],"downwards":[]}
{"upwards":["Natural number","Formal definitions","Set-theoretic definition"],"downwards":[]}
{"upwards":["Venn diagram"],"downwards":["Details","Example","History","Overview","Extensions to higher numbers of sets","Related concepts"]}
{"upwards":["Venn diagram","Details"],"downwards":[]}
{"upwards":["Venn diagram","Example"],"downwards":[]}
{"upwards":["Venn diagram","History"],"downwards":[]}
{"upwards":["Venn diagram","Overview"],"downwards":[]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets"],"downwards":["Edwards\u2013Venn diagrams","Other diagrams"]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets","Edwards\u2013Venn diagrams"],"downwards":[]}
{"upwards":["Venn diagram","Extensions to higher numbers of sets","Other diagrams"],"downwards":[]}
{"upwards":["Venn diagram","Related concepts"],"downwards":[]}
{"upwards":["Line segment"],"downwards":["In real or complex vector spaces","Properties","In proofs","As a degenerate ellipse","In other geometric shapes","Directed line segment","Generalizations","Types of line segments"]}
{"upwards":["Line segment","In real or complex vector spaces"],"downwards":[]}
{"upwards":["Line segment","Properties"],"downwards":[]}
{"upwards":["Line segment","In proofs"],"downwards":[]}
{"upwards":["Line segment","As a degenerate ellipse"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes"],"downwards":["Triangles","Quadrilaterals","Circles and ellipses"]}
{"upwards":["Line segment","In other geometric shapes","Triangles"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes","Quadrilaterals"],"downwards":[]}
{"upwards":["Line segment","In other geometric shapes","Circles and ellipses"],"downwards":[]}
{"upwards":["Line segment","Directed line segment"],"downwards":[]}
{"upwards":["Line segment","Generalizations"],"downwards":[]}
{"upwards":["Line segment","Types of line segments"],"downwards":[]}
{"upwards":["Intersection (set theory)"],"downwards":["Notation and terminology","Definition","Algebraic properties","Arbitrary intersections","Nullary intersection"]}
{"upwards":["Intersection (set theory)","Notation and terminology"],"downwards":[]}
{"upwards":["Intersection (set theory)","Definition"],"downwards":["Intersecting and disjoint sets"]}
{"upwards":["Intersection (set theory)","Definition","Intersecting and disjoint sets"],"downwards":[]}
{"upwards":["Intersection (set theory)","Algebraic properties"],"downwards":[]}
{"upwards":["Intersection (set theory)","Arbitrary intersections"],"downwards":[]}
{"upwards":["Intersection (set theory)","Nullary intersection"],"downwards":[]}
{"upwards":["Complex number"],"downwards":["Definition","Notation","Visualization","History","Relations and operations","Properties","Formal construction","Complex analysis","Applications","Generalizations and related notions"]}
{"upwards":["Complex number","Definition"],"downwards":[]}
{"upwards":["Complex number","Notation"],"downwards":[]}
{"upwards":["Complex number","Visualization"],"downwards":["Cartesian complex plane","Polar complex plane","Complex graphs"]}
{"upwards":["Complex number","Visualization","Cartesian complex plane"],"downwards":[]}
{"upwards":["Complex number","Visualization","Polar complex plane"],"downwards":["Modulus and argument"]}
{"upwards":["Complex number","Visualization","Polar complex plane","Modulus and argument"],"downwards":[]}
{"upwards":["Complex number","Visualization","Complex graphs"],"downwards":[]}
{"upwards":["Complex number","History"],"downwards":[]}
{"upwards":["Complex number","Relations and operations"],"downwards":["Equality","Ordering","Conjugate","Addition and subtraction","Multiplication and square","Reciprocal and division","Multiplication and division in polar form","Square root","Exponential function","Complex logarithm","Exponentiation"]}
{"upwards":["Complex number","Relations and operations","Equality"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Ordering"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Conjugate"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and square"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Reciprocal and division"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Multiplication and division in polar form"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Square root"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function"],"downwards":["Functional equation","Euler's formula"]}
{"upwards":["Complex number","Relations and operations","Exponential function","Functional equation"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponential function","Euler's formula"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Complex logarithm"],"downwards":[]}
{"upwards":["Complex number","Relations and operations","Exponentiation"],"downwards":["Integer and fractional exponents"]}
{"upwards":["Complex number","Relations and operations","Exponentiation","Integer and fractional exponents"],"downwards":[]}
{"upwards":["Complex number","Properties"],"downwards":["Field structure","Solutions of polynomial equations","Algebraic characterization","Characterization as a topological field"]}
{"upwards":["Complex number","Properties","Field structure"],"downwards":[]}
{"upwards":["Complex number","Properties","Solutions of polynomial equations"],"downwards":[]}
{"upwards":["Complex number","Properties","Algebraic characterization"],"downwards":[]}
{"upwards":["Complex number","Properties","Characterization as a topological field"],"downwards":[]}
{"upwards":["Complex number","Formal construction"],"downwards":["Construction as ordered pairs","Construction as a quotient field","Matrix representation of complex numbers"]}
{"upwards":["Complex number","Formal construction","Construction as ordered pairs"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Construction as a quotient field"],"downwards":[]}
{"upwards":["Complex number","Formal construction","Matrix representation of complex numbers"],"downwards":[]}
{"upwards":["Complex number","Complex analysis"],"downwards":["Complex exponential and related functions","Holomorphic functions"]}
{"upwards":["Complex number","Complex analysis","Complex exponential and related functions"],"downwards":[]}
{"upwards":["Complex number","Complex analysis","Holomorphic functions"],"downwards":[]}
{"upwards":["Complex number","Applications"],"downwards":["Geometry","Algebraic number theory","Analytic number theory","Improper integrals","Dynamic equations","Linear algebra","In applied mathematics","In physics"]}
{"upwards":["Complex number","Applications","Geometry"],"downwards":["Shapes","Fractal geometry","Triangles"]}
{"upwards":["Complex number","Applications","Geometry","Shapes"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Fractal geometry"],"downwards":[]}
{"upwards":["Complex number","Applications","Geometry","Triangles"],"downwards":[]}
{"upwards":["Complex number","Applications","Algebraic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Analytic number theory"],"downwards":[]}
{"upwards":["Complex number","Applications","Improper integrals"],"downwards":[]}
{"upwards":["Complex number","Applications","Dynamic equations"],"downwards":[]}
{"upwards":["Complex number","Applications","Linear algebra"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics"],"downwards":["Control theory","Signal analysis"]}
{"upwards":["Complex number","Applications","In applied mathematics","Control theory"],"downwards":[]}
{"upwards":["Complex number","Applications","In applied mathematics","Signal analysis"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics"],"downwards":["Electromagnetism and electrical engineering","Fluid dynamics","Quantum mechanics","Relativity"]}
{"upwards":["Complex number","Applications","In physics","Electromagnetism and electrical engineering"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Fluid dynamics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Quantum mechanics"],"downwards":[]}
{"upwards":["Complex number","Applications","In physics","Relativity"],"downwards":[]}
{"upwards":["Complex number","Generalizations and related notions"],"downwards":[]}
{"upwards":["Irrational number"],"downwards":["History","Examples","Types","Decimal expansions","Irrational powers","Open questions","In constructive mathematics","Set of all irrationals"]}
{"upwards":["Irrational number","History"],"downwards":["Ancient Greece","India","Middle Ages","Modern period"]}
{"upwards":["Irrational number","History","Ancient Greece"],"downwards":[]}
{"upwards":["Irrational number","History","India"],"downwards":[]}
{"upwards":["Irrational number","History","Middle Ages"],"downwards":[]}
{"upwards":["Irrational number","History","Modern period"],"downwards":[]}
{"upwards":["Irrational number","Examples"],"downwards":["Square roots","General roots","Logarithms"]}
{"upwards":["Irrational number","Examples","Square roots"],"downwards":[]}
{"upwards":["Irrational number","Examples","General roots"],"downwards":[]}
{"upwards":["Irrational number","Examples","Logarithms"],"downwards":[]}
{"upwards":["Irrational number","Types"],"downwards":["Transcendental\/algebraic"]}
{"upwards":["Irrational number","Types","Transcendental\/algebraic"],"downwards":[]}
{"upwards":["Irrational number","Decimal expansions"],"downwards":[]}
{"upwards":["Irrational number","Irrational powers"],"downwards":[]}
{"upwards":["Irrational number","Open questions"],"downwards":[]}
{"upwards":["Irrational number","In constructive mathematics"],"downwards":[]}
{"upwards":["Irrational number","Set of all irrationals"],"downwards":[]}
{"upwards":["e (mathematical constant)"],"downwards":["History","Applications","In calculus","Properties","Representations","Computing the digits","In computer culture","References","Further reading","External links"]}
{"upwards":["e (mathematical constant)","History"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications"],"downwards":["Compound interest","Bernoulli trials","Standard normal distribution","Derangements","Optimal planning problems","Asymptotics"]}
{"upwards":["e (mathematical constant)","Applications","Compound interest"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Bernoulli trials"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Standard normal distribution"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Derangements"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Optimal planning problems"],"downwards":[]}
{"upwards":["e (mathematical constant)","Applications","Asymptotics"],"downwards":[]}
{"upwards":["e (mathematical constant)","In calculus"],"downwards":["Alternative characterizations"]}
{"upwards":["e (mathematical constant)","In calculus","Alternative characterizations"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties"],"downwards":["Calculus","Inequalities","Exponential-like functions","Number theory","Complex numbers"]}
{"upwards":["e (mathematical constant)","Properties","Calculus"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Inequalities"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Exponential-like functions"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Number theory"],"downwards":[]}
{"upwards":["e (mathematical constant)","Properties","Complex numbers"],"downwards":[]}
{"upwards":["e (mathematical constant)","Representations"],"downwards":["Stochastic representations","Known digits"]}
{"upwards":["e (mathematical constant)","Representations","Stochastic representations"],"downwards":[]}
{"upwards":["e (mathematical constant)","Representations","Known digits"],"downwards":[]}
{"upwards":["e (mathematical constant)","Computing the digits"],"downwards":[]}
{"upwards":["e (mathematical constant)","In computer culture"],"downwards":[]}
{"upwards":["e (mathematical constant)","References"],"downwards":[]}
{"upwards":["e (mathematical constant)","Further reading"],"downwards":[]}
{"upwards":["e (mathematical constant)","External links"],"downwards":[]}
{"upwards":["Pi"],"downwards":["Fundamentals","History","Modern quest for more digits","Role and characterizations in mathematics","Outside mathematics"]}
{"upwards":["Pi","Fundamentals"],"downwards":["Name","Definition","Irrationality and normality","Transcendence","Continued fractions","Approximate value and digits","Complex numbers and Euler's identity"]}
{"upwards":["Pi","Fundamentals","Name"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Definition"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Irrationality and normality"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Transcendence"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Continued fractions"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Approximate value and digits"],"downwards":[]}
{"upwards":["Pi","Fundamentals","Complex numbers and Euler's identity"],"downwards":[]}
{"upwards":["Pi","History"],"downwards":["Antiquity","Polygon approximation era","Infinite series","Irrationality and transcendence","Adoption of the symbol \u03c0"]}
{"upwards":["Pi","History","Antiquity"],"downwards":[]}
{"upwards":["Pi","History","Polygon approximation era"],"downwards":[]}
{"upwards":["Pi","History","Infinite series"],"downwards":["Rate of convergence"]}
{"upwards":["Pi","History","Infinite series","Rate of convergence"],"downwards":[]}
{"upwards":["Pi","History","Irrationality and transcendence"],"downwards":[]}
{"upwards":["Pi","History","Adoption of the symbol \u03c0"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits"],"downwards":["Computer era and iterative algorithms","Motives for computing \u03c0","Rapidly convergent series","Monte Carlo methods","Spigot algorithms"]}
{"upwards":["Pi","Modern quest for more digits","Computer era and iterative algorithms"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Motives for computing \u03c0"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Rapidly convergent series"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Monte Carlo methods"],"downwards":[]}
{"upwards":["Pi","Modern quest for more digits","Spigot algorithms"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics"],"downwards":["Geometry and trigonometry","Units of angle","Eigenvalues","Inequalities","Fourier transform and Heisenberg uncertainty principle","Gaussian integrals","Topology","Cauchy's integral formula","Total curvature","The gamma function and Stirling's approximation","Number theory and Riemann zeta function","Fourier series","Modular forms and theta functions","Cauchy distribution and potential theory","In the Mandelbrot set"]}
{"upwards":["Pi","Role and characterizations in mathematics","Geometry and trigonometry"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Units of angle"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Eigenvalues"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Inequalities"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Fourier transform and Heisenberg uncertainty principle"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Gaussian integrals"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Topology"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Cauchy's integral formula"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Total curvature"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","The gamma function and Stirling's approximation"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Number theory and Riemann zeta function"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Fourier series"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Modular forms and theta functions"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","Cauchy distribution and potential theory"],"downwards":[]}
{"upwards":["Pi","Role and characterizations in mathematics","In the Mandelbrot set"],"downwards":[]}
{"upwards":["Pi","Outside mathematics"],"downwards":["Describing physical phenomena","Memorizing digits","In popular culture","In computer culture"]}
{"upwards":["Pi","Outside mathematics","Describing physical phenomena"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","Memorizing digits"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","In popular culture"],"downwards":[]}
{"upwards":["Pi","Outside mathematics","In computer culture"],"downwards":[]}
{"upwards":["Foundations of mathematics"],"downwards":["Historical context","Foundational crisis","Toward resolution of the crisis"]}
{"upwards":["Foundations of mathematics","Historical context"],"downwards":["Ancient Greek mathematics","Platonism as a philosophy of mathematics","Aristotelian realism","Middle Ages and Renaissance","19th century"]}
{"upwards":["Foundations of mathematics","Historical context","Ancient Greek mathematics"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Platonism as a philosophy of mathematics"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Aristotelian realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","Middle Ages and Renaissance"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century"],"downwards":["Real analysis","Group theory","Non-Euclidean geometries","Projective geometry","Boolean algebra and logic","Peano arithmetic"]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Real analysis"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Group theory"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Non-Euclidean geometries"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Projective geometry"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Boolean algebra and logic"],"downwards":[]}
{"upwards":["Foundations of mathematics","Historical context","19th century","Peano arithmetic"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis"],"downwards":["Philosophical views","More paradoxes"]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views"],"downwards":["Formalism","Intuitionism","Logicism","Set-theoretic Platonism","Indispensability argument for realism","Rough-and-ready realism","Philosophical consequences of G\u00f6del's completeness theorem"]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Formalism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Intuitionism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Logicism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Set-theoretic Platonism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Indispensability argument for realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Rough-and-ready realism"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","Philosophical views","Philosophical consequences of G\u00f6del's completeness theorem"],"downwards":[]}
{"upwards":["Foundations of mathematics","Foundational crisis","More paradoxes"],"downwards":[]}
{"upwards":["Foundations of mathematics","Toward resolution of the crisis"],"downwards":[]}
{"upwards":["Uncountable set"],"downwards":["Characterizations","Properties","Examples","Without the axiom of choice"]}
{"upwards":["Uncountable set","Characterizations"],"downwards":[]}
{"upwards":["Uncountable set","Properties"],"downwards":[]}
{"upwards":["Uncountable set","Examples"],"downwards":[]}
{"upwards":["Uncountable set","Without the axiom of choice"],"downwards":[]}
{"upwards":["Partition of a set"],"downwards":["Definition and notation","Examples","Partitions and equivalence relations","Refinement of partitions","Noncrossing partitions","Counting partitions"]}
{"upwards":["Partition of a set","Definition and notation"],"downwards":[]}
{"upwards":["Partition of a set","Examples"],"downwards":[]}
{"upwards":["Partition of a set","Partitions and equivalence relations"],"downwards":[]}
{"upwards":["Partition of a set","Refinement of partitions"],"downwards":[]}
{"upwards":["Partition of a set","Noncrossing partitions"],"downwards":[]}
{"upwards":["Partition of a set","Counting partitions"],"downwards":[]}
{"upwards":["Map (mathematics)"],"downwards":["Maps as functions","As morphisms"]}
{"upwards":["Map (mathematics)","Maps as functions"],"downwards":[]}
{"upwards":["Map (mathematics)","As morphisms"],"downwards":[]}
{"upwards":["Continuum hypothesis"],"downwards":["History","Cardinality of infinite sets","Independence from ZFC","Arguments for and against the continuum hypothesis","The generalized continuum hypothesis"]}
{"upwards":["Continuum hypothesis","History"],"downwards":[]}
{"upwards":["Continuum hypothesis","Cardinality of infinite sets"],"downwards":[]}
{"upwards":["Continuum hypothesis","Independence from ZFC"],"downwards":[]}
{"upwards":["Continuum hypothesis","Arguments for and against the continuum hypothesis"],"downwards":[]}
{"upwards":["Continuum hypothesis","The generalized continuum hypothesis"],"downwards":["Implications of GCH for cardinal exponentiation"]}
{"upwards":["Continuum hypothesis","The generalized continuum hypothesis","Implications of GCH for cardinal exponentiation"],"downwards":[]}
{"upwards":["Uncountable set"],"downwards":["Characterizations","Properties","Examples","Without the axiom of choice"]}
{"upwards":["Uncountable set","Characterizations"],"downwards":[]}
{"upwards":["Uncountable set","Properties"],"downwards":[]}
{"upwards":["Uncountable set","Examples"],"downwards":[]}
{"upwards":["Uncountable set","Without the axiom of choice"],"downwards":[]}
{"upwards":["Euclidean space"],"downwards":["Definition","Prototypical examples","Affine structure","Metric structure","Isometries","Topology","Axiomatic definitions","Usage","Other geometric spaces"]}
{"upwards":["Euclidean space","Definition"],"downwards":["History of the definition","Motivation of the modern definition","Technical definition"]}
{"upwards":["Euclidean space","Definition","History of the definition"],"downwards":[]}
{"upwards":["Euclidean space","Definition","Motivation of the modern definition"],"downwards":[]}
{"upwards":["Euclidean space","Definition","Technical definition"],"downwards":[]}
{"upwards":["Euclidean space","Prototypical examples"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure"],"downwards":["Subspaces","Lines and segments","Parallelism"]}
{"upwards":["Euclidean space","Affine structure","Subspaces"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure","Lines and segments"],"downwards":[]}
{"upwards":["Euclidean space","Affine structure","Parallelism"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure"],"downwards":["Distance and length","Orthogonality","Angle","Cartesian coordinates","Other coordinates"]}
{"upwards":["Euclidean space","Metric structure","Distance and length"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Orthogonality"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Angle"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Cartesian coordinates"],"downwards":[]}
{"upwards":["Euclidean space","Metric structure","Other coordinates"],"downwards":[]}
{"upwards":["Euclidean space","Isometries"],"downwards":["Isometry with prototypical examples","Euclidean group"]}
{"upwards":["Euclidean space","Isometries","Isometry with prototypical examples"],"downwards":[]}
{"upwards":["Euclidean space","Isometries","Euclidean group"],"downwards":[]}
{"upwards":["Euclidean space","Topology"],"downwards":[]}
{"upwards":["Euclidean space","Axiomatic definitions"],"downwards":[]}
{"upwards":["Euclidean space","Usage"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces"],"downwards":["Affine space","Projective space","Non-Euclidean geometries","Curved spaces","Pseudo-Euclidean space"]}
{"upwards":["Euclidean space","Other geometric spaces","Affine space"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Projective space"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Non-Euclidean geometries"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Curved spaces"],"downwards":[]}
{"upwards":["Euclidean space","Other geometric spaces","Pseudo-Euclidean space"],"downwards":[]}
{"upwards":["Aleph number"],"downwards":["Aleph-nought","Aleph-one","Continuum hypothesis","Aleph-omega","Aleph-\u03b1 for general \u03b1","Fixed points of omega","Role of axiom of choice"]}
{"upwards":["Aleph number","Aleph-nought"],"downwards":[]}
{"upwards":["Aleph number","Aleph-one"],"downwards":[]}
{"upwards":["Aleph number","Continuum hypothesis"],"downwards":[]}
{"upwards":["Aleph number","Aleph-omega"],"downwards":[]}
{"upwards":["Aleph number","Aleph-\u03b1 for general \u03b1"],"downwards":[]}
{"upwards":["Aleph number","Fixed points of omega"],"downwards":[]}
{"upwards":["Aleph number","Role of axiom of choice"],"downwards":[]}
{"upwards":["Injective function"],"downwards":["Definition","Examples","Injections can be undone","Injections may be made invertible","Other properties","Proving that functions are injective","Gallery"]}
{"upwards":["Injective function","Definition"],"downwards":[]}
{"upwards":["Injective function","Examples"],"downwards":[]}
{"upwards":["Injective function","Injections can be undone"],"downwards":[]}
{"upwards":["Injective function","Injections may be made invertible"],"downwards":[]}
{"upwards":["Injective function","Other properties"],"downwards":[]}
{"upwards":["Injective function","Proving that functions are injective"],"downwards":[]}
{"upwards":["Injective function","Gallery"],"downwards":[]}
{"upwards":["Axiom of choice"],"downwards":["Statement","Usage","Examples","Criticism and acceptance","In constructive mathematics","Independence","Stronger axioms","Equivalents","Weaker forms","Stronger forms of the negation of AC","Statements consistent with the negation of AC","Axiom of choice in type theory","Quotations","Notes","References","External links"]}
{"upwards":["Axiom of choice","Statement"],"downwards":["Nomenclature ZF, AC, and ZFC","Variants","Restriction to finite sets"]}
{"upwards":["Axiom of choice","Statement","Nomenclature ZF, AC, and ZFC"],"downwards":[]}
{"upwards":["Axiom of choice","Statement","Variants"],"downwards":[]}
{"upwards":["Axiom of choice","Statement","Restriction to finite sets"],"downwards":[]}
{"upwards":["Axiom of choice","Usage"],"downwards":[]}
{"upwards":["Axiom of choice","Examples"],"downwards":[]}
{"upwards":["Axiom of choice","Criticism and acceptance"],"downwards":[]}
{"upwards":["Axiom of choice","In constructive mathematics"],"downwards":[]}
{"upwards":["Axiom of choice","Independence"],"downwards":[]}
{"upwards":["Axiom of choice","Stronger axioms"],"downwards":[]}
{"upwards":["Axiom of choice","Equivalents"],"downwards":["Category theory"]}
{"upwards":["Axiom of choice","Equivalents","Category theory"],"downwards":[]}
{"upwards":["Axiom of choice","Weaker forms"],"downwards":["Results requiring AC (or weaker forms) but weaker than it","Possibly equivalent implications of AC"]}
{"upwards":["Axiom of choice","Weaker forms","Results requiring AC (or weaker forms) but weaker than it"],"downwards":[]}
{"upwards":["Axiom of choice","Weaker forms","Possibly equivalent implications of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Stronger forms of the negation of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Statements consistent with the negation of AC"],"downwards":[]}
{"upwards":["Axiom of choice","Axiom of choice in type theory"],"downwards":[]}
{"upwards":["Axiom of choice","Quotations"],"downwards":[]}
{"upwards":["Axiom of choice","Notes"],"downwards":[]}
{"upwards":["Axiom of choice","References"],"downwards":[]}
{"upwards":["Axiom of choice","External links"],"downwards":[]}
{"upwards":["Complement (set theory)"],"downwards":["Absolute complement","Relative complement","Complementary relation","LaTeX notation","In programming languages"]}
{"upwards":["Complement (set theory)","Absolute complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Absolute complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Relative complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Complementary relation"],"downwards":[]}
{"upwards":["Complement (set theory)","LaTeX notation"],"downwards":[]}
{"upwards":["Complement (set theory)","In programming languages"],"downwards":[]}
{"upwards":["Disjoint sets"],"downwards":["Generalizations","Intersections","Disjoint unions and partitions"]}
{"upwards":["Disjoint sets","Generalizations"],"downwards":[]}
{"upwards":["Disjoint sets","Intersections"],"downwards":[]}
{"upwards":["Disjoint sets","Disjoint unions and partitions"],"downwards":[]}
{"upwards":["Symmetric difference"],"downwards":["Properties","n-ary symmetric difference","Symmetric difference on measure spaces","Hausdorff distance vs. symmetric difference"]}
{"upwards":["Symmetric difference","Properties"],"downwards":[]}
{"upwards":["Symmetric difference","n-ary symmetric difference"],"downwards":[]}
{"upwards":["Symmetric difference","Symmetric difference on measure spaces"],"downwards":[]}
{"upwards":["Symmetric difference","Hausdorff distance vs. symmetric difference"],"downwards":[]}
{"upwards":["Algebra of sets"],"downwards":["Fundamentals","The fundamental properties of set algebra","The principle of duality","Some additional laws for unions and intersections","Some additional laws for complements","The algebra of inclusion","The algebra of relative complements"]}
{"upwards":["Algebra of sets","Fundamentals"],"downwards":[]}
{"upwards":["Algebra of sets","The fundamental properties of set algebra"],"downwards":[]}
{"upwards":["Algebra of sets","The principle of duality"],"downwards":[]}
{"upwards":["Algebra of sets","Some additional laws for unions and intersections"],"downwards":[]}
{"upwards":["Algebra of sets","Some additional laws for complements"],"downwards":[]}
{"upwards":["Algebra of sets","The algebra of inclusion"],"downwards":[]}
{"upwards":["Algebra of sets","The algebra of relative complements"],"downwards":[]}
{"upwards":["Relation (mathematics)"],"downwards":["Definition","Representation of relations","Properties of relations","Operations on relations","Examples","Generalizations"]}
{"upwards":["Relation (mathematics)","Definition"],"downwards":[]}
{"upwards":["Relation (mathematics)","Representation of relations"],"downwards":[]}
{"upwards":["Relation (mathematics)","Properties of relations"],"downwards":["Combinations of properties"]}
{"upwards":["Relation (mathematics)","Properties of relations","Combinations of properties"],"downwards":[]}
{"upwards":["Relation (mathematics)","Operations on relations"],"downwards":[]}
{"upwards":["Relation (mathematics)","Examples"],"downwards":[]}
{"upwards":["Relation (mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Universe (mathematics)"],"downwards":["In a specific context","In ordinary mathematics","In set theory","In predicate calculus","In category theory","In type theory"]}
{"upwards":["Universe (mathematics)","In a specific context"],"downwards":[]}
{"upwards":["Universe (mathematics)","In ordinary mathematics"],"downwards":[]}
{"upwards":["Universe (mathematics)","In set theory"],"downwards":[]}
{"upwards":["Universe (mathematics)","In predicate calculus"],"downwards":[]}
{"upwards":["Universe (mathematics)","In category theory"],"downwards":[]}
{"upwards":["Universe (mathematics)","In type theory"],"downwards":[]}
{"upwards":["Bijection"],"downwards":["Definition","Examples","More mathematical examples","Inverses","Composition","Cardinality","Properties","Category theory","Generalization to partial functions","Gallery"]}
{"upwards":["Bijection","Definition"],"downwards":[]}
{"upwards":["Bijection","Examples"],"downwards":["Batting line-up of a baseball or cricket team","Seats and students of a classroom"]}
{"upwards":["Bijection","Examples","Batting line-up of a baseball or cricket team"],"downwards":[]}
{"upwards":["Bijection","Examples","Seats and students of a classroom"],"downwards":[]}
{"upwards":["Bijection","More mathematical examples"],"downwards":[]}
{"upwards":["Bijection","Inverses"],"downwards":[]}
{"upwards":["Bijection","Composition"],"downwards":[]}
{"upwards":["Bijection","Cardinality"],"downwards":[]}
{"upwards":["Bijection","Properties"],"downwards":[]}
{"upwards":["Bijection","Category theory"],"downwards":[]}
{"upwards":["Bijection","Generalization to partial functions"],"downwards":[]}
{"upwards":["Bijection","Gallery"],"downwards":[]}
{"upwards":["Complement (set theory)"],"downwards":["Absolute complement","Relative complement","Complementary relation","LaTeX notation","In programming languages"]}
{"upwards":["Complement (set theory)","Absolute complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Absolute complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Absolute complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Complement (set theory)","Relative complement","Definition"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Examples"],"downwards":[]}
{"upwards":["Complement (set theory)","Relative complement","Properties"],"downwards":[]}
{"upwards":["Complement (set theory)","Complementary relation"],"downwards":[]}
{"upwards":["Complement (set theory)","LaTeX notation"],"downwards":[]}
{"upwards":["Complement (set theory)","In programming languages"],"downwards":[]}
{"upwards":["Cardinality"],"downwards":["History","Comparing sets","Cardinal numbers","Finite, countable and uncountable sets","Infinite sets","Examples and properties","Union and intersection"]}
{"upwards":["Cardinality","History"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets"],"downwards":["Definition 1: |A| = |B|","Definition 2: |A| \u2264 |B|","Definition 3: |A| < |B|"]}
{"upwards":["Cardinality","Comparing sets","Definition 1: |A| = |B|"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets","Definition 2: |A| \u2264 |B|"],"downwards":[]}
{"upwards":["Cardinality","Comparing sets","Definition 3: |A| < |B|"],"downwards":[]}
{"upwards":["Cardinality","Cardinal numbers"],"downwards":[]}
{"upwards":["Cardinality","Finite, countable and uncountable sets"],"downwards":[]}
{"upwards":["Cardinality","Infinite sets"],"downwards":["Cardinality of the continuum"]}
{"upwards":["Cardinality","Infinite sets","Cardinality of the continuum"],"downwards":[]}
{"upwards":["Cardinality","Examples and properties"],"downwards":[]}
{"upwards":["Cardinality","Union and intersection"],"downwards":[]}
{"upwards":["Dimension"],"downwards":["In mathematics","In physics","In computer graphics and spatial data","More dimensions","List of topics by dimension"]}
{"upwards":["Dimension","In mathematics"],"downwards":["Vector spaces","Manifolds","Varieties","Krull dimension","Topological spaces","Hausdorff dimension","Hilbert spaces"]}
{"upwards":["Dimension","In mathematics","Vector spaces"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Manifolds"],"downwards":["Complex dimension"]}
{"upwards":["Dimension","In mathematics","Manifolds","Complex dimension"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Varieties"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Krull dimension"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Topological spaces"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Hausdorff dimension"],"downwards":[]}
{"upwards":["Dimension","In mathematics","Hilbert spaces"],"downwards":[]}
{"upwards":["Dimension","In physics"],"downwards":["Spatial dimensions","Time","Additional dimensions"]}
{"upwards":["Dimension","In physics","Spatial dimensions"],"downwards":[]}
{"upwards":["Dimension","In physics","Time"],"downwards":[]}
{"upwards":["Dimension","In physics","Additional dimensions"],"downwards":[]}
{"upwards":["Dimension","In computer graphics and spatial data"],"downwards":[]}
{"upwards":["Dimension","More dimensions"],"downwards":[]}
{"upwards":["Dimension","List of topics by dimension"],"downwards":[]}
{"upwards":["Surjective function"],"downwards":["Definition","Examples","Properties","Space of surjections","Gallery"]}
{"upwards":["Surjective function","Definition"],"downwards":[]}
{"upwards":["Surjective function","Examples"],"downwards":[]}
{"upwards":["Surjective function","Properties"],"downwards":["Surjections as right invertible functions","Surjections as epimorphisms","Surjections as binary relations","Cardinality of the domain of a surjection","Composition and decomposition","Induced surjection and induced bijection"]}
{"upwards":["Surjective function","Properties","Surjections as right invertible functions"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as epimorphisms"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as binary relations"],"downwards":[]}
{"upwards":["Surjective function","Properties","Cardinality of the domain of a surjection"],"downwards":[]}
{"upwards":["Surjective function","Properties","Composition and decomposition"],"downwards":[]}
{"upwards":["Surjective function","Properties","Induced surjection and induced bijection"],"downwards":[]}
{"upwards":["Surjective function","Space of surjections"],"downwards":[]}
{"upwards":["Surjective function","Gallery"],"downwards":[]}
{"upwards":["Power set"],"downwards":["Example","Properties","Representing subsets as functions","Relation to binomial theorem","Recursive definition","Subsets of limited cardinality","Power object","Functors and quantifiers"]}
{"upwards":["Power set","Example"],"downwards":[]}
{"upwards":["Power set","Properties"],"downwards":[]}
{"upwards":["Power set","Representing subsets as functions"],"downwards":[]}
{"upwards":["Power set","Relation to binomial theorem"],"downwards":[]}
{"upwards":["Power set","Recursive definition"],"downwards":[]}
{"upwards":["Power set","Subsets of limited cardinality"],"downwards":[]}
{"upwards":["Power set","Power object"],"downwards":[]}
{"upwards":["Power set","Functors and quantifiers"],"downwards":[]}
{"upwards":["Colon (punctuation)"],"downwards":["History","Usage in English","Usage in other languages","Mathematics and logic","Computing"]}
{"upwards":["Colon (punctuation)","History"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English"],"downwards":["Syntactical-deductive","Syntactical-descriptive","Appositive","Segmental","Use of capitals","Spacing and parentheses"]}
{"upwards":["Colon (punctuation)","Usage in English","Syntactical-deductive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Syntactical-descriptive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Appositive"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Segmental"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Use of capitals"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in English","Spacing and parentheses"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages"],"downwards":["Suffix separator","Abbreviation mark","Word separator","End of sentence or verse","Score divider"]}
{"upwards":["Colon (punctuation)","Usage in other languages","Suffix separator"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Abbreviation mark"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Word separator"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","End of sentence or verse"],"downwards":[]}
{"upwards":["Colon (punctuation)","Usage in other languages","Score divider"],"downwards":[]}
{"upwards":["Colon (punctuation)","Mathematics and logic"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing"],"downwards":["Programming languages","Addresses","Text markup"]}
{"upwards":["Colon (punctuation)","Computing","Programming languages"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing","Addresses"],"downwards":[]}
{"upwards":["Colon (punctuation)","Computing","Text markup"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Boolean ring"],"downwards":["Notations","Examples","Relation to Boolean algebras","Properties of Boolean rings","Unification"]}
{"upwards":["Boolean ring","Notations"],"downwards":[]}
{"upwards":["Boolean ring","Examples"],"downwards":[]}
{"upwards":["Boolean ring","Relation to Boolean algebras"],"downwards":[]}
{"upwards":["Boolean ring","Properties of Boolean rings"],"downwards":[]}
{"upwards":["Boolean ring","Unification"],"downwards":[]}
{"upwards":["Ring (mathematics)"],"downwards":["Definition","Illustration","History","Basic examples","Basic concepts","Module","Constructions","Special kinds of rings","Rings with extra structure","Some examples of the ubiquity of rings","Category-theoretic description","Generalization","Other ring-like objects"]}
{"upwards":["Ring (mathematics)","Definition"],"downwards":["Notes on the definition"]}
{"upwards":["Ring (mathematics)","Definition","Notes on the definition"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration"],"downwards":["Some properties","Example: Integers modulo 4","Example: 2-by-2 matrices"]}
{"upwards":["Ring (mathematics)","Illustration","Some properties"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration","Example: Integers modulo 4"],"downwards":[]}
{"upwards":["Ring (mathematics)","Illustration","Example: 2-by-2 matrices"],"downwards":[]}
{"upwards":["Ring (mathematics)","History"],"downwards":["Dedekind","Hilbert","Fraenkel and Noether","Multiplicative identity and the term \"ring\""]}
{"upwards":["Ring (mathematics)","History","Dedekind"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Hilbert"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Fraenkel and Noether"],"downwards":[]}
{"upwards":["Ring (mathematics)","History","Multiplicative identity and the term \"ring\""],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples"],"downwards":["Commutative rings","Noncommutative rings","Non-rings"]}
{"upwards":["Ring (mathematics)","Basic examples","Commutative rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples","Noncommutative rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic examples","Non-rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts"],"downwards":["Products and powers","Elements in a ring","Subring","Ideal","Homomorphism","Quotient ring"]}
{"upwards":["Ring (mathematics)","Basic concepts","Products and powers"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Elements in a ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Subring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Ideal"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Homomorphism"],"downwards":[]}
{"upwards":["Ring (mathematics)","Basic concepts","Quotient ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Module"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions"],"downwards":["Direct product","Polynomial ring","Matrix ring and endomorphism ring","Limits and colimits of rings","Localization","Completion","Rings with generators and relations"]}
{"upwards":["Ring (mathematics)","Constructions","Direct product"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Polynomial ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Matrix ring and endomorphism ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Limits and colimits of rings"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Localization"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Completion"],"downwards":[]}
{"upwards":["Ring (mathematics)","Constructions","Rings with generators and relations"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings"],"downwards":["Domains","Division ring","Semisimple rings","Central simple algebra and Brauer group","Valuation ring"]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Domains"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Division ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings"],"downwards":["Examples","Properties"]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings","Examples"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Semisimple rings","Properties"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Central simple algebra and Brauer group"],"downwards":[]}
{"upwards":["Ring (mathematics)","Special kinds of rings","Valuation ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Rings with extra structure"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings"],"downwards":["Cohomology ring of a topological space","Burnside ring of a group","Representation ring of a group ring","Function field of an irreducible algebraic variety","Face ring of a simplicial complex"]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Cohomology ring of a topological space"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Burnside ring of a group"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Representation ring of a group ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Function field of an irreducible algebraic variety"],"downwards":[]}
{"upwards":["Ring (mathematics)","Some examples of the ubiquity of rings","Face ring of a simplicial complex"],"downwards":[]}
{"upwards":["Ring (mathematics)","Category-theoretic description"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization"],"downwards":["Rng","Nonassociative ring","Semiring"]}
{"upwards":["Ring (mathematics)","Generalization","Rng"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization","Nonassociative ring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Generalization","Semiring"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects"],"downwards":["Ring object in a category","Ring scheme","Ring spectrum"]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring object in a category"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring scheme"],"downwards":[]}
{"upwards":["Ring (mathematics)","Other ring-like objects","Ring spectrum"],"downwards":[]}
{"upwards":["Closure (mathematics)"],"downwards":["Definitions","In topology","Binary relations","Other examples","Closure operator","Notes","References"]}
{"upwards":["Closure (mathematics)","Definitions"],"downwards":["In algebraic structures"]}
{"upwards":["Closure (mathematics)","Definitions","In algebraic structures"],"downwards":[]}
{"upwards":["Closure (mathematics)","In topology"],"downwards":[]}
{"upwards":["Closure (mathematics)","Binary relations"],"downwards":[]}
{"upwards":["Closure (mathematics)","Other examples"],"downwards":[]}
{"upwards":["Closure (mathematics)","Closure operator"],"downwards":["Closure operator vs. closed sets"]}
{"upwards":["Closure (mathematics)","Closure operator","Closure operator vs. closed sets"],"downwards":[]}
{"upwards":["Closure (mathematics)","Notes"],"downwards":[]}
{"upwards":["Closure (mathematics)","References"],"downwards":[]}
{"upwards":["Domain of a function"],"downwards":["Natural domain","Other uses","Set theoretical notions"]}
{"upwards":["Domain of a function","Natural domain"],"downwards":["Examples"]}
{"upwards":["Domain of a function","Natural domain","Examples"],"downwards":[]}
{"upwards":["Domain of a function","Other uses"],"downwards":[]}
{"upwards":["Domain of a function","Set theoretical notions"],"downwards":[]}
{"upwards":["Codomain"],"downwards":["Examples"]}
{"upwards":["Codomain","Examples"],"downwards":[]}
{"upwards":["Rock paper scissors"],"downwards":["Etymology","Gameplay","History","Strategies","Variations","Analogues in real life","Instances of usage","Tournaments"]}
{"upwards":["Rock paper scissors","Etymology"],"downwards":[]}
{"upwards":["Rock paper scissors","Gameplay"],"downwards":[]}
{"upwards":["Rock paper scissors","History"],"downwards":["Origins","Spread beyond East Asia"]}
{"upwards":["Rock paper scissors","History","Origins"],"downwards":[]}
{"upwards":["Rock paper scissors","History","Spread beyond East Asia"],"downwards":[]}
{"upwards":["Rock paper scissors","Strategies"],"downwards":["Algorithms"]}
{"upwards":["Rock paper scissors","Strategies","Algorithms"],"downwards":[]}
{"upwards":["Rock paper scissors","Variations"],"downwards":["Adapted rules","Different weapons","Additional weapons"]}
{"upwards":["Rock paper scissors","Variations","Adapted rules"],"downwards":[]}
{"upwards":["Rock paper scissors","Variations","Different weapons"],"downwards":[]}
{"upwards":["Rock paper scissors","Variations","Additional weapons"],"downwards":[]}
{"upwards":["Rock paper scissors","Analogues in real life"],"downwards":["Lizard mating strategies","Bacteria","Mechanical devices and geometrical constructions"]}
{"upwards":["Rock paper scissors","Analogues in real life","Lizard mating strategies"],"downwards":[]}
{"upwards":["Rock paper scissors","Analogues in real life","Bacteria"],"downwards":[]}
{"upwards":["Rock paper scissors","Analogues in real life","Mechanical devices and geometrical constructions"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage"],"downwards":["American court case","Auction house selection","FA Women's Super League match","Play by chimpanzees","Game design"]}
{"upwards":["Rock paper scissors","Instances of usage","American court case"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","Auction house selection"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","FA Women's Super League match"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","Play by chimpanzees"],"downwards":[]}
{"upwards":["Rock paper scissors","Instances of usage","Game design"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments"],"downwards":["World Rock Paper Scissors Association","World Rock Paper Scissors Society","UK championships","USARPS tournaments","National XtremeRPS Competition 2007\u20132008","Guinness Book of World Records","World Series","Jackpot En Poy of Eat Bulaga!"]}
{"upwards":["Rock paper scissors","Tournaments","World Rock Paper Scissors Association"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","World Rock Paper Scissors Society"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","UK championships"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","USARPS tournaments"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","National XtremeRPS Competition 2007\u20132008"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","Guinness Book of World Records"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","World Series"],"downwards":[]}
{"upwards":["Rock paper scissors","Tournaments","Jackpot En Poy of Eat Bulaga!"],"downwards":[]}
{"upwards":["Ostensive definition"],"downwards":["Overview"]}
{"upwards":["Ostensive definition","Overview"],"downwards":[]}
{"upwards":["Blackboard bold"],"downwards":["History","Encoding","Usage"]}
{"upwards":["Blackboard bold","History"],"downwards":["Origin","Use in textbooks"]}
{"upwards":["Blackboard bold","History","Origin"],"downwards":[]}
{"upwards":["Blackboard bold","History","Use in textbooks"],"downwards":[]}
{"upwards":["Blackboard bold","Encoding"],"downwards":[]}
{"upwards":["Blackboard bold","Usage"],"downwards":[]}
{"upwards":["Alternative set theory"],"downwards":["Vop\u011bnka's Alternative Set Theory","Other alternative set theories"]}
{"upwards":["Alternative set theory","Vop\u011bnka's Alternative Set Theory"],"downwards":[]}
{"upwards":["Alternative set theory","Other alternative set theories"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle"],"downwards":["Formula","Examples","A special case","Formula generalization","In probability","Other formulas","Applications","Diluted inclusion\u2013exclusion principle","Proof of main statement"]}
{"upwards":["Inclusion\u2013exclusion principle","Formula"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Examples"],"downwards":["Counting integers","Counting derangements"]}
{"upwards":["Inclusion\u2013exclusion principle","Examples","Counting integers"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Examples","Counting derangements"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","A special case"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Formula generalization"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","In probability"],"downwards":["Special case"]}
{"upwards":["Inclusion\u2013exclusion principle","In probability","Special case"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Other formulas"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications"],"downwards":["Counting derangements","Counting intersections","Graph coloring","Bipartite graph perfect matchings","Number of onto functions","Permutations with forbidden positions","Stirling numbers of the second kind","Rook polynomials","Euler's phi function"]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Counting derangements"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Counting intersections"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Graph coloring"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Bipartite graph perfect matchings"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Number of onto functions"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Permutations with forbidden positions"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Stirling numbers of the second kind"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Rook polynomials"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Applications","Euler's phi function"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Diluted inclusion\u2013exclusion principle"],"downwards":[]}
{"upwards":["Inclusion\u2013exclusion principle","Proof of main statement"],"downwards":["Algebraic proof"]}
{"upwards":["Inclusion\u2013exclusion principle","Proof of main statement","Algebraic proof"],"downwards":[]}
{"upwards":["Category of sets"],"downwards":["Properties of the category of sets","Foundations for the category of sets"]}
{"upwards":["Category of sets","Properties of the category of sets"],"downwards":[]}
{"upwards":["Category of sets","Foundations for the category of sets"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Plane (mathematics)"],"downwards":["Euclidean plane","Elliptic plane","Projective plane","Further generalizations","Topological and differential geometric notions"]}
{"upwards":["Plane (mathematics)","Euclidean plane"],"downwards":["Embedding in three-dimensional space"]}
{"upwards":["Plane (mathematics)","Euclidean plane","Embedding in three-dimensional space"],"downwards":[]}
{"upwards":["Plane (mathematics)","Elliptic plane"],"downwards":[]}
{"upwards":["Plane (mathematics)","Projective plane"],"downwards":[]}
{"upwards":["Plane (mathematics)","Further generalizations"],"downwards":[]}
{"upwards":["Plane (mathematics)","Topological and differential geometric notions"],"downwards":[]}
{"upwards":["Dense set"],"downwards":["Definition","Examples","Properties","Related notions"]}
{"upwards":["Dense set","Definition"],"downwards":["Density in metric spaces"]}
{"upwards":["Dense set","Definition","Density in metric spaces"],"downwards":[]}
{"upwards":["Dense set","Examples"],"downwards":[]}
{"upwards":["Dense set","Properties"],"downwards":[]}
{"upwards":["Dense set","Related notions"],"downwards":[]}
{"upwards":["Family of sets"],"downwards":["Examples","Properties","Related concepts","Special types of set families"]}
{"upwards":["Family of sets","Examples"],"downwards":[]}
{"upwards":["Family of sets","Properties"],"downwards":[]}
{"upwards":["Family of sets","Related concepts"],"downwards":["Covers and topologies"]}
{"upwards":["Family of sets","Related concepts","Covers and topologies"],"downwards":[]}
{"upwards":["Family of sets","Special types of set families"],"downwards":[]}
{"upwards":["Fuzzy set"],"downwards":["Definition","Fuzzy logic","Fuzzy number and only number","Fuzzy categories","Fuzzy relation equation","Entropy","Extensions"]}
{"upwards":["Fuzzy set","Definition"],"downwards":["Crisp sets related to a fuzzy set","Other definitions","Fuzzy set operations","Disjoint fuzzy sets","Scalar cardinality","Distance and similarity","L-fuzzy sets","Neutrosophic fuzzy sets","Pythagorean fuzzy sets"]}
{"upwards":["Fuzzy set","Definition","Crisp sets related to a fuzzy set"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Other definitions"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Fuzzy set operations"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Disjoint fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Scalar cardinality"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Distance and similarity"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","L-fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Neutrosophic fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Definition","Pythagorean fuzzy sets"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy logic"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy number and only number"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy categories"],"downwards":[]}
{"upwards":["Fuzzy set","Fuzzy relation equation"],"downwards":[]}
{"upwards":["Fuzzy set","Entropy"],"downwards":[]}
{"upwards":["Fuzzy set","Extensions"],"downwards":[]}
{"upwards":["Internal set"],"downwards":["Internal sets in the ultrapower construction","Internal subsets of the reals"]}
{"upwards":["Internal set","Internal sets in the ultrapower construction"],"downwards":[]}
{"upwards":["Internal set","Internal subsets of the reals"],"downwards":[]}
{"upwards":["Flag of France"],"downwards":["Design","Symbolism","History","Regimental flags","Naval flags","Colonial flags","Other","Gallery"]}
{"upwards":["Flag of France","Design"],"downwards":[]}
{"upwards":["Flag of France","Symbolism"],"downwards":[]}
{"upwards":["Flag of France","History"],"downwards":["Kingdom of France","The Tricolore"]}
{"upwards":["Flag of France","History","Kingdom of France"],"downwards":[]}
{"upwards":["Flag of France","History","The Tricolore"],"downwards":[]}
{"upwards":["Flag of France","Regimental flags"],"downwards":[]}
{"upwards":["Flag of France","Naval flags"],"downwards":[]}
{"upwards":["Flag of France","Colonial flags"],"downwards":[]}
{"upwards":["Flag of France","Other"],"downwards":[]}
{"upwards":["Flag of France","Gallery"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems"],"downwards":["Formal systems: completeness, consistency, and effective axiomatization","First incompleteness theorem","Second incompleteness theorem","Examples of undecidable statements","Relationship with computability","Proof sketch for the first theorem","Proof sketch for the second theorem","Discussion and implications","History"]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization"],"downwards":["Effective axiomatization","Completeness","Consistency","Systems which contain arithmetic","Conflicting goals"]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Effective axiomatization"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Completeness"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Consistency"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Systems which contain arithmetic"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Formal systems: completeness, consistency, and effective axiomatization","Conflicting goals"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem"],"downwards":["Syntactic form of the G\u00f6del sentence","Truth of the G\u00f6del sentence","Relationship with the liar paradox","Extensions of G\u00f6del's original result"]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Syntactic form of the G\u00f6del sentence"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Truth of the G\u00f6del sentence"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Relationship with the liar paradox"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","First incompleteness theorem","Extensions of G\u00f6del's original result"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem"],"downwards":["Expressing consistency","The Hilbert\u2013Bernays conditions","Implications for consistency proofs"]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","Expressing consistency"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","The Hilbert\u2013Bernays conditions"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Second incompleteness theorem","Implications for consistency proofs"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Examples of undecidable statements"],"downwards":["Undecidable statements provable in larger systems"]}
{"upwards":["G\u00f6del's incompleteness theorems","Examples of undecidable statements","Undecidable statements provable in larger systems"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Relationship with computability"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem"],"downwards":["Arithmetization of syntax","Construction of a statement about \"provability\"","Diagonalization","Proof via Berry's paradox","Computer verified proofs"]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Arithmetization of syntax"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Construction of a statement about \"provability\""],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Diagonalization"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Proof via Berry's paradox"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the first theorem","Computer verified proofs"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Proof sketch for the second theorem"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications"],"downwards":["Consequences for logicism and Hilbert's second problem","Minds and machines","Paraconsistent logic","Appeals to the incompleteness theorems in other fields"]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Consequences for logicism and Hilbert's second problem"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Minds and machines"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Paraconsistent logic"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","Discussion and implications","Appeals to the incompleteness theorems in other fields"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History"],"downwards":["Announcement","Generalization and acceptance","Criticisms"]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Announcement"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Generalization and acceptance"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms"],"downwards":["Finsler","Zermelo","Wittgenstein"]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Finsler"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Zermelo"],"downwards":[]}
{"upwards":["G\u00f6del's incompleteness theorems","History","Criticisms","Wittgenstein"],"downwards":[]}
{"upwards":["Multiset"],"downwards":["History","Examples","Definition","Basic properties and operations","Counting multisets","Applications","Generalizations"]}
{"upwards":["Multiset","History"],"downwards":[]}
{"upwards":["Multiset","Examples"],"downwards":[]}
{"upwards":["Multiset","Definition"],"downwards":[]}
{"upwards":["Multiset","Basic properties and operations"],"downwards":[]}
{"upwards":["Multiset","Counting multisets"],"downwards":["Recurrence relation","Generating series","Generalization and connection to the negative binomial series"]}
{"upwards":["Multiset","Counting multisets","Recurrence relation"],"downwards":[]}
{"upwards":["Multiset","Counting multisets","Generating series"],"downwards":[]}
{"upwards":["Multiset","Counting multisets","Generalization and connection to the negative binomial series"],"downwards":[]}
{"upwards":["Multiset","Applications"],"downwards":[]}
{"upwards":["Multiset","Generalizations"],"downwards":[]}
{"upwards":["De Morgan's laws"],"downwards":["Formal notation","History","Informal proof","Formal proof","Generalising De Morgan duality","Extension to predicate and modal logic","In intuitionistic logic","In computer engineering"]}
{"upwards":["De Morgan's laws","Formal notation"],"downwards":["Substitution form","Set theory and Boolean algebra","Engineering","Text searching"]}
{"upwards":["De Morgan's laws","Formal notation","Substitution form"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Set theory and Boolean algebra"],"downwards":["Unions and intersections of any number of sets"]}
{"upwards":["De Morgan's laws","Formal notation","Set theory and Boolean algebra","Unions and intersections of any number of sets"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Engineering"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal notation","Text searching"],"downwards":[]}
{"upwards":["De Morgan's laws","History"],"downwards":[]}
{"upwards":["De Morgan's laws","Informal proof"],"downwards":["Negation of a disjunction","Negation of a conjunction"]}
{"upwards":["De Morgan's laws","Informal proof","Negation of a disjunction"],"downwards":[]}
{"upwards":["De Morgan's laws","Informal proof","Negation of a conjunction"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof"],"downwards":["Part 1","Part 2","Conclusion"]}
{"upwards":["De Morgan's laws","Formal proof","Part 1"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof","Part 2"],"downwards":[]}
{"upwards":["De Morgan's laws","Formal proof","Conclusion"],"downwards":[]}
{"upwards":["De Morgan's laws","Generalising De Morgan duality"],"downwards":[]}
{"upwards":["De Morgan's laws","Extension to predicate and modal logic"],"downwards":[]}
{"upwards":["De Morgan's laws","In intuitionistic logic"],"downwards":[]}
{"upwards":["De Morgan's laws","In computer engineering"],"downwards":[]}
{"upwards":["Bertrand Russell"],"downwards":["Biography","Views","Selected works"]}
{"upwards":["Bertrand Russell","Biography"],"downwards":["Early life and background","Childhood and adolescence","University and first marriage","Early career","First World War","G. H. Hardy on the Trinity controversy","Between the wars","Second World War","Later life","Political causes","Final years, death and legacy","Titles and honours from birth"]}
{"upwards":["Bertrand Russell","Biography","Early life and background"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Childhood and adolescence"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","University and first marriage"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Early career"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","First World War"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","G. H. Hardy on the Trinity controversy"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Between the wars"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Second World War"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Later life"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Political causes"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Final years, death and legacy"],"downwards":["Marriages and issue"]}
{"upwards":["Bertrand Russell","Biography","Final years, death and legacy","Marriages and issue"],"downwards":[]}
{"upwards":["Bertrand Russell","Biography","Titles and honours from birth"],"downwards":[]}
{"upwards":["Bertrand Russell","Views"],"downwards":["Philosophy","Religion","Society","Freedom of opinion and expression","Education"]}
{"upwards":["Bertrand Russell","Views","Philosophy"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Religion"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Society"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Freedom of opinion and expression"],"downwards":[]}
{"upwards":["Bertrand Russell","Views","Education"],"downwards":[]}
{"upwards":["Bertrand Russell","Selected works"],"downwards":[]}
{"upwards":["Georg Cantor"],"downwards":["Biography","Mathematical work","Philosophy, religion, literature and Cantor's mathematics","Cantor's ancestry","Biographies"]}
{"upwards":["Georg Cantor","Biography"],"downwards":["Youth and studies","Teacher and researcher","Later years and death"]}
{"upwards":["Georg Cantor","Biography","Youth and studies"],"downwards":[]}
{"upwards":["Georg Cantor","Biography","Teacher and researcher"],"downwards":[]}
{"upwards":["Georg Cantor","Biography","Later years and death"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work"],"downwards":["Number theory, trigonometric series and ordinals","Set theory"]}
{"upwards":["Georg Cantor","Mathematical work","Number theory, trigonometric series and ordinals"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory"],"downwards":["One-to-one correspondence","Continuum hypothesis","Absolute infinite, well-ordering theorem, and paradoxes"]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","One-to-one correspondence"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","Continuum hypothesis"],"downwards":[]}
{"upwards":["Georg Cantor","Mathematical work","Set theory","Absolute infinite, well-ordering theorem, and paradoxes"],"downwards":[]}
{"upwards":["Georg Cantor","Philosophy, religion, literature and Cantor's mathematics"],"downwards":[]}
{"upwards":["Georg Cantor","Cantor's ancestry"],"downwards":[]}
{"upwards":["Georg Cantor","Biographies"],"downwards":[]}
{"upwards":["Principia Mathematica"],"downwards":["Scope of foundations laid","Theoretical basis","Ramified types and the axiom of reducibility","Notation","Consistency and criticisms","Contents","Comparison with set theory","Differences between editions","Editions"]}
{"upwards":["Principia Mathematica","Scope of foundations laid"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis"],"downwards":["Contemporary construction of a formal theory","Construction","Primitive ideas","Primitive propositions"]}
{"upwards":["Principia Mathematica","Theoretical basis","Contemporary construction of a formal theory"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Construction"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Primitive ideas"],"downwards":[]}
{"upwards":["Principia Mathematica","Theoretical basis","Primitive propositions"],"downwards":[]}
{"upwards":["Principia Mathematica","Ramified types and the axiom of reducibility"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation"],"downwards":["An introduction to the notation of \"Section A Mathematical Logic\" (formulas \u27381\u2013\u27385.71)","An introduction to the notation of \"Section B Theory of Apparent Variables\" (formulas \u27388\u2013\u273814.34)","Introduction to the notation of the theory of classes and relations"]}
{"upwards":["Principia Mathematica","Notation","An introduction to the notation of \"Section A Mathematical Logic\" (formulas \u27381\u2013\u27385.71)"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation","An introduction to the notation of \"Section B Theory of Apparent Variables\" (formulas \u27388\u2013\u273814.34)"],"downwards":[]}
{"upwards":["Principia Mathematica","Notation","Introduction to the notation of the theory of classes and relations"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms"],"downwards":["G\u00f6del 1930, 1931","Wittgenstein 1919, 1939","G\u00f6del 1944"]}
{"upwards":["Principia Mathematica","Consistency and criticisms","G\u00f6del 1930, 1931"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms","Wittgenstein 1919, 1939"],"downwards":[]}
{"upwards":["Principia Mathematica","Consistency and criticisms","G\u00f6del 1944"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents"],"downwards":["Part I Mathematical logic. Volume I \u27381 to \u273843","Part II Prolegomena to cardinal arithmetic. Volume I \u273850 to \u273897","Part III Cardinal arithmetic. Volume II \u2738100 to \u2738126","Part IV Relation-arithmetic. Volume II \u2738150 to \u2738186","Part V Series. Volume II \u2738200 to \u2738234 and volume III \u2738250 to \u2738276","Part VI Quantity. Volume III \u2738300 to \u2738375"]}
{"upwards":["Principia Mathematica","Contents","Part I Mathematical logic. Volume I \u27381 to \u273843"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part II Prolegomena to cardinal arithmetic. Volume I \u273850 to \u273897"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part III Cardinal arithmetic. Volume II \u2738100 to \u2738126"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part IV Relation-arithmetic. Volume II \u2738150 to \u2738186"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part V Series. Volume II \u2738200 to \u2738234 and volume III \u2738250 to \u2738276"],"downwards":[]}
{"upwards":["Principia Mathematica","Contents","Part VI Quantity. Volume III \u2738300 to \u2738375"],"downwards":[]}
{"upwards":["Principia Mathematica","Comparison with set theory"],"downwards":[]}
{"upwards":["Principia Mathematica","Differences between editions"],"downwards":[]}
{"upwards":["Principia Mathematica","Editions"],"downwards":[]}
{"upwards":["Integer"],"downwards":["History","Algebraic properties","Order-theoretic properties","Construction","Computer science","Cardinality"]}
{"upwards":["Integer","History"],"downwards":[]}
{"upwards":["Integer","Algebraic properties"],"downwards":[]}
{"upwards":["Integer","Order-theoretic properties"],"downwards":[]}
{"upwards":["Integer","Construction"],"downwards":["Traditional development","Equivalence classes of ordered pairs","Other approaches"]}
{"upwards":["Integer","Construction","Traditional development"],"downwards":[]}
{"upwards":["Integer","Construction","Equivalence classes of ordered pairs"],"downwards":[]}
{"upwards":["Integer","Construction","Other approaches"],"downwards":[]}
{"upwards":["Integer","Computer science"],"downwards":[]}
{"upwards":["Integer","Cardinality"],"downwards":[]}
{"upwards":["Mereology"],"downwards":["History","Axioms and primitive notions","Various systems","Set theory","Mathematics","Natural language","Metaphysics","Important surveys"]}
{"upwards":["Mereology","History"],"downwards":[]}
{"upwards":["Mereology","Axioms and primitive notions"],"downwards":[]}
{"upwards":["Mereology","Various systems"],"downwards":[]}
{"upwards":["Mereology","Set theory"],"downwards":[]}
{"upwards":["Mereology","Mathematics"],"downwards":[]}
{"upwards":["Mereology","Natural language"],"downwards":[]}
{"upwards":["Mereology","Metaphysics"],"downwards":["Mereological constitution","Mereological composition"]}
{"upwards":["Mereology","Metaphysics","Mereological constitution"],"downwards":[]}
{"upwards":["Mereology","Metaphysics","Mereological composition"],"downwards":["Fundamentality","Special composition question (SCQ)"]}
{"upwards":["Mereology","Metaphysics","Mereological composition","Fundamentality"],"downwards":[]}
{"upwards":["Mereology","Metaphysics","Mereological composition","Special composition question (SCQ)"],"downwards":[]}
{"upwards":["Mereology","Important surveys"],"downwards":[]}
{"upwards":["Rough set"],"downwards":["Definitions","Rule extraction","Incomplete data","Applications","History","Extensions and generalizations"]}
{"upwards":["Rough set","Definitions"],"downwards":["Information system framework","Example: equivalence-class structure","Definition of a rough set","Definability","Reduct and core","Attribute dependency"]}
{"upwards":["Rough set","Definitions","Information system framework"],"downwards":[]}
{"upwards":["Rough set","Definitions","Example: equivalence-class structure"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set"],"downwards":["Lower approximation and positive region","Upper approximation and negative region","Boundary region","The rough set","Objective analysis"]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Lower approximation and positive region"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Upper approximation and negative region"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Boundary region"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","The rough set"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definition of a rough set","Objective analysis"],"downwards":[]}
{"upwards":["Rough set","Definitions","Definability"],"downwards":[]}
{"upwards":["Rough set","Definitions","Reduct and core"],"downwards":[]}
{"upwards":["Rough set","Definitions","Attribute dependency"],"downwards":[]}
{"upwards":["Rough set","Rule extraction"],"downwards":["Decision matrices","LERS rule induction system"]}
{"upwards":["Rough set","Rule extraction","Decision matrices"],"downwards":[]}
{"upwards":["Rough set","Rule extraction","LERS rule induction system"],"downwards":[]}
{"upwards":["Rough set","Incomplete data"],"downwards":[]}
{"upwards":["Rough set","Applications"],"downwards":[]}
{"upwards":["Rough set","History"],"downwards":[]}
{"upwards":["Rough set","Extensions and generalizations"],"downwards":["Rough membership","Other generalizations"]}
{"upwards":["Rough set","Extensions and generalizations","Rough membership"],"downwards":[]}
{"upwards":["Rough set","Extensions and generalizations","Other generalizations"],"downwards":[]}
{"upwards":["ISBN"],"downwards":["History","Overview","Check digits","EAN format used in barcodes, and upgrading"]}
{"upwards":["ISBN","History"],"downwards":[]}
{"upwards":["ISBN","Overview"],"downwards":["Issuing process","Registration group element","Registrant element"]}
{"upwards":["ISBN","Overview","Issuing process"],"downwards":[]}
{"upwards":["ISBN","Overview","Registration group element"],"downwards":[]}
{"upwards":["ISBN","Overview","Registrant element"],"downwards":["English language pattern"]}
{"upwards":["ISBN","Overview","Registrant element","English language pattern"],"downwards":[]}
{"upwards":["ISBN","Check digits"],"downwards":["ISBN-10 check digits","ISBN-10 check digit calculation","ISBN-13 check digit calculation","ISBN-10 to ISBN-13 conversion","Errors in usage","eISBN"]}
{"upwards":["ISBN","Check digits","ISBN-10 check digits"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-10 check digit calculation"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-13 check digit calculation"],"downwards":[]}
{"upwards":["ISBN","Check digits","ISBN-10 to ISBN-13 conversion"],"downwards":[]}
{"upwards":["ISBN","Check digits","Errors in usage"],"downwards":[]}
{"upwards":["ISBN","Check digits","eISBN"],"downwards":[]}
{"upwards":["ISBN","EAN format used in barcodes, and upgrading"],"downwards":[]}
{"upwards":["The Principles of Mathematics"],"downwards":["Contents","Early reviews","Second edition","Later reviews"]}
{"upwards":["The Principles of Mathematics","Contents"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Early reviews"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Second edition"],"downwards":[]}
{"upwards":["The Principles of Mathematics","Later reviews"],"downwards":[]}
{"upwards":["Crelle's Journal"],"downwards":["History","Past editors","References","External links"]}
{"upwards":["Crelle's Journal","History"],"downwards":[]}
{"upwards":["Crelle's Journal","Past editors"],"downwards":[]}
{"upwards":["Crelle's Journal","References"],"downwards":[]}
{"upwards":["Crelle's Journal","External links"],"downwards":[]}
{"upwards":["Digital object identifier"],"downwards":["Nomenclature and syntax","Content","Features and benefits","Comparison with other identifier schemes","Resolution","IDF organizational structure","Standardization"]}
{"upwards":["Digital object identifier","Nomenclature and syntax"],"downwards":["Display"]}
{"upwards":["Digital object identifier","Nomenclature and syntax","Display"],"downwards":[]}
{"upwards":["Digital object identifier","Content"],"downwards":[]}
{"upwards":["Digital object identifier","Features and benefits"],"downwards":[]}
{"upwards":["Digital object identifier","Comparison with other identifier schemes"],"downwards":[]}
{"upwards":["Digital object identifier","Resolution"],"downwards":[]}
{"upwards":["Digital object identifier","IDF organizational structure"],"downwards":[]}
{"upwards":["Digital object identifier","Standardization"],"downwards":[]}
{"upwards":["Bibcode"],"downwards":["Adoption","Format","Examples"]}
{"upwards":["Bibcode","Adoption"],"downwards":[]}
{"upwards":["Bibcode","Format"],"downwards":[]}
{"upwards":["Bibcode","Examples"],"downwards":[]}
{"upwards":["Abstract algebra"],"downwards":["History","Basic concepts","Branches of abstract algebra","Applications"]}
{"upwards":["Abstract algebra","History"],"downwards":["Elementary algebra","Early group theory","Early ring theory","Early field theory","Other major areas","Modern algebra"]}
{"upwards":["Abstract algebra","History","Elementary algebra"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early group theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early ring theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Early field theory"],"downwards":[]}
{"upwards":["Abstract algebra","History","Other major areas"],"downwards":[]}
{"upwards":["Abstract algebra","History","Modern algebra"],"downwards":[]}
{"upwards":["Abstract algebra","Basic concepts"],"downwards":[]}
{"upwards":["Abstract algebra","Branches of abstract algebra"],"downwards":["Group theory","Ring theory"]}
{"upwards":["Abstract algebra","Branches of abstract algebra","Group theory"],"downwards":[]}
{"upwards":["Abstract algebra","Branches of abstract algebra","Ring theory"],"downwards":[]}
{"upwards":["Abstract algebra","Applications"],"downwards":[]}
{"upwards":["Algebraic structure"],"downwards":["Introduction","Common axioms","Common algebraic structures","Hybrid structures","Universal algebra","Category theory","Different meanings of \"structure\""]}
{"upwards":["Algebraic structure","Introduction"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms"],"downwards":["Equational axioms","Existential axioms","Non-equational axioms"]}
{"upwards":["Algebraic structure","Common axioms","Equational axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms","Existential axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common axioms","Non-equational axioms"],"downwards":[]}
{"upwards":["Algebraic structure","Common algebraic structures"],"downwards":["One set with operations","Two sets with operations"]}
{"upwards":["Algebraic structure","Common algebraic structures","One set with operations"],"downwards":[]}
{"upwards":["Algebraic structure","Common algebraic structures","Two sets with operations"],"downwards":[]}
{"upwards":["Algebraic structure","Hybrid structures"],"downwards":[]}
{"upwards":["Algebraic structure","Universal algebra"],"downwards":[]}
{"upwards":["Algebraic structure","Category theory"],"downwards":[]}
{"upwards":["Algebraic structure","Different meanings of \"structure\""],"downwards":[]}
{"upwards":["Group (mathematics)"],"downwards":["Definition and illustration","History","Elementary consequences of the group axioms","Basic concepts","Examples and applications","Finite groups","Groups with additional structure","Generalizations"]}
{"upwards":["Group (mathematics)","Definition and illustration"],"downwards":["First example: the integers","Definition","Notation and terminology","Second example: a symmetry group"]}
{"upwards":["Group (mathematics)","Definition and illustration","First example: the integers"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Definition"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Notation and terminology"],"downwards":[]}
{"upwards":["Group (mathematics)","Definition and illustration","Second example: a symmetry group"],"downwards":[]}
{"upwards":["Group (mathematics)","History"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms"],"downwards":["Uniqueness of identity element","Uniqueness of inverses","Division","Equivalent definition with relaxed axioms"]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Uniqueness of identity element"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Uniqueness of inverses"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Division"],"downwards":[]}
{"upwards":["Group (mathematics)","Elementary consequences of the group axioms","Equivalent definition with relaxed axioms"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts"],"downwards":["Group homomorphisms","Subgroups","Cosets","Quotient groups","Presentations"]}
{"upwards":["Group (mathematics)","Basic concepts","Group homomorphisms"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Subgroups"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Cosets"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Quotient groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Basic concepts","Presentations"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications"],"downwards":["Numbers","Modular arithmetic","Cyclic groups","Symmetry groups","General linear group and representation theory","Galois groups"]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers"],"downwards":["Integers","Rationals"]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers","Integers"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Numbers","Rationals"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Modular arithmetic"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Cyclic groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Symmetry groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","General linear group and representation theory"],"downwards":[]}
{"upwards":["Group (mathematics)","Examples and applications","Galois groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups"],"downwards":["Finite abelian groups","Simple groups","Classification of finite simple groups"]}
{"upwards":["Group (mathematics)","Finite groups","Finite abelian groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups","Simple groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Finite groups","Classification of finite simple groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Groups with additional structure"],"downwards":["Topological groups","Lie groups"]}
{"upwards":["Group (mathematics)","Groups with additional structure","Topological groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Groups with additional structure","Lie groups"],"downwards":[]}
{"upwards":["Group (mathematics)","Generalizations"],"downwards":[]}
{"upwards":["Field (mathematics)"],"downwards":["Definition","Examples","Elementary notions","Finite fields","History","Constructing fields","Fields with additional structure","Galois theory","Invariants of fields","Applications","Related notions","Notes","References","External links"]}
{"upwards":["Field (mathematics)","Definition"],"downwards":["Classic definition","Alternative definition"]}
{"upwards":["Field (mathematics)","Definition","Classic definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Definition","Alternative definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples"],"downwards":["Rational numbers","Real and complex numbers","Constructible numbers","A field with four elements"]}
{"upwards":["Field (mathematics)","Examples","Rational numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","Real and complex numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","Constructible numbers"],"downwards":[]}
{"upwards":["Field (mathematics)","Examples","A field with four elements"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions"],"downwards":["Consequences of the definition","The additive and the multiplicative group of a field","Characteristic","Subfields and prime fields"]}
{"upwards":["Field (mathematics)","Elementary notions","Consequences of the definition"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","The additive and the multiplicative group of a field"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","Characteristic"],"downwards":[]}
{"upwards":["Field (mathematics)","Elementary notions","Subfields and prime fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Finite fields"],"downwards":[]}
{"upwards":["Field (mathematics)","History"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields"],"downwards":["Constructing fields from rings","Constructing fields within a bigger field","Field extensions","Closure operations"]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings"],"downwards":["Field of fractions","Residue fields"]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings","Field of fractions"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields from rings","Residue fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Constructing fields within a bigger field"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions"],"downwards":["Algebraic extensions","Transcendence bases"]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions","Algebraic extensions"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Field extensions","Transcendence bases"],"downwards":[]}
{"upwards":["Field (mathematics)","Constructing fields","Closure operations"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure"],"downwards":["Ordered fields","Topological fields","Differential fields"]}
{"upwards":["Field (mathematics)","Fields with additional structure","Ordered fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure","Topological fields"],"downwards":["Local fields"]}
{"upwards":["Field (mathematics)","Fields with additional structure","Topological fields","Local fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Fields with additional structure","Differential fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Galois theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields"],"downwards":["Model theory of fields","The absolute Galois group","K-theory"]}
{"upwards":["Field (mathematics)","Invariants of fields","Model theory of fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields","The absolute Galois group"],"downwards":[]}
{"upwards":["Field (mathematics)","Invariants of fields","K-theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications"],"downwards":["Linear algebra and commutative algebra","Finite fields: cryptography and coding theory","Geometry: field of functions","Number theory: global fields"]}
{"upwards":["Field (mathematics)","Applications","Linear algebra and commutative algebra"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Finite fields: cryptography and coding theory"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Geometry: field of functions"],"downwards":[]}
{"upwards":["Field (mathematics)","Applications","Number theory: global fields"],"downwards":[]}
{"upwards":["Field (mathematics)","Related notions"],"downwards":["Division rings"]}
{"upwards":["Field (mathematics)","Related notions","Division rings"],"downwards":[]}
{"upwards":["Field (mathematics)","Notes"],"downwards":[]}
{"upwards":["Field (mathematics)","References"],"downwards":[]}
{"upwards":["Field (mathematics)","External links"],"downwards":[]}
{"upwards":["JSTOR"],"downwards":["History","Content","Access","Usage"]}
{"upwards":["JSTOR","History"],"downwards":[]}
{"upwards":["JSTOR","Content"],"downwards":[]}
{"upwards":["JSTOR","Access"],"downwards":["Aaron Swartz incident","Limitations","Increasing public access"]}
{"upwards":["JSTOR","Access","Aaron Swartz incident"],"downwards":[]}
{"upwards":["JSTOR","Access","Limitations"],"downwards":[]}
{"upwards":["JSTOR","Access","Increasing public access"],"downwards":[]}
{"upwards":["JSTOR","Usage"],"downwards":[]}
{"upwards":["Real number"],"downwards":["Characterizing properties","Arithmetic","Dedekind completeness","Decimal representation","Topological completeness","Cardinality","Other properties","History","Formal definitions","Applications and connections","Vocabulary and notation","Generalizations and extensions"]}
{"upwards":["Real number","Characterizing properties"],"downwards":[]}
{"upwards":["Real number","Arithmetic"],"downwards":["Auxiliary operations","Auxiliary order relations","Integers and fractions as real numbers"]}
{"upwards":["Real number","Arithmetic","Auxiliary operations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Auxiliary order relations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Integers and fractions as real numbers"],"downwards":[]}
{"upwards":["Real number","Dedekind completeness"],"downwards":[]}
{"upwards":["Real number","Decimal representation"],"downwards":[]}
{"upwards":["Real number","Topological completeness"],"downwards":["\"The complete ordered field\""]}
{"upwards":["Real number","Topological completeness","\"The complete ordered field\""],"downwards":[]}
{"upwards":["Real number","Cardinality"],"downwards":[]}
{"upwards":["Real number","Other properties"],"downwards":[]}
{"upwards":["Real number","History"],"downwards":[]}
{"upwards":["Real number","Formal definitions"],"downwards":["Axiomatic approach","Construction from the rational numbers"]}
{"upwards":["Real number","Formal definitions","Axiomatic approach"],"downwards":[]}
{"upwards":["Real number","Formal definitions","Construction from the rational numbers"],"downwards":[]}
{"upwards":["Real number","Applications and connections"],"downwards":["Physics","Logic","Computation","Set theory"]}
{"upwards":["Real number","Applications and connections","Physics"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Logic"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Computation"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Set theory"],"downwards":[]}
{"upwards":["Real number","Vocabulary and notation"],"downwards":[]}
{"upwards":["Real number","Generalizations and extensions"],"downwards":[]}
{"upwards":["Countable set"],"downwards":["A note on terminology","Definition","History","Introduction","Formal overview","Minimal model of set theory is countable","Total orders"]}
{"upwards":["Countable set","A note on terminology"],"downwards":[]}
{"upwards":["Countable set","Definition"],"downwards":[]}
{"upwards":["Countable set","History"],"downwards":[]}
{"upwards":["Countable set","Introduction"],"downwards":[]}
{"upwards":["Countable set","Formal overview"],"downwards":[]}
{"upwards":["Countable set","Minimal model of set theory is countable"],"downwards":[]}
{"upwards":["Countable set","Total orders"],"downwards":[]}
{"upwards":["Real number"],"downwards":["Characterizing properties","Arithmetic","Dedekind completeness","Decimal representation","Topological completeness","Cardinality","Other properties","History","Formal definitions","Applications and connections","Vocabulary and notation","Generalizations and extensions"]}
{"upwards":["Real number","Characterizing properties"],"downwards":[]}
{"upwards":["Real number","Arithmetic"],"downwards":["Auxiliary operations","Auxiliary order relations","Integers and fractions as real numbers"]}
{"upwards":["Real number","Arithmetic","Auxiliary operations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Auxiliary order relations"],"downwards":[]}
{"upwards":["Real number","Arithmetic","Integers and fractions as real numbers"],"downwards":[]}
{"upwards":["Real number","Dedekind completeness"],"downwards":[]}
{"upwards":["Real number","Decimal representation"],"downwards":[]}
{"upwards":["Real number","Topological completeness"],"downwards":["\"The complete ordered field\""]}
{"upwards":["Real number","Topological completeness","\"The complete ordered field\""],"downwards":[]}
{"upwards":["Real number","Cardinality"],"downwards":[]}
{"upwards":["Real number","Other properties"],"downwards":[]}
{"upwards":["Real number","History"],"downwards":[]}
{"upwards":["Real number","Formal definitions"],"downwards":["Axiomatic approach","Construction from the rational numbers"]}
{"upwards":["Real number","Formal definitions","Axiomatic approach"],"downwards":[]}
{"upwards":["Real number","Formal definitions","Construction from the rational numbers"],"downwards":[]}
{"upwards":["Real number","Applications and connections"],"downwards":["Physics","Logic","Computation","Set theory"]}
{"upwards":["Real number","Applications and connections","Physics"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Logic"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Computation"],"downwards":[]}
{"upwards":["Real number","Applications and connections","Set theory"],"downwards":[]}
{"upwards":["Real number","Vocabulary and notation"],"downwards":[]}
{"upwards":["Real number","Generalizations and extensions"],"downwards":[]}
{"upwards":["Bijection"],"downwards":["Definition","Examples","More mathematical examples","Inverses","Composition","Cardinality","Properties","Category theory","Generalization to partial functions","Gallery"]}
{"upwards":["Bijection","Definition"],"downwards":[]}
{"upwards":["Bijection","Examples"],"downwards":["Batting line-up of a baseball or cricket team","Seats and students of a classroom"]}
{"upwards":["Bijection","Examples","Batting line-up of a baseball or cricket team"],"downwards":[]}
{"upwards":["Bijection","Examples","Seats and students of a classroom"],"downwards":[]}
{"upwards":["Bijection","More mathematical examples"],"downwards":[]}
{"upwards":["Bijection","Inverses"],"downwards":[]}
{"upwards":["Bijection","Composition"],"downwards":[]}
{"upwards":["Bijection","Cardinality"],"downwards":[]}
{"upwards":["Bijection","Properties"],"downwards":[]}
{"upwards":["Bijection","Category theory"],"downwards":[]}
{"upwards":["Bijection","Generalization to partial functions"],"downwards":[]}
{"upwards":["Bijection","Gallery"],"downwards":[]}
{"upwards":["Transcendental number"],"downwards":["History","Properties","Numbers proven to be transcendental","Possible transcendental numbers","Sketch of a proof that e is transcendental"]}
{"upwards":["Transcendental number","History"],"downwards":[]}
{"upwards":["Transcendental number","Properties"],"downwards":[]}
{"upwards":["Transcendental number","Numbers proven to be transcendental"],"downwards":[]}
{"upwards":["Transcendental number","Possible transcendental numbers"],"downwards":[]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental"],"downwards":["The transcendence of \u03c0"]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental","The transcendence of \u03c0"],"downwards":[]}
{"upwards":["Rational number"],"downwards":["Terminology","Arithmetic","Continued fraction representation","Other representations","Formal construction","Properties","Real numbers and topological properties","p-adic numbers"]}
{"upwards":["Rational number","Terminology"],"downwards":["Etymology"]}
{"upwards":["Rational number","Terminology","Etymology"],"downwards":[]}
{"upwards":["Rational number","Arithmetic"],"downwards":["Irreducible fraction","Embedding of integers","Equality","Ordering","Addition","Subtraction","Multiplication","Inverse","Division","Exponentiation to integer power"]}
{"upwards":["Rational number","Arithmetic","Irreducible fraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Embedding of integers"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Equality"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Ordering"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Addition"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Multiplication"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Inverse"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Division"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Exponentiation to integer power"],"downwards":[]}
{"upwards":["Rational number","Continued fraction representation"],"downwards":[]}
{"upwards":["Rational number","Other representations"],"downwards":[]}
{"upwards":["Rational number","Formal construction"],"downwards":[]}
{"upwards":["Rational number","Properties"],"downwards":["Countability"]}
{"upwards":["Rational number","Properties","Countability"],"downwards":[]}
{"upwards":["Rational number","Real numbers and topological properties"],"downwards":[]}
{"upwards":["Rational number","p-adic numbers"],"downwards":[]}
{"upwards":["Rational number"],"downwards":["Terminology","Arithmetic","Continued fraction representation","Other representations","Formal construction","Properties","Real numbers and topological properties","p-adic numbers"]}
{"upwards":["Rational number","Terminology"],"downwards":["Etymology"]}
{"upwards":["Rational number","Terminology","Etymology"],"downwards":[]}
{"upwards":["Rational number","Arithmetic"],"downwards":["Irreducible fraction","Embedding of integers","Equality","Ordering","Addition","Subtraction","Multiplication","Inverse","Division","Exponentiation to integer power"]}
{"upwards":["Rational number","Arithmetic","Irreducible fraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Embedding of integers"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Equality"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Ordering"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Addition"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Multiplication"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Inverse"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Division"],"downwards":[]}
{"upwards":["Rational number","Arithmetic","Exponentiation to integer power"],"downwards":[]}
{"upwards":["Rational number","Continued fraction representation"],"downwards":[]}
{"upwards":["Rational number","Other representations"],"downwards":[]}
{"upwards":["Rational number","Formal construction"],"downwards":[]}
{"upwards":["Rational number","Properties"],"downwards":["Countability"]}
{"upwards":["Rational number","Properties","Countability"],"downwards":[]}
{"upwards":["Rational number","Real numbers and topological properties"],"downwards":[]}
{"upwards":["Rational number","p-adic numbers"],"downwards":[]}
{"upwards":["Line (geometry)"],"downwards":["Properties","Definition","Other representations","Generalizations of the Euclidean line","Extensions"]}
{"upwards":["Line (geometry)","Properties"],"downwards":["In higher dimensions","Types","In axiomatic systems"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions"],"downwards":["Collinear points"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions","Collinear points"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","Types"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","In axiomatic systems"],"downwards":[]}
{"upwards":["Line (geometry)","Definition"],"downwards":["Linear equation","Parametric equation","Hesse normal form"]}
{"upwards":["Line (geometry)","Definition","Linear equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Parametric equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Hesse normal form"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations"],"downwards":["Vectors","Polar coordinates"]}
{"upwards":["Line (geometry)","Other representations","Vectors"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations","Polar coordinates"],"downwards":[]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line"],"downwards":["Projective geometry"]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line","Projective geometry"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions"],"downwards":["Ray","Line segment","Number line"]}
{"upwards":["Line (geometry)","Extensions","Ray"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Line segment"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Number line"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
{"upwards":["Subset","Definition"],"downwards":[]}
{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["PubMed Central"],"downwards":["Adoption","Technology","Reception","PMCID"]}
{"upwards":["PubMed Central","Adoption"],"downwards":[]}
{"upwards":["PubMed Central","Technology"],"downwards":[]}
{"upwards":["PubMed Central","Reception"],"downwards":[]}
{"upwards":["PubMed Central","PMCID"],"downwards":[]}
{"upwards":["PubMed"],"downwards":["Content","Characteristics"]}
{"upwards":["PubMed","Content"],"downwards":[]}
{"upwards":["PubMed","Characteristics"],"downwards":["Website design","Search","Journal article parameters"]}
{"upwards":["PubMed","Characteristics","Website design"],"downwards":["PubMed for handhelds\/mobiles"]}
{"upwards":["PubMed","Characteristics","Website design","PubMed for handhelds\/mobiles"],"downwards":[]}
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{"upwards":["PubMed","Characteristics","Search","Standard search"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Search","Comprehensive search"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Journal article parameters"],"downwards":["Publication Type: Clinical queries\/systematic reviews","Secondary ID"]}
{"upwards":["PubMed","Characteristics","Journal article parameters","Publication Type: Clinical queries\/systematic reviews"],"downwards":[]}
{"upwards":["PubMed","Characteristics","Journal article parameters","Secondary ID"],"downwards":[]}
{"upwards":["First-order logic"],"downwards":["Introduction","Syntax","Semantics","Deductive systems","Equality and its axioms","Metalogical properties","Limitations","Restrictions, extensions, and variations","Automated theorem proving and formal methods"]}
{"upwards":["First-order logic","Introduction"],"downwards":[]}
{"upwards":["First-order logic","Syntax"],"downwards":["Alphabet","Formation rules","Free and bound variables","Example: ordered abelian groups"]}
{"upwards":["First-order logic","Syntax","Alphabet"],"downwards":["Logical symbols","Non-logical symbols"]}
{"upwards":["First-order logic","Syntax","Alphabet","Logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Alphabet","Non-logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules"],"downwards":["Terms","Formulas","Notational conventions"]}
{"upwards":["First-order logic","Syntax","Formation rules","Terms"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Formulas"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Notational conventions"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Free and bound variables"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Example: ordered abelian groups"],"downwards":[]}
{"upwards":["First-order logic","Semantics"],"downwards":["First-order structures","Evaluation of truth values","Validity, satisfiability, and logical consequence","Algebraizations","First-order theories, models, and elementary classes","Empty domains"]}
{"upwards":["First-order logic","Semantics","First-order structures"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Evaluation of truth values"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Validity, satisfiability, and logical consequence"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Algebraizations"],"downwards":[]}
{"upwards":["First-order logic","Semantics","First-order theories, models, and elementary classes"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Empty domains"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems"],"downwards":["Rules of inference","Hilbert-style systems and natural deduction","Sequent calculus","Tableaux method","Resolution","Provable identities"]}
{"upwards":["First-order logic","Deductive systems","Rules of inference"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Hilbert-style systems and natural deduction"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Sequent calculus"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Tableaux method"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Resolution"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Provable identities"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms"],"downwards":["First-order logic without equality","Defining equality within a theory"]}
{"upwards":["First-order logic","Equality and its axioms","First-order logic without equality"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms","Defining equality within a theory"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties"],"downwards":["Completeness and undecidability","The L\u00f6wenheim\u2013Skolem theorem","The compactness theorem","Lindstr\u00f6m's theorem"]}
{"upwards":["First-order logic","Metalogical properties","Completeness and undecidability"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The compactness theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","Lindstr\u00f6m's theorem"],"downwards":[]}
{"upwards":["First-order logic","Limitations"],"downwards":["Expressiveness","Formalizing natural languages"]}
{"upwards":["First-order logic","Limitations","Expressiveness"],"downwards":[]}
{"upwards":["First-order logic","Limitations","Formalizing natural languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations"],"downwards":["Restricted languages","Many-sorted logic","Additional quantifiers","Infinitary logics","Non-classical and modal logics","Fixpoint logic","Higher-order logics"]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Restricted languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Many-sorted logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Additional quantifiers"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Infinitary logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Non-classical and modal logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Fixpoint logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Higher-order logics"],"downwards":[]}
{"upwards":["First-order logic","Automated theorem proving and formal methods"],"downwards":[]}
{"upwards":["Joseph Dauben"],"downwards":["Positions","Publications","Articles, reviews, and essays"]}
{"upwards":["Joseph Dauben","Positions"],"downwards":[]}
{"upwards":["Joseph Dauben","Publications"],"downwards":[]}
{"upwards":["Joseph Dauben","Articles, reviews, and essays"],"downwards":[]}
{"upwards":["Harvard University Press"],"downwards":["Related publishers, imprints, and series","Awards","Publications"]}
{"upwards":["Harvard University Press","Related publishers, imprints, and series"],"downwards":[]}
{"upwards":["Harvard University Press","Awards"],"downwards":[]}
{"upwards":["Harvard University Press","Publications"],"downwards":[]}
{"upwards":["Paul Halmos"],"downwards":["Early life and education","Career","Accomplishments","Books by Halmos"]}
{"upwards":["Paul Halmos","Early life and education"],"downwards":[]}
{"upwards":["Paul Halmos","Career"],"downwards":[]}
{"upwards":["Paul Halmos","Accomplishments"],"downwards":[]}
{"upwards":["Paul Halmos","Books by Halmos"],"downwards":[]}
{"upwards":["Dover Publications"],"downwards":["Classic reprints","History","Publications","References","External links"]}
{"upwards":["Dover Publications","Classic reprints"],"downwards":[]}
{"upwards":["Dover Publications","History"],"downwards":[]}
{"upwards":["Dover Publications","Publications"],"downwards":[]}
{"upwards":["Dover Publications","References"],"downwards":[]}
{"upwards":["Dover Publications","External links"],"downwards":[]}
{"upwards":["Cambridge University Press"],"downwards":["History","Print and typographic heritage","Current publications","Open access","Nobel prize winners published by Cambridge University Press","Organisational governance and operational structure","Cambridge University Press partnerships and acquisitions","Digital developments","Controversies","Corporate social responsibility","References","External links"]}
{"upwards":["Cambridge University Press","History"],"downwards":[]}
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{"upwards":["Cambridge University Press","Print and typographic heritage","People"],"downwards":[]}
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{"upwards":["Cambridge University Press","Nobel prize winners published by Cambridge University Press"],"downwards":[]}
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{"upwards":["Cambridge University Press","Controversies"],"downwards":["Tax exemption controversy","Alms for Jihad","Cambridge University Press v. Patton","The China Quarterly","The Cambridge Handbook of Privatization"]}
{"upwards":["Cambridge University Press","Controversies","Tax exemption controversy"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","Alms for Jihad"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","Cambridge University Press v. Patton"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","The China Quarterly"],"downwards":[]}
{"upwards":["Cambridge University Press","Controversies","The Cambridge Handbook of Privatization"],"downwards":[]}
{"upwards":["Cambridge University Press","Corporate social responsibility"],"downwards":["Community","Environment"]}
{"upwards":["Cambridge University Press","Corporate social responsibility","Community"],"downwards":[]}
{"upwards":["Cambridge University Press","Corporate social responsibility","Environment"],"downwards":[]}
{"upwards":["Cambridge University Press","References"],"downwards":["Citations","Sources"]}
{"upwards":["Cambridge University Press","References","Citations"],"downwards":[]}
{"upwards":["Cambridge University Press","References","Sources"],"downwards":[]}
{"upwards":["Cambridge University Press","External links"],"downwards":[]}
{"upwards":["Mathematical logic"],"downwards":["Subfields and scope","History","Formal logical systems","Set theory","Model theory","Recursion theory","Proof theory and constructive mathematics","Applications","Connections with computer science","Foundations of mathematics"]}
{"upwards":["Mathematical logic","Subfields and scope"],"downwards":[]}
{"upwards":["Mathematical logic","History"],"downwards":["Early history","19th century","20th century"]}
{"upwards":["Mathematical logic","History","Early history"],"downwards":[]}
{"upwards":["Mathematical logic","History","19th century"],"downwards":["Foundational theories"]}
{"upwards":["Mathematical logic","History","19th century","Foundational theories"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century"],"downwards":["Set theory and paradoxes","Symbolic logic","Beginnings of the other branches"]}
{"upwards":["Mathematical logic","History","20th century","Set theory and paradoxes"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Symbolic logic"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Beginnings of the other branches"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems"],"downwards":["First-order logic","Other classical logics","Nonclassical and modal logic","Algebraic logic"]}
{"upwards":["Mathematical logic","Formal logical systems","First-order logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Other classical logics"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Nonclassical and modal logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Algebraic logic"],"downwards":[]}
{"upwards":["Mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Mathematical logic","Model theory"],"downwards":[]}
{"upwards":["Mathematical logic","Recursion theory"],"downwards":["Algorithmically unsolvable problems"]}
{"upwards":["Mathematical logic","Recursion theory","Algorithmically unsolvable problems"],"downwards":[]}
{"upwards":["Mathematical logic","Proof theory and constructive mathematics"],"downwards":[]}
{"upwards":["Mathematical logic","Applications"],"downwards":[]}
{"upwards":["Mathematical logic","Connections with computer science"],"downwards":[]}
{"upwards":["Mathematical logic","Foundations of mathematics"],"downwards":[]}
{"upwards":["Axiom"],"downwards":["Etymology","Historical development","Mathematical logic"]}
{"upwards":["Axiom","Etymology"],"downwards":[]}
{"upwards":["Axiom","Historical development"],"downwards":["Early Greeks","Modern development","Other sciences"]}
{"upwards":["Axiom","Historical development","Early Greeks"],"downwards":[]}
{"upwards":["Axiom","Historical development","Modern development"],"downwards":[]}
{"upwards":["Axiom","Historical development","Other sciences"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic"],"downwards":["Logical axioms","Non-logical axioms","Role in mathematical logic","Further discussion"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms"],"downwards":["Examples"]}
{"upwards":["Axiom","Mathematical logic","Non-logical axioms","Examples"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic"],"downwards":["Deductive systems and completeness"]}
{"upwards":["Axiom","Mathematical logic","Role in mathematical logic","Deductive systems and completeness"],"downwards":[]}
{"upwards":["Axiom","Mathematical logic","Further discussion"],"downwards":[]}
{"upwards":["List of axioms"],"downwards":["ZF (the Zermelo\u2013Fraenkel axioms without the axiom of choice)","Axiom of choice","Other axioms of mathematical logic","Geometry","Other axioms"]}
{"upwards":["List of axioms","ZF (the Zermelo\u2013Fraenkel axioms without the axiom of choice)"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice"],"downwards":["Equivalents of AC","Stronger than AC","Weaker than AC","Alternates incompatible with AC"]}
{"upwards":["List of axioms","Axiom of choice","Equivalents of AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Stronger than AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Weaker than AC"],"downwards":[]}
{"upwards":["List of axioms","Axiom of choice","Alternates incompatible with AC"],"downwards":[]}
{"upwards":["List of axioms","Other axioms of mathematical logic"],"downwards":[]}
{"upwards":["List of axioms","Geometry"],"downwards":[]}
{"upwards":["List of axioms","Other axioms"],"downwards":[]}
{"upwards":["Formal proof"],"downwards":["Background"]}
{"upwards":["Formal proof","Background"],"downwards":["Formal language","Formal grammar","Formal systems","Interpretations"]}
{"upwards":["Formal proof","Background","Formal language"],"downwards":[]}
{"upwards":["Formal proof","Background","Formal grammar"],"downwards":[]}
{"upwards":["Formal proof","Background","Formal systems"],"downwards":[]}
{"upwards":["Formal proof","Background","Interpretations"],"downwards":[]}
{"upwards":["Semantics of logic"],"downwards":["Overview"]}
{"upwards":["Semantics of logic","Overview"],"downwards":[]}
{"upwards":["Information theory"],"downwards":["Overview","Historical background","Quantities of information","Coding theory","Applications to other fields"]}
{"upwards":["Information theory","Overview"],"downwards":[]}
{"upwards":["Information theory","Historical background"],"downwards":[]}
{"upwards":["Information theory","Quantities of information"],"downwards":["Entropy of an information source","Joint entropy","Conditional entropy (equivocation)","Mutual information (transinformation)","Kullback\u2013Leibler divergence (information gain)","Directed Information","Other quantities"]}
{"upwards":["Information theory","Quantities of information","Entropy of an information source"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Joint entropy"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Conditional entropy (equivocation)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Mutual information (transinformation)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Kullback\u2013Leibler divergence (information gain)"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Directed Information"],"downwards":[]}
{"upwards":["Information theory","Quantities of information","Other quantities"],"downwards":[]}
{"upwards":["Information theory","Coding theory"],"downwards":["Source theory","Channel capacity"]}
{"upwards":["Information theory","Coding theory","Source theory"],"downwards":["Rate"]}
{"upwards":["Information theory","Coding theory","Source theory","Rate"],"downwards":[]}
{"upwards":["Information theory","Coding theory","Channel capacity"],"downwards":["Capacity of particular channel models","Channels with memory and directed information"]}
{"upwards":["Information theory","Coding theory","Channel capacity","Capacity of particular channel models"],"downwards":[]}
{"upwards":["Information theory","Coding theory","Channel capacity","Channels with memory and directed information"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields"],"downwards":["Intelligence uses and secrecy applications","Pseudorandom number generation","Seismic exploration","Semiotics","Integrated process organization of neural information","Miscellaneous applications"]}
{"upwards":["Information theory","Applications to other fields","Intelligence uses and secrecy applications"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Pseudorandom number generation"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Seismic exploration"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Semiotics"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Integrated process organization of neural information"],"downwards":[]}
{"upwards":["Information theory","Applications to other fields","Miscellaneous applications"],"downwards":[]}
{"upwards":["Lemma (mathematics)"],"downwards":["Etymology","Comparison with theorem","Well-known lemmas"]}
{"upwards":["Lemma (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Lemma (mathematics)","Comparison with theorem"],"downwards":[]}
{"upwards":["Lemma (mathematics)","Well-known lemmas"],"downwards":[]}
{"upwards":["Structure (mathematical logic)"],"downwards":["Definition","Induced substructures and closed subsets","Homomorphisms and embeddings","Structures and first-order logic","Many-sorted structures","Other generalizations"]}
{"upwards":["Structure (mathematical logic)","Definition"],"downwards":["Domain","Signature","Interpretation function","Examples"]}
{"upwards":["Structure (mathematical logic)","Definition","Domain"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Signature"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Interpretation function"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets"],"downwards":["Examples"]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings"],"downwards":["Homomorphisms","Embeddings","Example","Homomorphism problem"]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphisms"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Embeddings"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Example"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphism problem"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic"],"downwards":["Satisfaction relation","Definable relations"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Satisfaction relation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations"],"downwards":["Definability with parameters","Implicit definability"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Definability with parameters"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Implicit definability"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Many-sorted structures"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations"],"downwards":["Partial algebras","Structures for typed languages","Higher-order languages","Structures that are proper classes"]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Partial algebras"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures for typed languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Higher-order languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures that are proper classes"],"downwards":[]}
{"upwards":["Theorem"],"downwards":["Theoremhood and truth","Epistemological considerations","Informal account of theorems","Relation with scientific theories","Terminology","Layout","Lore","Theorems in logic"]}
{"upwards":["Theorem","Theoremhood and truth"],"downwards":[]}
{"upwards":["Theorem","Epistemological considerations"],"downwards":[]}
{"upwards":["Theorem","Informal account of theorems"],"downwards":[]}
{"upwards":["Theorem","Relation with scientific theories"],"downwards":[]}
{"upwards":["Theorem","Terminology"],"downwards":[]}
{"upwards":["Theorem","Layout"],"downwards":[]}
{"upwards":["Theorem","Lore"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic"],"downwards":["Syntax and semantics","Interpretation of a formal theorem","Theorems and theories"]}
{"upwards":["Theorem","Theorems in logic","Syntax and semantics"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Interpretation of a formal theorem"],"downwards":[]}
{"upwards":["Theorem","Theorems in logic","Theorems and theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)"],"downwards":["General theories (as expressed in formal language)","First-order theories","Examples"]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)"],"downwards":["Subtheories and extensions","Deductive theories","Consistency and completeness","Interpretation of a theory","Theories associated with a structure"]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Subtheories and extensions"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Deductive theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Consistency and completeness"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Interpretation of a theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","General theories (as expressed in formal language)","Theories associated with a structure"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories"],"downwards":["Derivation in a first-order theory","Syntactic consequence in a first-order theory","Interpretation of a first-order theory","First-order theories with identity","Topics related to first-order theories"]}
{"upwards":["Theory (mathematical logic)","First-order theories","Derivation in a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Syntactic consequence in a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Interpretation of a first-order theory"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","First-order theories with identity"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","First-order theories","Topics related to first-order theories"],"downwards":[]}
{"upwards":["Theory (mathematical logic)","Examples"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem"],"downwards":["Preliminaries","Statement","Consequences","Relationship to the incompleteness theorems","Relationship to the compactness theorem","Completeness in other logics","Proofs"]}
{"upwards":["G\u00f6del's completeness theorem","Preliminaries"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement"],"downwards":["G\u00f6del's original formulation","More general form","Model existence theorem","As a theorem of arithmetic"]}
{"upwards":["G\u00f6del's completeness theorem","Statement","G\u00f6del's original formulation"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","More general form"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","Model existence theorem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Statement","As a theorem of arithmetic"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Consequences"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Relationship to the incompleteness theorems"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Relationship to the compactness theorem"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Completeness in other logics"],"downwards":[]}
{"upwards":["G\u00f6del's completeness theorem","Proofs"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem"],"downwards":["History","Statement","General form","Discussion"]}
{"upwards":["Tarski's undefinability theorem","History"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","Statement"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","General form"],"downwards":[]}
{"upwards":["Tarski's undefinability theorem","Discussion"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox"],"downwards":["Banach and Tarski publication","Formal treatment","Connection with earlier work and the role of the axiom of choice","A sketch of the proof","Obtaining infinitely many balls from one","Von Neumann paradox in the Euclidean plane"]}
{"upwards":["Banach\u2013Tarski paradox","Banach and Tarski publication"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Formal treatment"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Connection with earlier work and the role of the axiom of choice"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof"],"downwards":["Step 1","Step 2","Step 3","Step 4","Some details, fleshed out"]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 1"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 2"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 3"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Step 4"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","A sketch of the proof","Some details, fleshed out"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Obtaining infinitely many balls from one"],"downwards":[]}
{"upwards":["Banach\u2013Tarski paradox","Von Neumann paradox in the Euclidean plane"],"downwards":["Recent progress"]}
{"upwards":["Banach\u2013Tarski paradox","Von Neumann paradox in the Euclidean plane","Recent progress"],"downwards":[]}
{"upwards":["Cantor's theorem"],"downwards":["Proof","When A is countably infinite","Related paradoxes","History","Generalizations"]}
{"upwards":["Cantor's theorem","Proof"],"downwards":[]}
{"upwards":["Cantor's theorem","When A is countably infinite"],"downwards":[]}
{"upwards":["Cantor's theorem","Related paradoxes"],"downwards":[]}
{"upwards":["Cantor's theorem","History"],"downwards":[]}
{"upwards":["Cantor's theorem","Generalizations"],"downwards":[]}
{"upwards":["Cantor's diagonal argument"],"downwards":["Uncountable set","Consequences","Version for Quine's New Foundations"]}
{"upwards":["Cantor's diagonal argument","Uncountable set"],"downwards":["Real numbers","General sets"]}
{"upwards":["Cantor's diagonal argument","Uncountable set","Real numbers"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Uncountable set","General sets"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences"],"downwards":["Ordering of cardinals","Open questions","Diagonalization in broader context"]}
{"upwards":["Cantor's diagonal argument","Consequences","Ordering of cardinals"],"downwards":["In the absence of excluded middle"]}
{"upwards":["Cantor's diagonal argument","Consequences","Ordering of cardinals","In the absence of excluded middle"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences","Open questions"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Consequences","Diagonalization in broader context"],"downwards":[]}
{"upwards":["Cantor's diagonal argument","Version for Quine's New Foundations"],"downwards":[]}
{"upwards":["Compactness theorem"],"downwards":["History","Applications","Proofs"]}
{"upwards":["Compactness theorem","History"],"downwards":[]}
{"upwards":["Compactness theorem","Applications"],"downwards":["Robinson's principle","Upward L\u00f6wenheim\u2013Skolem theorem","Non-standard analysis"]}
{"upwards":["Compactness theorem","Applications","Robinson's principle"],"downwards":[]}
{"upwards":["Compactness theorem","Applications","Upward L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["Compactness theorem","Applications","Non-standard analysis"],"downwards":[]}
{"upwards":["Compactness theorem","Proofs"],"downwards":[]}
{"upwards":["Halting problem"],"downwards":["Background","History","Formalization","Computability theory","Generalization"]}
{"upwards":["Halting problem","Background"],"downwards":["Programming consequences","Common pitfalls"]}
{"upwards":["Halting problem","Background","Programming consequences"],"downwards":[]}
{"upwards":["Halting problem","Background","Common pitfalls"],"downwards":[]}
{"upwards":["Halting problem","History"],"downwards":["Timeline","Origin of the halting problem"]}
{"upwards":["Halting problem","History","Timeline"],"downwards":[]}
{"upwards":["Halting problem","History","Origin of the halting problem"],"downwards":[]}
{"upwards":["Halting problem","Formalization"],"downwards":["Representation as a set","Proof concept","Sketch of rigorous proof"]}
{"upwards":["Halting problem","Formalization","Representation as a set"],"downwards":[]}
{"upwards":["Halting problem","Formalization","Proof concept"],"downwards":[]}
{"upwards":["Halting problem","Formalization","Sketch of rigorous proof"],"downwards":[]}
{"upwards":["Halting problem","Computability theory"],"downwards":["Approximations","G\u00f6del's incompleteness theorems"]}
{"upwards":["Halting problem","Computability theory","Approximations"],"downwards":[]}
{"upwards":["Halting problem","Computability theory","G\u00f6del's incompleteness theorems"],"downwards":[]}
{"upwards":["Halting problem","Generalization"],"downwards":["Halting on all inputs","Recognizing partial solutions","Lossy computation","Oracle machines"]}
{"upwards":["Halting problem","Generalization","Halting on all inputs"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Recognizing partial solutions"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Lossy computation"],"downwards":[]}
{"upwards":["Halting problem","Generalization","Oracle machines"],"downwards":[]}
{"upwards":["Lindstr\u00f6m's theorem"],"downwards":["Notes","References"]}
{"upwards":["Lindstr\u00f6m's theorem","Notes"],"downwards":[]}
{"upwards":["Lindstr\u00f6m's theorem","References"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem"],"downwards":["Theorem","Discussion","Proof sketch","In other logics","Historical notes","References","Sources","External links"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Theorem"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion"],"downwards":["Concepts","Consequences"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts"],"downwards":["Signatures","Structures \/ Models"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts","Signatures"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Concepts","Structures \/ Models"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Discussion","Consequences"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch"],"downwards":["Downward part","Upward part"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch","Downward part"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Proof sketch","Upward part"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","In other logics"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Historical notes"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","References"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources"],"downwards":["Historical publications","Secondary sources"]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources","Historical publications"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","Sources","Secondary sources"],"downwards":[]}
{"upwards":["L\u00f6wenheim\u2013Skolem theorem","External links"],"downwards":[]}
{"upwards":["Logic"],"downwards":["Definition","Basic concepts","Systems of logic","Areas of research","History"]}
{"upwards":["Logic","Definition"],"downwards":["Formal logic","Informal logic"]}
{"upwards":["Logic","Definition","Formal logic"],"downwards":[]}
{"upwards":["Logic","Definition","Informal logic"],"downwards":[]}
{"upwards":["Logic","Basic concepts"],"downwards":["Premises, conclusions, and truth","Arguments and inferences","Fallacies","Definitory and strategic rules","Formal systems"]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth"],"downwards":["Premises and conclusions","Internal structure","Logical truth","Truth tables"]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Premises and conclusions"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Internal structure"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Logical truth"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Premises, conclusions, and truth","Truth tables"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Arguments and inferences"],"downwards":["Deductive","Ampliative"]}
{"upwards":["Logic","Basic concepts","Arguments and inferences","Deductive"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Arguments and inferences","Ampliative"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Fallacies"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Definitory and strategic rules"],"downwards":[]}
{"upwards":["Logic","Basic concepts","Formal systems"],"downwards":[]}
{"upwards":["Logic","Systems of logic"],"downwards":["Aristotelian","Classical","Extended","Deviant","Informal"]}
{"upwards":["Logic","Systems of logic","Aristotelian"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Classical"],"downwards":["Propositional logic","First-order logic"]}
{"upwards":["Logic","Systems of logic","Classical","Propositional logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Classical","First-order logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Extended"],"downwards":["Modal logic","Higher order logic"]}
{"upwards":["Logic","Systems of logic","Extended","Modal logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Extended","Higher order logic"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Deviant"],"downwards":[]}
{"upwards":["Logic","Systems of logic","Informal"],"downwards":[]}
{"upwards":["Logic","Areas of research"],"downwards":["Philosophy of logic and philosophical logic","Metalogic","Mathematical logic","Computational logic","Formal semantics of natural language","Epistemology of logic"]}
{"upwards":["Logic","Areas of research","Philosophy of logic and philosophical logic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Metalogic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Mathematical logic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Computational logic"],"downwards":[]}
{"upwards":["Logic","Areas of research","Formal semantics of natural language"],"downwards":[]}
{"upwards":["Logic","Areas of research","Epistemology of logic"],"downwards":[]}
{"upwards":["Logic","History"],"downwards":[]}
{"upwards":["Term logic"],"downwards":["Aristotle's system","Basics","Term","Proposition","Singular terms","Syllogism in the first figure","Syllogism in the second figure","Syllogism in the third figure","Table of syllogisms","Decline of term logic","Revival","Boole\u2019s acceptance of Aristotle"]}
{"upwards":["Term logic","Aristotle's system"],"downwards":[]}
{"upwards":["Term logic","Basics"],"downwards":[]}
{"upwards":["Term logic","Term"],"downwards":[]}
{"upwards":["Term logic","Proposition"],"downwards":[]}
{"upwards":["Term logic","Singular terms"],"downwards":["The three figures","The fourth figure"]}
{"upwards":["Term logic","Singular terms","The three figures"],"downwards":[]}
{"upwards":["Term logic","Singular terms","The fourth figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the first figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the second figure"],"downwards":[]}
{"upwards":["Term logic","Syllogism in the third figure"],"downwards":[]}
{"upwards":["Term logic","Table of syllogisms"],"downwards":[]}
{"upwards":["Term logic","Decline of term logic"],"downwards":[]}
{"upwards":["Term logic","Revival"],"downwards":[]}
{"upwards":["Term logic","Boole\u2019s acceptance of Aristotle"],"downwards":[]}
{"upwards":["Classical logic"],"downwards":["Characteristics","History","Generalized semantics","References","Further reading"]}
{"upwards":["Classical logic","Characteristics"],"downwards":[]}
{"upwards":["Classical logic","History"],"downwards":[]}
{"upwards":["Classical logic","Generalized semantics"],"downwards":[]}
{"upwards":["Classical logic","References"],"downwards":[]}
{"upwards":["Classical logic","Further reading"],"downwards":[]}
{"upwards":["Logical truth"],"downwards":["Logical truths and analytic truths","Truth values and tautologies","Logical truth and logical constants","Logical truth and rules of inference","Logical truth and logical positivism","Non-classical logics"]}
{"upwards":["Logical truth","Logical truths and analytic truths"],"downwards":[]}
{"upwards":["Logical truth","Truth values and tautologies"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and logical constants"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and rules of inference"],"downwards":[]}
{"upwards":["Logical truth","Logical truth and logical positivism"],"downwards":[]}
{"upwards":["Logical truth","Non-classical logics"],"downwards":[]}
{"upwards":["Tautology (logic)"],"downwards":["History","Background","Definition and examples","Verifying tautologies","Tautological implication","Substitution","Semantic completeness and soundness","Efficient verification and the Boolean satisfiability problem","Tautologies versus validities in first-order logic"]}
{"upwards":["Tautology (logic)","History"],"downwards":[]}
{"upwards":["Tautology (logic)","Background"],"downwards":[]}
{"upwards":["Tautology (logic)","Definition and examples"],"downwards":[]}
{"upwards":["Tautology (logic)","Verifying tautologies"],"downwards":[]}
{"upwards":["Tautology (logic)","Tautological implication"],"downwards":[]}
{"upwards":["Tautology (logic)","Substitution"],"downwards":[]}
{"upwards":["Tautology (logic)","Semantic completeness and soundness"],"downwards":[]}
{"upwards":["Tautology (logic)","Efficient verification and the Boolean satisfiability problem"],"downwards":[]}
{"upwards":["Tautology (logic)","Tautologies versus validities in first-order logic"],"downwards":[]}
{"upwards":["Proposition"],"downwards":["Historical usage","Relation to the mind","Treatment in logic","Objections to propositions"]}
{"upwards":["Proposition","Historical usage"],"downwards":["By Aristotle","By the logical positivists","By Russell"]}
{"upwards":["Proposition","Historical usage","By Aristotle"],"downwards":[]}
{"upwards":["Proposition","Historical usage","By the logical positivists"],"downwards":[]}
{"upwards":["Proposition","Historical usage","By Russell"],"downwards":[]}
{"upwards":["Proposition","Relation to the mind"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic"],"downwards":["Aristotelian logic","Syntactic characterization","Semantic characterization"]}
{"upwards":["Proposition","Treatment in logic","Aristotelian logic"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic","Syntactic characterization"],"downwards":[]}
{"upwards":["Proposition","Treatment in logic","Semantic characterization"],"downwards":[]}
{"upwards":["Proposition","Objections to propositions"],"downwards":[]}
{"upwards":["Inference"],"downwards":["Definition","Examples","Incorrect inference","Applications"]}
{"upwards":["Inference","Definition"],"downwards":[]}
{"upwards":["Inference","Examples"],"downwards":["Example for definition #1","Example for definition #2"]}
{"upwards":["Inference","Examples","Example for definition #1"],"downwards":[]}
{"upwards":["Inference","Examples","Example for definition #2"],"downwards":[]}
{"upwards":["Inference","Incorrect inference"],"downwards":[]}
{"upwards":["Inference","Applications"],"downwards":["Inference engines","Semantic web","Bayesian statistics and probability logic","Fuzzy logic","Non-monotonic logic"]}
{"upwards":["Inference","Applications","Inference engines"],"downwards":["Prolog engine"]}
{"upwards":["Inference","Applications","Inference engines","Prolog engine"],"downwards":[]}
{"upwards":["Inference","Applications","Semantic web"],"downwards":[]}
{"upwards":["Inference","Applications","Bayesian statistics and probability logic"],"downwards":[]}
{"upwards":["Inference","Applications","Fuzzy logic"],"downwards":[]}
{"upwards":["Inference","Applications","Non-monotonic logic"],"downwards":[]}
{"upwards":["Logical equivalence"],"downwards":["Logical equivalences","Examples","Relation to material equivalence"]}
{"upwards":["Logical equivalence","Logical equivalences"],"downwards":["General logical equivalences","Logical equivalences involving conditional statements","Logical equivalences involving biconditionals"]}
{"upwards":["Logical equivalence","Logical equivalences","General logical equivalences"],"downwards":[]}
{"upwards":["Logical equivalence","Logical equivalences","Logical equivalences involving conditional statements"],"downwards":[]}
{"upwards":["Logical equivalence","Logical equivalences","Logical equivalences involving biconditionals"],"downwards":[]}
{"upwards":["Logical equivalence","Examples"],"downwards":["In logic"]}
{"upwards":["Logical equivalence","Examples","In logic"],"downwards":[]}
{"upwards":["Logical equivalence","Relation to material equivalence"],"downwards":[]}
{"upwards":["Logical consequence"],"downwards":["Formal accounts","A priori property of logical consequence","Proofs and models","Modal accounts"]}
{"upwards":["Logical consequence","Formal accounts"],"downwards":[]}
{"upwards":["Logical consequence","A priori property of logical consequence"],"downwards":[]}
{"upwards":["Logical consequence","Proofs and models"],"downwards":["Syntactic consequence","Semantic consequence"]}
{"upwards":["Logical consequence","Proofs and models","Syntactic consequence"],"downwards":[]}
{"upwards":["Logical consequence","Proofs and models","Semantic consequence"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts"],"downwards":["Modal-formal accounts","Warrant-based accounts","Non-monotonic logical consequence"]}
{"upwards":["Logical consequence","Modal accounts","Modal-formal accounts"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts","Warrant-based accounts"],"downwards":[]}
{"upwards":["Logical consequence","Modal accounts","Non-monotonic logical consequence"],"downwards":[]}
{"upwards":["Consistency"],"downwards":["Consistency and completeness in arithmetic and set theory","First-order logic","Model theory"]}
{"upwards":["Consistency","Consistency and completeness in arithmetic and set theory"],"downwards":[]}
{"upwards":["Consistency","First-order logic"],"downwards":["Notation","Definition","Basic results","Henkin's theorem","Sketch of proof"]}
{"upwards":["Consistency","First-order logic","Notation"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Definition"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Basic results"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Henkin's theorem"],"downwards":[]}
{"upwards":["Consistency","First-order logic","Sketch of proof"],"downwards":[]}
{"upwards":["Consistency","Model theory"],"downwards":[]}
{"upwards":["Square of opposition"],"downwards":["Summary","The problem of existential import","Modern squares of opposition","Logical hexagons and other bi-simplexes","Square of opposition (or logical square) and modal logic","Set-theoretical interpretation of categorical statements"]}
{"upwards":["Square of opposition","Summary"],"downwards":[]}
{"upwards":["Square of opposition","The problem of existential import"],"downwards":[]}
{"upwards":["Square of opposition","Modern squares of opposition"],"downwards":[]}
{"upwards":["Square of opposition","Logical hexagons and other bi-simplexes"],"downwards":[]}
{"upwards":["Square of opposition","Square of opposition (or logical square) and modal logic"],"downwards":[]}
{"upwards":["Square of opposition","Set-theoretical interpretation of categorical statements"],"downwards":[]}
{"upwards":["Propositional calculus"],"downwards":["Explanation","History","Terminology","Basic concepts","Generic description of a propositional calculus","Example 1. Simple axiom system","Example 2. Natural deduction system","Basic and derived argument forms","Proofs in propositional calculus","Soundness and completeness of the rules","Interpretation of a truth-functional propositional calculus","Alternative calculus","Equivalence to equational logics","Graphical calculi","Other logical calculi","Solvers"]}
{"upwards":["Propositional calculus","Explanation"],"downwards":[]}
{"upwards":["Propositional calculus","History"],"downwards":[]}
{"upwards":["Propositional calculus","Terminology"],"downwards":[]}
{"upwards":["Propositional calculus","Basic concepts"],"downwards":["Closure under operations","Argument"]}
{"upwards":["Propositional calculus","Basic concepts","Closure under operations"],"downwards":[]}
{"upwards":["Propositional calculus","Basic concepts","Argument"],"downwards":[]}
{"upwards":["Propositional calculus","Generic description of a propositional calculus"],"downwards":[]}
{"upwards":["Propositional calculus","Example 1. Simple axiom system"],"downwards":[]}
{"upwards":["Propositional calculus","Example 2. Natural deduction system"],"downwards":[]}
{"upwards":["Propositional calculus","Basic and derived argument forms"],"downwards":[]}
{"upwards":["Propositional calculus","Proofs in propositional calculus"],"downwards":["Example of a proof in natural deduction system","Example of a proof in a classical propositional calculus system"]}
{"upwards":["Propositional calculus","Proofs in propositional calculus","Example of a proof in natural deduction system"],"downwards":[]}
{"upwards":["Propositional calculus","Proofs in propositional calculus","Example of a proof in a classical propositional calculus system"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules"],"downwards":["Sketch of a soundness proof","Sketch of completeness proof","Another outline for a completeness proof"]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of a soundness proof"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof"],"downwards":["Example","Verifying completeness for the classical propositional calculus system"]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof","Example"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Sketch of completeness proof","Verifying completeness for the classical propositional calculus system"],"downwards":[]}
{"upwards":["Propositional calculus","Soundness and completeness of the rules","Another outline for a completeness proof"],"downwards":[]}
{"upwards":["Propositional calculus","Interpretation of a truth-functional propositional calculus"],"downwards":["Interpretation of a sentence of truth-functional propositional logic"]}
{"upwards":["Propositional calculus","Interpretation of a truth-functional propositional calculus","Interpretation of a sentence of truth-functional propositional logic"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus"],"downwards":["Axioms","Inference rule","Meta-inference rule","Example of a proof"]}
{"upwards":["Propositional calculus","Alternative calculus","Axioms"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Inference rule"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Meta-inference rule"],"downwards":[]}
{"upwards":["Propositional calculus","Alternative calculus","Example of a proof"],"downwards":[]}
{"upwards":["Propositional calculus","Equivalence to equational logics"],"downwards":[]}
{"upwards":["Propositional calculus","Graphical calculi"],"downwards":[]}
{"upwards":["Propositional calculus","Other logical calculi"],"downwards":[]}
{"upwards":["Propositional calculus","Solvers"],"downwards":[]}
{"upwards":["List of formal systems"],"downwards":["Mathematical","Other formal systems"]}
{"upwards":["List of formal systems","Mathematical"],"downwards":[]}
{"upwards":["List of formal systems","Other formal systems"],"downwards":[]}
{"upwards":["Formal language"],"downwards":["History","Words over an alphabet","Definition","Examples","Language-specification formalisms","Operations on languages","Applications"]}
{"upwards":["Formal language","History"],"downwards":[]}
{"upwards":["Formal language","Words over an alphabet"],"downwards":[]}
{"upwards":["Formal language","Definition"],"downwards":[]}
{"upwards":["Formal language","Examples"],"downwards":["Constructions"]}
{"upwards":["Formal language","Examples","Constructions"],"downwards":[]}
{"upwards":["Formal language","Language-specification formalisms"],"downwards":[]}
{"upwards":["Formal language","Operations on languages"],"downwards":[]}
{"upwards":["Formal language","Applications"],"downwards":["Programming languages","Formal theories, systems, and proofs"]}
{"upwards":["Formal language","Applications","Programming languages"],"downwards":[]}
{"upwards":["Formal language","Applications","Formal theories, systems, and proofs"],"downwards":["Interpretations and models"]}
{"upwards":["Formal language","Applications","Formal theories, systems, and proofs","Interpretations and models"],"downwards":[]}
{"upwards":["Axiom schema"],"downwards":["Formal definition","Finite axiomatization","Examples","Finitely axiomatized theories","In higher-order logic"]}
{"upwards":["Axiom schema","Formal definition"],"downwards":[]}
{"upwards":["Axiom schema","Finite axiomatization"],"downwards":[]}
{"upwards":["Axiom schema","Examples"],"downwards":[]}
{"upwards":["Axiom schema","Finitely axiomatized theories"],"downwards":[]}
{"upwards":["Axiom schema","In higher-order logic"],"downwards":[]}
{"upwards":["Arity"],"downwards":["Examples","Terminology"]}
{"upwards":["Arity","Examples"],"downwards":["Nullary","Unary","Binary","Ternary","n-ary","Varying arity"]}
{"upwards":["Arity","Examples","Nullary"],"downwards":[]}
{"upwards":["Arity","Examples","Unary"],"downwards":[]}
{"upwards":["Arity","Examples","Binary"],"downwards":[]}
{"upwards":["Arity","Examples","Ternary"],"downwards":[]}
{"upwards":["Arity","Examples","n-ary"],"downwards":[]}
{"upwards":["Arity","Examples","Varying arity"],"downwards":[]}
{"upwards":["Arity","Terminology"],"downwards":[]}
{"upwards":["Automata theory"],"downwards":["History","Automata","Variant definitions of automata","Types of automata","Hierarchy in terms of powers","Applications","Automata simulators","Connection to category theory"]}
{"upwards":["Automata theory","History"],"downwards":[]}
{"upwards":["Automata theory","Automata"],"downwards":["Informal description","Formal definition"]}
{"upwards":["Automata theory","Automata","Informal description"],"downwards":[]}
{"upwards":["Automata theory","Automata","Formal definition"],"downwards":[]}
{"upwards":["Automata theory","Variant definitions of automata"],"downwards":[]}
{"upwards":["Automata theory","Types of automata"],"downwards":["Discrete, continuous, and hybrid automata"]}
{"upwards":["Automata theory","Types of automata","Discrete, continuous, and hybrid automata"],"downwards":[]}
{"upwards":["Automata theory","Hierarchy in terms of powers"],"downwards":[]}
{"upwards":["Automata theory","Applications"],"downwards":[]}
{"upwards":["Automata theory","Automata simulators"],"downwards":[]}
{"upwards":["Automata theory","Connection to category theory"],"downwards":[]}
{"upwards":["Syntax (logic)"],"downwards":["Syntactic entities"]}
{"upwards":["Syntax (logic)","Syntactic entities"],"downwards":["Symbols","Formal language","Formation rules","Propositions","Formal theories","Formal systems","Interpretations"]}
{"upwards":["Syntax (logic)","Syntactic entities","Symbols"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal language"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formation rules"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Propositions"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal theories"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems"],"downwards":["Syntactic consequence within a formal system","Syntactic completeness of a formal system"]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems","Syntactic consequence within a formal system"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Formal systems","Syntactic completeness of a formal system"],"downwards":[]}
{"upwards":["Syntax (logic)","Syntactic entities","Interpretations"],"downwards":[]}
{"upwards":["Alphabet (formal languages)"],"downwards":["Notation","Applications"]}
{"upwards":["Alphabet (formal languages)","Notation"],"downwards":[]}
{"upwards":["Alphabet (formal languages)","Applications"],"downwards":[]}
{"upwards":["Expression (mathematics)"],"downwards":["Examples","Syntax versus semantics","Variables"]}
{"upwards":["Expression (mathematics)","Examples"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics"],"downwards":["Syntax","Semantics","Formal languages and lambda calculus"]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Syntax"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Semantics"],"downwards":[]}
{"upwards":["Expression (mathematics)","Syntax versus semantics","Formal languages and lambda calculus"],"downwards":[]}
{"upwards":["Expression (mathematics)","Variables"],"downwards":[]}
{"upwards":["Ground expression"],"downwards":["Examples","Formal definitions"]}
{"upwards":["Ground expression","Examples"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions"],"downwards":["Ground term","Ground atom","Ground formula"]}
{"upwards":["Ground expression","Formal definitions","Ground term"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground atom"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground formula"],"downwards":[]}
{"upwards":["Conservative extension"],"downwards":["Examples","Model-theoretic conservative extension"]}
{"upwards":["Conservative extension","Examples"],"downwards":[]}
{"upwards":["Conservative extension","Model-theoretic conservative extension"],"downwards":[]}
{"upwards":["Formal grammar"],"downwards":["History","Introductory example","Formal definition","The Chomsky hierarchy","Analytic grammars"]}
{"upwards":["Formal grammar","History"],"downwards":[]}
{"upwards":["Formal grammar","Introductory example"],"downwards":["Example 1","Examples 2 and 3"]}
{"upwards":["Formal grammar","Introductory example","Example 1"],"downwards":[]}
{"upwards":["Formal grammar","Introductory example","Examples 2 and 3"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition"],"downwards":["The syntax of grammars","Some mathematical constructs regarding formal grammars","Example"]}
{"upwards":["Formal grammar","Formal definition","The syntax of grammars"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition","Some mathematical constructs regarding formal grammars"],"downwards":[]}
{"upwards":["Formal grammar","Formal definition","Example"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy"],"downwards":["Context-free grammars","Regular grammars","Other forms of generative grammars","Recursive grammars"]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Context-free grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Regular grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Other forms of generative grammars"],"downwards":[]}
{"upwards":["Formal grammar","The Chomsky hierarchy","Recursive grammars"],"downwards":[]}
{"upwards":["Formal grammar","Analytic grammars"],"downwards":[]}
{"upwards":["Extension by definitions"],"downwards":["Definition of relation symbols","Definition of function symbols","Extensions by definitions","Examples"]}
{"upwards":["Extension by definitions","Definition of relation symbols"],"downwards":[]}
{"upwards":["Extension by definitions","Definition of function symbols"],"downwards":[]}
{"upwards":["Extension by definitions","Extensions by definitions"],"downwards":[]}
{"upwards":["Extension by definitions","Examples"],"downwards":[]}
{"upwards":["Extension by new constant and function names"],"downwards":[]}
{"upwards":["Atomic formula"],"downwards":["Atomic formula in first-order logic"]}
{"upwards":["Atomic formula","Atomic formula in first-order logic"],"downwards":[]}
{"upwards":["Formation rule"],"downwards":["Formal language","Formal systems","Propositional and predicate logic"]}
{"upwards":["Formation rule","Formal language"],"downwards":[]}
{"upwards":["Formation rule","Formal systems"],"downwards":[]}
{"upwards":["Formation rule","Propositional and predicate logic"],"downwards":[]}
{"upwards":["Well-formed formula"],"downwards":["Introduction","Propositional calculus","Predicate logic","Atomic and open formulas","Closed formulas","Properties applicable to formulas","Usage of the terminology"]}
{"upwards":["Well-formed formula","Introduction"],"downwards":[]}
{"upwards":["Well-formed formula","Propositional calculus"],"downwards":[]}
{"upwards":["Well-formed formula","Predicate logic"],"downwards":[]}
{"upwards":["Well-formed formula","Atomic and open formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Closed formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Properties applicable to formulas"],"downwards":[]}
{"upwards":["Well-formed formula","Usage of the terminology"],"downwards":[]}
{"upwards":["Ground expression"],"downwards":["Examples","Formal definitions"]}
{"upwards":["Ground expression","Examples"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions"],"downwards":["Ground term","Ground atom","Ground formula"]}
{"upwards":["Ground expression","Formal definitions","Ground term"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground atom"],"downwards":[]}
{"upwards":["Ground expression","Formal definitions","Ground formula"],"downwards":[]}
{"upwards":["Sentence (mathematical logic)"],"downwards":["Example"]}
{"upwards":["Sentence (mathematical logic)","Example"],"downwards":[]}
{"upwards":["Metalanguage"],"downwards":["Types of metalanguage","In natural language","Types of expressions","Role in metaphor","Metaprogramming"]}
{"upwards":["Metalanguage","Types of metalanguage"],"downwards":["Embedded","Ordered","Nested"]}
{"upwards":["Metalanguage","Types of metalanguage","Embedded"],"downwards":[]}
{"upwards":["Metalanguage","Types of metalanguage","Ordered"],"downwards":[]}
{"upwards":["Metalanguage","Types of metalanguage","Nested"],"downwards":[]}
{"upwards":["Metalanguage","In natural language"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions"],"downwards":["Deductive systems","Metavariables","Metatheories and metatheorems","Interpretations"]}
{"upwards":["Metalanguage","Types of expressions","Deductive systems"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Metavariables"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Metatheories and metatheorems"],"downwards":[]}
{"upwards":["Metalanguage","Types of expressions","Interpretations"],"downwards":[]}
{"upwards":["Metalanguage","Role in metaphor"],"downwards":[]}
{"upwards":["Metalanguage","Metaprogramming"],"downwards":[]}
{"upwards":["Free variables and bound variables"],"downwards":["Examples","Formal explanation","Natural language"]}
{"upwards":["Free variables and bound variables","Examples"],"downwards":["Variable-binding operators"]}
{"upwards":["Free variables and bound variables","Examples","Variable-binding operators"],"downwards":[]}
{"upwards":["Free variables and bound variables","Formal explanation"],"downwards":["Function expressions"]}
{"upwards":["Free variables and bound variables","Formal explanation","Function expressions"],"downwards":[]}
{"upwards":["Free variables and bound variables","Natural language"],"downwards":[]}
{"upwards":["Open formula"],"downwards":[]}
{"upwards":["Logical disjunction"],"downwards":["Inclusive and exclusive disjunction","Notation","Classical disjunction","Applications in computer science","Set theory","Natural language"]}
{"upwards":["Logical disjunction","Inclusive and exclusive disjunction"],"downwards":[]}
{"upwards":["Logical disjunction","Notation"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction"],"downwards":["Semantics","Defined by other operators","Properties"]}
{"upwards":["Logical disjunction","Classical disjunction","Semantics"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction","Defined by other operators"],"downwards":[]}
{"upwards":["Logical disjunction","Classical disjunction","Properties"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science"],"downwards":["Bitwise operation","Logical operation","Constructive disjunction"]}
{"upwards":["Logical disjunction","Applications in computer science","Bitwise operation"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science","Logical operation"],"downwards":[]}
{"upwards":["Logical disjunction","Applications in computer science","Constructive disjunction"],"downwards":[]}
{"upwards":["Logical disjunction","Set theory"],"downwards":[]}
{"upwards":["Logical disjunction","Natural language"],"downwards":[]}
{"upwards":["Universal quantification"],"downwards":["Basics","Properties","Universal closure","As adjoint"]}
{"upwards":["Universal quantification","Basics"],"downwards":["Notation"]}
{"upwards":["Universal quantification","Basics","Notation"],"downwards":[]}
{"upwards":["Universal quantification","Properties"],"downwards":["Negation","Other connectives","Rules of inference","The empty set"]}
{"upwards":["Universal quantification","Properties","Negation"],"downwards":[]}
{"upwards":["Universal quantification","Properties","Other connectives"],"downwards":[]}
{"upwards":["Universal quantification","Properties","Rules of inference"],"downwards":[]}
{"upwards":["Universal quantification","Properties","The empty set"],"downwards":[]}
{"upwards":["Universal quantification","Universal closure"],"downwards":[]}
{"upwards":["Universal quantification","As adjoint"],"downwards":[]}
{"upwards":["Logical biconditional"],"downwards":["Definition","Properties","Rules of inference","Colloquial usage"]}
{"upwards":["Logical biconditional","Definition"],"downwards":["Truth table","Venn diagrams"]}
{"upwards":["Logical biconditional","Definition","Truth table"],"downwards":[]}
{"upwards":["Logical biconditional","Definition","Venn diagrams"],"downwards":[]}
{"upwards":["Logical biconditional","Properties"],"downwards":[]}
{"upwards":["Logical biconditional","Rules of inference"],"downwards":["Biconditional introduction","Biconditional elimination"]}
{"upwards":["Logical biconditional","Rules of inference","Biconditional introduction"],"downwards":[]}
{"upwards":["Logical biconditional","Rules of inference","Biconditional elimination"],"downwards":[]}
{"upwards":["Logical biconditional","Colloquial usage"],"downwards":[]}
{"upwards":["Material conditional"],"downwards":["Notation","Definitions","Formal properties","Discrepancies with natural language"]}
{"upwards":["Material conditional","Notation"],"downwards":["History"]}
{"upwards":["Material conditional","Notation","History"],"downwards":[]}
{"upwards":["Material conditional","Definitions"],"downwards":["Semantics","Truth table","Deductive definition"]}
{"upwards":["Material conditional","Definitions","Semantics"],"downwards":[]}
{"upwards":["Material conditional","Definitions","Truth table"],"downwards":[]}
{"upwards":["Material conditional","Definitions","Deductive definition"],"downwards":[]}
{"upwards":["Material conditional","Formal properties"],"downwards":[]}
{"upwards":["Material conditional","Discrepancies with natural language"],"downwards":[]}
{"upwards":["Propositional variable"],"downwards":["Uses","Predicate logic"]}
{"upwards":["Propositional variable","Uses"],"downwards":[]}
{"upwards":["Propositional variable","Predicate logic"],"downwards":[]}
{"upwards":["Logical equality"],"downwards":["Definition","Alternative descriptions"]}
{"upwards":["Logical equality","Definition"],"downwards":[]}
{"upwards":["Logical equality","Alternative descriptions"],"downwards":[]}
{"upwards":["Predicate variable"],"downwards":["Notation","Usage"]}
{"upwards":["Predicate variable","Notation"],"downwards":[]}
{"upwards":["Predicate variable","Usage"],"downwards":[]}
{"upwards":["Functional predicate"],"downwards":["Introducing new function symbols","Doing without functional predicates"]}
{"upwards":["Functional predicate","Introducing new function symbols"],"downwards":[]}
{"upwards":["Functional predicate","Doing without functional predicates"],"downwards":[]}
{"upwards":["Logical conjunction"],"downwards":["Notation","Definition","Introduction and elimination rules","Negation","Properties","Applications in computer engineering","Set-theoretic correspondence","Natural language"]}
{"upwards":["Logical conjunction","Notation"],"downwards":[]}
{"upwards":["Logical conjunction","Definition"],"downwards":["Truth table","Defined by other operators"]}
{"upwards":["Logical conjunction","Definition","Truth table"],"downwards":[]}
{"upwards":["Logical conjunction","Definition","Defined by other operators"],"downwards":[]}
{"upwards":["Logical conjunction","Introduction and elimination rules"],"downwards":[]}
{"upwards":["Logical conjunction","Negation"],"downwards":["Definition","Other proof strategies"]}
{"upwards":["Logical conjunction","Negation","Definition"],"downwards":[]}
{"upwards":["Logical conjunction","Negation","Other proof strategies"],"downwards":[]}
{"upwards":["Logical conjunction","Properties"],"downwards":[]}
{"upwards":["Logical conjunction","Applications in computer engineering"],"downwards":[]}
{"upwards":["Logical conjunction","Set-theoretic correspondence"],"downwards":[]}
{"upwards":["Logical conjunction","Natural language"],"downwards":[]}
{"upwards":["Uniqueness quantification"],"downwards":["Proving uniqueness","Reduction to ordinary existential and universal quantification","Generalizations"]}
{"upwards":["Uniqueness quantification","Proving uniqueness"],"downwards":[]}
{"upwards":["Uniqueness quantification","Reduction to ordinary existential and universal quantification"],"downwards":[]}
{"upwards":["Uniqueness quantification","Generalizations"],"downwards":[]}
{"upwards":["Existential quantification"],"downwards":["Basics","Properties","As adjoint","Encoding","Origin"]}
{"upwards":["Existential quantification","Basics"],"downwards":[]}
{"upwards":["Existential quantification","Properties"],"downwards":["Negation","Rules of inference","The empty set"]}
{"upwards":["Existential quantification","Properties","Negation"],"downwards":[]}
{"upwards":["Existential quantification","Properties","Rules of inference"],"downwards":[]}
{"upwards":["Existential quantification","Properties","The empty set"],"downwards":[]}
{"upwards":["Existential quantification","As adjoint"],"downwards":[]}
{"upwards":["Existential quantification","Encoding"],"downwards":[]}
{"upwards":["Existential quantification","Origin"],"downwards":[]}
{"upwards":["Finitary relation"],"downwards":["Example","Definitions","History"]}
{"upwards":["Finitary relation","Example"],"downwards":[]}
{"upwards":["Finitary relation","Definitions"],"downwards":[]}
{"upwards":["Finitary relation","History"],"downwards":[]}
{"upwards":["Negation"],"downwards":["Definition","Notation","Properties","Rules of inference","Programming language and ordinary language","Kripke semantics"]}
{"upwards":["Negation","Definition"],"downwards":[]}
{"upwards":["Negation","Notation"],"downwards":[]}
{"upwards":["Negation","Properties"],"downwards":["Double negation","Distributivity","Linearity","Self dual","Negations of quantifiers"]}
{"upwards":["Negation","Properties","Double negation"],"downwards":[]}
{"upwards":["Negation","Properties","Distributivity"],"downwards":[]}
{"upwards":["Negation","Properties","Linearity"],"downwards":[]}
{"upwards":["Negation","Properties","Self dual"],"downwards":[]}
{"upwards":["Negation","Properties","Negations of quantifiers"],"downwards":[]}
{"upwards":["Negation","Rules of inference"],"downwards":[]}
{"upwards":["Negation","Programming language and ordinary language"],"downwards":[]}
{"upwards":["Negation","Kripke semantics"],"downwards":[]}
{"upwards":["Quantifier rank"],"downwards":["Definition","Examples"]}
{"upwards":["Quantifier rank","Definition"],"downwards":[]}
{"upwards":["Quantifier rank","Examples"],"downwards":[]}
{"upwards":["Atomic sentence"],"downwards":["Examples","Interpretations","Translating sentences from a natural language into an artificial language","Philosophical significance"]}
{"upwards":["Atomic sentence","Examples"],"downwards":["Assumptions","Atomic sentences","Atomic formulae","Compound sentences","Compound formulae"]}
{"upwards":["Atomic sentence","Examples","Assumptions"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Atomic sentences"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Atomic formulae"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Compound sentences"],"downwards":[]}
{"upwards":["Atomic sentence","Examples","Compound formulae"],"downwards":[]}
{"upwards":["Atomic sentence","Interpretations"],"downwards":[]}
{"upwards":["Atomic sentence","Translating sentences from a natural language into an artificial language"],"downwards":[]}
{"upwards":["Atomic sentence","Philosophical significance"],"downwards":[]}
{"upwards":["Spectrum of a sentence"],"downwards":["Definition","Examples","Properties"]}
{"upwards":["Spectrum of a sentence","Definition"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Examples"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Properties"],"downwards":["Equivalence to Turing machines","Other properties"]}
{"upwards":["Spectrum of a sentence","Properties","Equivalence to Turing machines"],"downwards":[]}
{"upwards":["Spectrum of a sentence","Properties","Other properties"],"downwards":[]}
{"upwards":["Signature (logic)"],"downwards":["Definition","Other conventions","Use of signatures in logic and algebra","Many-sorted signatures"]}
{"upwards":["Signature (logic)","Definition"],"downwards":[]}
{"upwards":["Signature (logic)","Other conventions"],"downwards":[]}
{"upwards":["Signature (logic)","Use of signatures in logic and algebra"],"downwards":[]}
{"upwards":["Signature (logic)","Many-sorted signatures"],"downwards":["Symbol types","Signature"]}
{"upwards":["Signature (logic)","Many-sorted signatures","Symbol types"],"downwards":[]}
{"upwards":["Signature (logic)","Many-sorted signatures","Signature"],"downwards":[]}
{"upwards":["Uninterpreted function"],"downwards":["Example","Discussion"]}
{"upwards":["Uninterpreted function","Example"],"downwards":[]}
{"upwards":["Uninterpreted function","Discussion"],"downwards":[]}
{"upwards":["Symbol (formal)"],"downwards":["Overview","Can words be modeled as formal symbols?","References"]}
{"upwards":["Symbol (formal)","Overview"],"downwards":[]}
{"upwards":["Symbol (formal)","Can words be modeled as formal symbols?"],"downwards":[]}
{"upwards":["Symbol (formal)","References"],"downwards":[]}
{"upwards":["Substitution (logic)"],"downwards":["Propositional logic","First-order logic","Algebra"]}
{"upwards":["Substitution (logic)","Propositional logic"],"downwards":["Definition","Tautologies"]}
{"upwards":["Substitution (logic)","Propositional logic","Definition"],"downwards":[]}
{"upwards":["Substitution (logic)","Propositional logic","Tautologies"],"downwards":[]}
{"upwards":["Substitution (logic)","First-order logic"],"downwards":[]}
{"upwards":["Substitution (logic)","Algebra"],"downwards":[]}
{"upwards":["String (computer science)"],"downwards":["Purpose","History","String datatypes","Literal strings","Non-text strings","String processing algorithms","Character string-oriented languages and utilities","Character string functions","Formal theory"]}
{"upwards":["String (computer science)","Purpose"],"downwards":[]}
{"upwards":["String (computer science)","History"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes"],"downwards":["String length","Character encoding","Implementations","Representations","Security concerns"]}
{"upwards":["String (computer science)","String datatypes","String length"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Character encoding"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Implementations"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations"],"downwards":["Null-terminated","Byte- and bit-terminated","Length-prefixed","Strings as records","Other representations"]}
{"upwards":["String (computer science)","String datatypes","Representations","Null-terminated"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Byte- and bit-terminated"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Length-prefixed"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Strings as records"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Representations","Other representations"],"downwards":[]}
{"upwards":["String (computer science)","String datatypes","Security concerns"],"downwards":[]}
{"upwards":["String (computer science)","Literal strings"],"downwards":[]}
{"upwards":["String (computer science)","Non-text strings"],"downwards":[]}
{"upwards":["String (computer science)","String processing algorithms"],"downwards":[]}
{"upwards":["String (computer science)","Character string-oriented languages and utilities"],"downwards":[]}
{"upwards":["String (computer science)","Character string functions"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory"],"downwards":["Concatenation and substrings","Prefixes and suffixes","Reversal","Rotations","Lexicographical ordering","String operations","Topology"]}
{"upwards":["String (computer science)","Formal theory","Concatenation and substrings"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Prefixes and suffixes"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Reversal"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Rotations"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Lexicographical ordering"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","String operations"],"downwards":[]}
{"upwards":["String (computer science)","Formal theory","Topology"],"downwards":[]}
{"upwards":["True arithmetic"],"downwards":["Definition","Arithmetic undefinability","Computability properties","Model-theoretic properties","True theory of second-order arithmetic","Notes","References","External links"]}
{"upwards":["True arithmetic","Definition"],"downwards":[]}
{"upwards":["True arithmetic","Arithmetic undefinability"],"downwards":[]}
{"upwards":["True arithmetic","Computability properties"],"downwards":[]}
{"upwards":["True arithmetic","Model-theoretic properties"],"downwards":[]}
{"upwards":["True arithmetic","True theory of second-order arithmetic"],"downwards":[]}
{"upwards":["True arithmetic","Notes"],"downwards":[]}
{"upwards":["True arithmetic","References"],"downwards":[]}
{"upwards":["True arithmetic","External links"],"downwards":[]}
{"upwards":["Peano axioms"],"downwards":["Historical second-order formulation","Peano arithmetic as first-order theory"]}
{"upwards":["Peano axioms","Historical second-order formulation"],"downwards":["Defining arithmetic operations and relations","Models","Consistency"]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations"],"downwards":["Addition","Multiplication","Inequalities"]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Addition"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Multiplication"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Defining arithmetic operations and relations","Inequalities"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Models"],"downwards":["Set-theoretic models","Interpretation in category theory"]}
{"upwards":["Peano axioms","Historical second-order formulation","Models","Set-theoretic models"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Models","Interpretation in category theory"],"downwards":[]}
{"upwards":["Peano axioms","Historical second-order formulation","Consistency"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory"],"downwards":["Equivalent axiomatizations","Undecidability and incompleteness","Nonstandard models"]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Equivalent axiomatizations"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Undecidability and incompleteness"],"downwards":[]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Nonstandard models"],"downwards":["Overspill"]}
{"upwards":["Peano axioms","Peano arithmetic as first-order theory","Nonstandard models","Overspill"],"downwards":[]}
{"upwards":["Non-logical symbol"],"downwards":["Signatures","Models","Informal semantics","Descriptive signs"]}
{"upwards":["Non-logical symbol","Signatures"],"downwards":[]}
{"upwards":["Non-logical symbol","Models"],"downwards":[]}
{"upwards":["Non-logical symbol","Informal semantics"],"downwards":[]}
{"upwards":["Non-logical symbol","Descriptive signs"],"downwards":[]}
{"upwards":["Variable (mathematics)"],"downwards":["History","Notation","Specific kinds of variables","Moduli spaces","Conventional variable names"]}
{"upwards":["Variable (mathematics)","History"],"downwards":[]}
{"upwards":["Variable (mathematics)","Notation"],"downwards":[]}
{"upwards":["Variable (mathematics)","Specific kinds of variables"],"downwards":["Dependent and independent variables","Examples"]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Dependent and independent variables"],"downwards":[]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Examples"],"downwards":["Example: the ideal gas law"]}
{"upwards":["Variable (mathematics)","Specific kinds of variables","Examples","Example: the ideal gas law"],"downwards":[]}
{"upwards":["Variable (mathematics)","Moduli spaces"],"downwards":[]}
{"upwards":["Variable (mathematics)","Conventional variable names"],"downwards":[]}
{"upwards":["Skolem arithmetic"],"downwards":["Expressive power","Idea of decidability","Complexity","Decidable extensions","Undecidable extensions"]}
{"upwards":["Skolem arithmetic","Expressive power"],"downwards":[]}
{"upwards":["Skolem arithmetic","Idea of decidability"],"downwards":[]}
{"upwards":["Skolem arithmetic","Complexity"],"downwards":[]}
{"upwards":["Skolem arithmetic","Decidable extensions"],"downwards":[]}
{"upwards":["Skolem arithmetic","Undecidable extensions"],"downwards":[]}
{"upwards":["Term (logic)"],"downwards":["Formal definition","Operations with terms","Related concepts"]}
{"upwards":["Term (logic)","Formal definition"],"downwards":["Term structure vs. representation","Structural equality","Ground and linear terms","Building formulas from terms"]}
{"upwards":["Term (logic)","Formal definition","Term structure vs. representation"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Structural equality"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Ground and linear terms"],"downwards":[]}
{"upwards":["Term (logic)","Formal definition","Building formulas from terms"],"downwards":[]}
{"upwards":["Term (logic)","Operations with terms"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts"],"downwards":["Sorted terms","Lambda terms"]}
{"upwards":["Term (logic)","Related concepts","Sorted terms"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts","Lambda terms"],"downwards":["Motivation","Definition"]}
{"upwards":["Term (logic)","Related concepts","Lambda terms","Motivation"],"downwards":[]}
{"upwards":["Term (logic)","Related concepts","Lambda terms","Definition"],"downwards":[]}
{"upwards":["Logical constant"],"downwards":[]}
{"upwards":["Elementary function arithmetic"],"downwards":["Definition","Friedman's grand conjecture","Related systems"]}
{"upwards":["Elementary function arithmetic","Definition"],"downwards":[]}
{"upwards":["Elementary function arithmetic","Friedman's grand conjecture"],"downwards":[]}
{"upwards":["Elementary function arithmetic","Related systems"],"downwards":[]}
{"upwards":["Robinson arithmetic"],"downwards":["Axioms","Metamathematics"]}
{"upwards":["Robinson arithmetic","Axioms"],"downwards":["Variant axiomatizations"]}
{"upwards":["Robinson arithmetic","Axioms","Variant axiomatizations"],"downwards":[]}
{"upwards":["Robinson arithmetic","Metamathematics"],"downwards":[]}
{"upwards":["Primitive recursive arithmetic"],"downwards":["Language and axioms","Logic-free calculus"]}
{"upwards":["Primitive recursive arithmetic","Language and axioms"],"downwards":[]}
{"upwards":["Primitive recursive arithmetic","Logic-free calculus"],"downwards":[]}
{"upwards":["Second-order arithmetic"],"downwards":["Definition","Models","Subsystems","Projective determinacy","Coding mathematics"]}
{"upwards":["Second-order arithmetic","Definition"],"downwards":["Syntax","Semantics","Axioms","The full system"]}
{"upwards":["Second-order arithmetic","Definition","Syntax"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Semantics"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Axioms"],"downwards":["Basic","Induction and comprehension schema"]}
{"upwards":["Second-order arithmetic","Definition","Axioms","Basic"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","Axioms","Induction and comprehension schema"],"downwards":[]}
{"upwards":["Second-order arithmetic","Definition","The full system"],"downwards":[]}
{"upwards":["Second-order arithmetic","Models"],"downwards":["Definable functions","More types of models"]}
{"upwards":["Second-order arithmetic","Models","Definable functions"],"downwards":[]}
{"upwards":["Second-order arithmetic","Models","More types of models"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems"],"downwards":["Arithmetical comprehension","The arithmetical hierarchy for formulas","Recursive comprehension","Weaker systems","Stronger systems"]}
{"upwards":["Second-order arithmetic","Subsystems","Arithmetical comprehension"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","The arithmetical hierarchy for formulas"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Recursive comprehension"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Weaker systems"],"downwards":[]}
{"upwards":["Second-order arithmetic","Subsystems","Stronger systems"],"downwards":[]}
{"upwards":["Second-order arithmetic","Projective determinacy"],"downwards":[]}
{"upwards":["Second-order arithmetic","Coding mathematics"],"downwards":[]}
{"upwards":["Euclid's Elements"],"downwards":["History","Influence","Contents","Euclid's method and style of presentation","Criticism","Apocrypha","Editions","References","External links"]}
{"upwards":["Euclid's Elements","History"],"downwards":["Basis in earlier work","Transmission of the text"]}
{"upwards":["Euclid's Elements","History","Basis in earlier work"],"downwards":[]}
{"upwards":["Euclid's Elements","History","Transmission of the text"],"downwards":[]}
{"upwards":["Euclid's Elements","Influence"],"downwards":["In modern mathematics"]}
{"upwards":["Euclid's Elements","Influence","In modern mathematics"],"downwards":[]}
{"upwards":["Euclid's Elements","Contents"],"downwards":[]}
{"upwards":["Euclid's Elements","Euclid's method and style of presentation"],"downwards":[]}
{"upwards":["Euclid's Elements","Criticism"],"downwards":[]}
{"upwards":["Euclid's Elements","Apocrypha"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions"],"downwards":["Translations","Currently in print","Free versions"]}
{"upwards":["Euclid's Elements","Editions","Translations"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions","Currently in print"],"downwards":[]}
{"upwards":["Euclid's Elements","Editions","Free versions"],"downwards":[]}
{"upwards":["Euclid's Elements","References"],"downwards":["Notes","Citations","Sources"]}
{"upwards":["Euclid's Elements","References","Notes"],"downwards":[]}
{"upwards":["Euclid's Elements","References","Citations"],"downwards":[]}
{"upwards":["Euclid's Elements","References","Sources"],"downwards":[]}
{"upwards":["Euclid's Elements","External links"],"downwards":[]}
{"upwards":["Foundations of geometry"],"downwards":["Axiomatic systems","Euclidean geometry","Non-Euclidean geometry","Other geometries"]}
{"upwards":["Foundations of geometry","Axiomatic systems"],"downwards":["Properties of axiomatic systems"]}
{"upwards":["Foundations of geometry","Axiomatic systems","Properties of axiomatic systems"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry"],"downwards":["Euclid's Elements","A critique of Euclid","Pasch and Peano","Pieri and the Italian school of geometers","Hilbert's axioms","Birkhoff's axioms","School geometry","Other systems"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Euclid's Elements"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","A critique of Euclid"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Pasch and Peano"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Pieri and the Italian school of geometers"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms"],"downwards":["Changes in Hilbert's axioms","Consistency and independence"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms","Changes in Hilbert's axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Hilbert's axioms","Consistency and independence"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Birkhoff's axioms"],"downwards":["Postulates"]}
{"upwards":["Foundations of geometry","Euclidean geometry","Birkhoff's axioms","Postulates"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry"],"downwards":["Mac Lane's axioms","SMSG (School Mathematics Study Group) axioms","UCSMP (University of Chicago School Mathematics Project) axioms"]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","Mac Lane's axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","SMSG (School Mathematics Study Group) axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","School geometry","UCSMP (University of Chicago School Mathematics Project) axioms"],"downwards":[]}
{"upwards":["Foundations of geometry","Euclidean geometry","Other systems"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry"],"downwards":["Euclid's parallel postulate","Neutral (or absolute) geometry","Hyperbolic geometry","Elliptic geometry"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Euclid's parallel postulate"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry"],"downwards":["Relation to other geometries","Incompleteness"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry","Relation to other geometries"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Neutral (or absolute) geometry","Incompleteness"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Hyperbolic geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Elliptic geometry"],"downwards":["Spherical geometry"]}
{"upwards":["Foundations of geometry","Non-Euclidean geometry","Elliptic geometry","Spherical geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries"],"downwards":["Projective geometry","Affine geometry","Ordered geometry","Finite geometry"]}
{"upwards":["Foundations of geometry","Other geometries","Projective geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Affine geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Ordered geometry"],"downwards":[]}
{"upwards":["Foundations of geometry","Other geometries","Finite geometry"],"downwards":[]}
{"upwards":["Boolean algebra (structure)"],"downwards":["History","Definition","Examples","Homomorphisms and isomorphisms","Boolean rings","Ideals and filters","Representations","Axiomatics","Generalizations"]}
{"upwards":["Boolean algebra (structure)","History"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Definition"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Examples"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Homomorphisms and isomorphisms"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Boolean rings"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Ideals and filters"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Representations"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Axiomatics"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Generalizations"],"downwards":[]}
{"upwards":["Construction of the real numbers"],"downwards":["Axiomatic definitions","Explicit constructions of models"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions"],"downwards":["Axioms","Tarski's axiomatization of the reals"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms"],"downwards":["On the least upper bound property","On models"]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms","On the least upper bound property"],"downwards":[]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Axioms","On models"],"downwards":[]}
{"upwards":["Construction of the real numbers","Axiomatic definitions","Tarski's axiomatization of the reals"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models"],"downwards":["Construction from Cauchy sequences","Construction by Dedekind cuts","Construction using hyperreal numbers","Construction from surreal numbers","Construction from integers (Eudoxus reals)","Other constructions"]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from Cauchy sequences"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction by Dedekind cuts"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction using hyperreal numbers"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from surreal numbers"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Construction from integers (Eudoxus reals)"],"downwards":[]}
{"upwards":["Construction of the real numbers","Explicit constructions of models","Other constructions"],"downwards":[]}
{"upwards":["Minimal axioms for Boolean algebra"],"downwards":["References"]}
{"upwards":["Minimal axioms for Boolean algebra","References"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined"],"downwards":["Definition","Basis","Truth tables","Examples","Boolean algebras of Boolean operations","Axiomatizing Boolean algebras","Underlying lattice structure","Boolean homomorphisms","Infinitary extensions","Other definitions of Boolean algebra"]}
{"upwards":["Boolean algebras canonically defined","Definition"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Basis"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Truth tables"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples"],"downwards":["Bit vectors","Power set algebra","Representation theorems","Other examples"]}
{"upwards":["Boolean algebras canonically defined","Examples","Bit vectors"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Power set algebra"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Representation theorems"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Examples","Other examples"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Boolean algebras of Boolean operations"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Axiomatizing Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Underlying lattice structure"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Boolean homomorphisms"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Infinitary extensions"],"downwards":[]}
{"upwards":["Boolean algebras canonically defined","Other definitions of Boolean algebra"],"downwards":[]}
{"upwards":["List of mathematical theories"],"downwards":[]}
{"upwards":["Natural deduction"],"downwards":["Motivation","Judgments and propositions","Introduction and elimination","Hypothetical derivations","Consistency, completeness, and normal forms","First and higher-order extensions","Different presentations of natural deduction","Proofs and type theory","Classical and modal logics","Comparison with other foundational approaches"]}
{"upwards":["Natural deduction","Motivation"],"downwards":[]}
{"upwards":["Natural deduction","Judgments and propositions"],"downwards":[]}
{"upwards":["Natural deduction","Introduction and elimination"],"downwards":[]}
{"upwards":["Natural deduction","Hypothetical derivations"],"downwards":[]}
{"upwards":["Natural deduction","Consistency, completeness, and normal forms"],"downwards":[]}
{"upwards":["Natural deduction","First and higher-order extensions"],"downwards":[]}
{"upwards":["Natural deduction","Different presentations of natural deduction"],"downwards":["Tree-like presentations","Sequential presentations"]}
{"upwards":["Natural deduction","Different presentations of natural deduction","Tree-like presentations"],"downwards":[]}
{"upwards":["Natural deduction","Different presentations of natural deduction","Sequential presentations"],"downwards":[]}
{"upwards":["Natural deduction","Proofs and type theory"],"downwards":["Example: Dependent Type Theory"]}
{"upwards":["Natural deduction","Proofs and type theory","Example: Dependent Type Theory"],"downwards":[]}
{"upwards":["Natural deduction","Classical and modal logics"],"downwards":[]}
{"upwards":["Natural deduction","Comparison with other foundational approaches"],"downwards":["Sequent calculus"]}
{"upwards":["Natural deduction","Comparison with other foundational approaches","Sequent calculus"],"downwards":[]}
{"upwards":["Rule of inference"],"downwards":["Standard form","Example: Hilbert systems for two propositional logics","Admissibility and derivability"]}
{"upwards":["Rule of inference","Standard form"],"downwards":[]}
{"upwards":["Rule of inference","Example: Hilbert systems for two propositional logics"],"downwards":[]}
{"upwards":["Rule of inference","Admissibility and derivability"],"downwards":[]}
{"upwards":["Tarski's axioms"],"downwards":["Overview","The axioms","Discussion","Comparison with Hilbert"]}
{"upwards":["Tarski's axioms","Overview"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms"],"downwards":["Fundamental relations","Congruence axioms","Betweenness axioms","Congruence and betweenness"]}
{"upwards":["Tarski's axioms","The axioms","Fundamental relations"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Congruence axioms"],"downwards":["Commentary"]}
{"upwards":["Tarski's axioms","The axioms","Congruence axioms","Commentary"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Betweenness axioms"],"downwards":[]}
{"upwards":["Tarski's axioms","The axioms","Congruence and betweenness"],"downwards":[]}
{"upwards":["Tarski's axioms","Discussion"],"downwards":[]}
{"upwards":["Tarski's axioms","Comparison with Hilbert"],"downwards":[]}
{"upwards":["Non-Euclidean geometry"],"downwards":["Principles","History","Axiomatic basis of non-Euclidean geometry","Models","Uncommon properties","Importance","Planar algebras","Kinematic geometries","Fiction"]}
{"upwards":["Non-Euclidean geometry","Principles"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History"],"downwards":["Background","Discovery of non-Euclidean geometry","Terminology"]}
{"upwards":["Non-Euclidean geometry","History","Background"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History","Discovery of non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","History","Terminology"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Axiomatic basis of non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models"],"downwards":["Elliptic geometry","Hyperbolic geometry","Three-dimensional non-Euclidean geometry"]}
{"upwards":["Non-Euclidean geometry","Models","Elliptic geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models","Hyperbolic geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Models","Three-dimensional non-Euclidean geometry"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Uncommon properties"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Importance"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Planar algebras"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Kinematic geometries"],"downwards":[]}
{"upwards":["Non-Euclidean geometry","Fiction"],"downwards":[]}
{"upwards":["Proof theory"],"downwards":["History","Structural proof theory","Ordinal analysis","Provability logic","Reverse mathematics","Functional interpretations","Formal and informal proof","Proof-theoretic semantics"]}
{"upwards":["Proof theory","History"],"downwards":[]}
{"upwards":["Proof theory","Structural proof theory"],"downwards":[]}
{"upwards":["Proof theory","Ordinal analysis"],"downwards":[]}
{"upwards":["Proof theory","Provability logic"],"downwards":[]}
{"upwards":["Proof theory","Reverse mathematics"],"downwards":[]}
{"upwards":["Proof theory","Functional interpretations"],"downwards":[]}
{"upwards":["Proof theory","Formal and informal proof"],"downwards":[]}
{"upwards":["Proof theory","Proof-theoretic semantics"],"downwards":[]}
{"upwards":["Euclidean geometry"],"downwards":["The Elements","Notation and terminology","Some important or well known results","System of measurement and arithmetic","Applications","Later history","As a description of the structure of space","Treatment of infinity","Logical basis"]}
{"upwards":["Euclidean geometry","The Elements"],"downwards":["Axioms","Parallel postulate","Methods of proof"]}
{"upwards":["Euclidean geometry","The Elements","Axioms"],"downwards":[]}
{"upwards":["Euclidean geometry","The Elements","Parallel postulate"],"downwards":[]}
{"upwards":["Euclidean geometry","The Elements","Methods of proof"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology"],"downwards":["Naming of points and figures","Complementary and supplementary angles","Modern versions of Euclid's notation"]}
{"upwards":["Euclidean geometry","Notation and terminology","Naming of points and figures"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology","Complementary and supplementary angles"],"downwards":[]}
{"upwards":["Euclidean geometry","Notation and terminology","Modern versions of Euclid's notation"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results"],"downwards":["Pons asinorum","Congruence of triangles","Triangle angle sum","Pythagorean theorem","Thales' theorem","Scaling of area and volume"]}
{"upwards":["Euclidean geometry","Some important or well known results","Pons asinorum"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Congruence of triangles"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Triangle angle sum"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Pythagorean theorem"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Thales' theorem"],"downwards":[]}
{"upwards":["Euclidean geometry","Some important or well known results","Scaling of area and volume"],"downwards":[]}
{"upwards":["Euclidean geometry","System of measurement and arithmetic"],"downwards":[]}
{"upwards":["Euclidean geometry","Applications"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history"],"downwards":["Archimedes and Apollonius","17th century: Descartes","18th century","19th century","20th century and relativity"]}
{"upwards":["Euclidean geometry","Later history","Archimedes and Apollonius"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","17th century: Descartes"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","18th century"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","19th century"],"downwards":["Higher dimensions","Non-Euclidean geometry"]}
{"upwards":["Euclidean geometry","Later history","19th century","Higher dimensions"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","19th century","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Euclidean geometry","Later history","20th century and relativity"],"downwards":[]}
{"upwards":["Euclidean geometry","As a description of the structure of space"],"downwards":[]}
{"upwards":["Euclidean geometry","Treatment of infinity"],"downwards":["Infinite objects","Infinite processes"]}
{"upwards":["Euclidean geometry","Treatment of infinity","Infinite objects"],"downwards":[]}
{"upwards":["Euclidean geometry","Treatment of infinity","Infinite processes"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis"],"downwards":["Classical logic","Modern standards of rigor","Axiomatic formulations"]}
{"upwards":["Euclidean geometry","Logical basis","Classical logic"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis","Modern standards of rigor"],"downwards":[]}
{"upwards":["Euclidean geometry","Logical basis","Axiomatic formulations"],"downwards":[]}
{"upwards":["Sequent calculus"],"downwards":["Overview","Proving logical formulas","The system LK","Variants"]}
{"upwards":["Sequent calculus","Overview"],"downwards":["Hilbert-style deduction systems","Natural deduction systems","Sequent calculus systems","Distinction between natural deduction and sequent calculus","Origin of word \"sequent\""]}
{"upwards":["Sequent calculus","Overview","Hilbert-style deduction systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Natural deduction systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Sequent calculus systems"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Distinction between natural deduction and sequent calculus"],"downwards":[]}
{"upwards":["Sequent calculus","Overview","Origin of word \"sequent\""],"downwards":[]}
{"upwards":["Sequent calculus","Proving logical formulas"],"downwards":["Reduction trees","Relation to standard axiomatizations"]}
{"upwards":["Sequent calculus","Proving logical formulas","Reduction trees"],"downwards":[]}
{"upwards":["Sequent calculus","Proving logical formulas","Relation to standard axiomatizations"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK"],"downwards":["Inference rules","An intuitive explanation","Example derivations","Relation to analytic tableaux","Structural rules","Properties of the system LK"]}
{"upwards":["Sequent calculus","The system LK","Inference rules"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","An intuitive explanation"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Example derivations"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Relation to analytic tableaux"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Structural rules"],"downwards":[]}
{"upwards":["Sequent calculus","The system LK","Properties of the system LK"],"downwards":[]}
{"upwards":["Sequent calculus","Variants"],"downwards":["Minor structural alternatives","Absurdity","Substructural logics","Intuitionistic sequent calculus: System LJ"]}
{"upwards":["Sequent calculus","Variants","Minor structural alternatives"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Absurdity"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Substructural logics"],"downwards":[]}
{"upwards":["Sequent calculus","Variants","Intuitionistic sequent calculus: System LJ"],"downwards":[]}
{"upwards":["Axiomatic system"],"downwards":["Properties","Relative consistency","Models","Axiomatic method"]}
{"upwards":["Axiomatic system","Properties"],"downwards":[]}
{"upwards":["Axiomatic system","Relative consistency"],"downwards":[]}
{"upwards":["Axiomatic system","Models"],"downwards":["Example"]}
{"upwards":["Axiomatic system","Models","Example"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method"],"downwards":["History","Issues","Example: The Peano axiomatization of natural numbers","Axiomatization"]}
{"upwards":["Axiomatic system","Axiomatic method","History"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Issues"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Example: The Peano axiomatization of natural numbers"],"downwards":[]}
{"upwards":["Axiomatic system","Axiomatic method","Axiomatization"],"downwards":[]}
{"upwards":["Tarski's axiomatization of the reals"],"downwards":["The axioms","How these axioms imply a field"]}
{"upwards":["Tarski's axiomatization of the reals","The axioms"],"downwards":[]}
{"upwards":["Tarski's axiomatization of the reals","How these axioms imply a field"],"downwards":[]}
{"upwards":["Hilbert's axioms"],"downwards":["The axioms","Hilbert's discarded axiom","Editions and translations of Grundlagen der Geometrie","Application"]}
{"upwards":["Hilbert's axioms","The axioms"],"downwards":["I. Incidence","II. Order","III. Congruence","IV. Parallels","V. Continuity"]}
{"upwards":["Hilbert's axioms","The axioms","I. Incidence"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","II. Order"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","III. Congruence"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","IV. Parallels"],"downwards":[]}
{"upwards":["Hilbert's axioms","The axioms","V. Continuity"],"downwards":[]}
{"upwards":["Hilbert's axioms","Hilbert's discarded axiom"],"downwards":[]}
{"upwards":["Hilbert's axioms","Editions and translations of Grundlagen der Geometrie"],"downwards":[]}
{"upwards":["Hilbert's axioms","Application"],"downwards":[]}
{"upwards":["Formal system"],"downwards":["Background","History","Examples","Variants"]}
{"upwards":["Formal system","Background"],"downwards":["Recursive","Inference and entailment"]}
{"upwards":["Formal system","Background","Recursive"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment"],"downwards":["Formal language","Deductive system","Logical system"]}
{"upwards":["Formal system","Background","Inference and entailment","Formal language"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Deductive system"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Logical system"],"downwards":[]}
{"upwards":["Formal system","History"],"downwards":["Formalism","QED manifesto"]}
{"upwards":["Formal system","History","Formalism"],"downwards":["Hilbert's program"]}
{"upwards":["Formal system","History","Formalism","Hilbert's program"],"downwards":[]}
{"upwards":["Formal system","History","QED manifesto"],"downwards":[]}
{"upwards":["Formal system","Examples"],"downwards":[]}
{"upwards":["Formal system","Variants"],"downwards":["Proof system"]}
{"upwards":["Formal system","Variants","Proof system"],"downwards":[]}
{"upwards":["Hilbert system"],"downwards":["Formal deductions","Conservative extensions","Metatheorems","Some useful theorems and their proofs","Alternative axiomatizations","Further connections"]}
{"upwards":["Hilbert system","Formal deductions"],"downwards":["Logical axioms"]}
{"upwards":["Hilbert system","Formal deductions","Logical axioms"],"downwards":[]}
{"upwards":["Hilbert system","Conservative extensions"],"downwards":["Existential quantification","Conjunction and disjunction"]}
{"upwards":["Hilbert system","Conservative extensions","Existential quantification"],"downwards":[]}
{"upwards":["Hilbert system","Conservative extensions","Conjunction and disjunction"],"downwards":[]}
{"upwards":["Hilbert system","Metatheorems"],"downwards":[]}
{"upwards":["Hilbert system","Some useful theorems and their proofs"],"downwards":[]}
{"upwards":["Hilbert system","Alternative axiomatizations"],"downwards":[]}
{"upwards":["Hilbert system","Further connections"],"downwards":[]}
{"upwards":["List of Hilbert systems"],"downwards":["Classical propositional calculus systems","Implicational propositional calculus","Intuitionistic and intermediate logics","Positive implicational calculus","Positive propositional calculus","Equivalential calculus"]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems"],"downwards":["Implication and negation","Implication and falsum","Negation and disjunction","Sheffer's stroke"]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Implication and negation"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Implication and falsum"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Negation and disjunction"],"downwards":[]}
{"upwards":["List of Hilbert systems","Classical propositional calculus systems","Sheffer's stroke"],"downwards":[]}
{"upwards":["List of Hilbert systems","Implicational propositional calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Intuitionistic and intermediate logics"],"downwards":[]}
{"upwards":["List of Hilbert systems","Positive implicational calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Positive propositional calculus"],"downwards":[]}
{"upwards":["List of Hilbert systems","Equivalential calculus"],"downwards":[]}
{"upwards":["Ordinal analysis"],"downwards":["History","Definition","Upper bound","Examples","Table of ordinal analyses"]}
{"upwards":["Ordinal analysis","History"],"downwards":[]}
{"upwards":["Ordinal analysis","Definition"],"downwards":["Ordinal notations"]}
{"upwards":["Ordinal analysis","Definition","Ordinal notations"],"downwards":[]}
{"upwards":["Ordinal analysis","Upper bound"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples"],"downwards":["Theories with proof-theoretic ordinal \u03c9","Theories with proof-theoretic ordinal \u03c92","Theories with proof-theoretic ordinal \u03c93","Theories with proof-theoretic ordinal \u03c9n (for n = 2, 3, ... \u03c9)","Theories with proof-theoretic ordinal \u03c9\u03c9","Theories with proof-theoretic ordinal \u03b50","Theories with proof-theoretic ordinal the Feferman\u2013Sch\u00fctte ordinal \u03930","Theories with proof-theoretic ordinal the Bachmann\u2013Howard ordinal","Theories with larger proof-theoretic ordinals"]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c92"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c93"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9n (for n = 2, 3, ... \u03c9)"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03c9\u03c9"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal \u03b50"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal the Feferman\u2013Sch\u00fctte ordinal \u03930"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with proof-theoretic ordinal the Bachmann\u2013Howard ordinal"],"downwards":[]}
{"upwards":["Ordinal analysis","Examples","Theories with larger proof-theoretic ordinals"],"downwards":[]}
{"upwards":["Ordinal analysis","Table of ordinal analyses"],"downwards":["Key"]}
{"upwards":["Ordinal analysis","Table of ordinal analyses","Key"],"downwards":[]}
{"upwards":["Self-verifying theories"],"downwards":["References","External links"]}
{"upwards":["Self-verifying theories","References"],"downwards":[]}
{"upwards":["Self-verifying theories","External links"],"downwards":[]}
{"upwards":["Complete theory"],"downwards":["Examples"]}
{"upwards":["Complete theory","Examples"],"downwards":[]}
{"upwards":["Model theory"],"downwards":["Overview","Fundamental notions of first-order model theory","Definability","Types","Constructing models","Categoricity","Stability theory","Non-elementary model theory","Selected applications","History","Connections to related branches of mathematical logic"]}
{"upwards":["Model theory","Overview"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory"],"downwards":["First-order logic","Basic model-theoretic concepts","Compactness and the L\u00f6wenheim-Skolem theorem"]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","First-order logic"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","Basic model-theoretic concepts"],"downwards":[]}
{"upwards":["Model theory","Fundamental notions of first-order model theory","Compactness and the L\u00f6wenheim-Skolem theorem"],"downwards":[]}
{"upwards":["Model theory","Definability"],"downwards":["Definable sets","Eliminating quantifiers","Minimality","Definable and interpretable structures"]}
{"upwards":["Model theory","Definability","Definable sets"],"downwards":[]}
{"upwards":["Model theory","Definability","Eliminating quantifiers"],"downwards":[]}
{"upwards":["Model theory","Definability","Minimality"],"downwards":[]}
{"upwards":["Model theory","Definability","Definable and interpretable structures"],"downwards":[]}
{"upwards":["Model theory","Types"],"downwards":["Basic notions","Structures and types","Stone spaces"]}
{"upwards":["Model theory","Types","Basic notions"],"downwards":[]}
{"upwards":["Model theory","Types","Structures and types"],"downwards":[]}
{"upwards":["Model theory","Types","Stone spaces"],"downwards":[]}
{"upwards":["Model theory","Constructing models"],"downwards":["Realising and omitting types","Ultraproducts"]}
{"upwards":["Model theory","Constructing models","Realising and omitting types"],"downwards":[]}
{"upwards":["Model theory","Constructing models","Ultraproducts"],"downwards":[]}
{"upwards":["Model theory","Categoricity"],"downwards":["\u03c9-categoricity","Uncountable categoricity"]}
{"upwards":["Model theory","Categoricity","\u03c9-categoricity"],"downwards":[]}
{"upwards":["Model theory","Categoricity","Uncountable categoricity"],"downwards":[]}
{"upwards":["Model theory","Stability theory"],"downwards":["The stability hierarchy","Geometric stability theory"]}
{"upwards":["Model theory","Stability theory","The stability hierarchy"],"downwards":[]}
{"upwards":["Model theory","Stability theory","Geometric stability theory"],"downwards":[]}
{"upwards":["Model theory","Non-elementary model theory"],"downwards":[]}
{"upwards":["Model theory","Selected applications"],"downwards":[]}
{"upwards":["Model theory","History"],"downwards":[]}
{"upwards":["Model theory","Connections to related branches of mathematical logic"],"downwards":["Finite model theory","Set theory"]}
{"upwards":["Model theory","Connections to related branches of mathematical logic","Finite model theory"],"downwards":[]}
{"upwards":["Model theory","Connections to related branches of mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Structure (mathematical logic)"],"downwards":["Definition","Induced substructures and closed subsets","Homomorphisms and embeddings","Structures and first-order logic","Many-sorted structures","Other generalizations"]}
{"upwards":["Structure (mathematical logic)","Definition"],"downwards":["Domain","Signature","Interpretation function","Examples"]}
{"upwards":["Structure (mathematical logic)","Definition","Domain"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Signature"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Interpretation function"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Definition","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets"],"downwards":["Examples"]}
{"upwards":["Structure (mathematical logic)","Induced substructures and closed subsets","Examples"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings"],"downwards":["Homomorphisms","Embeddings","Example","Homomorphism problem"]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphisms"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Embeddings"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Example"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Homomorphisms and embeddings","Homomorphism problem"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic"],"downwards":["Satisfaction relation","Definable relations"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Satisfaction relation"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations"],"downwards":["Definability with parameters","Implicit definability"]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Definability with parameters"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Structures and first-order logic","Definable relations","Implicit definability"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Many-sorted structures"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations"],"downwards":["Partial algebras","Structures for typed languages","Higher-order languages","Structures that are proper classes"]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Partial algebras"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures for typed languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Higher-order languages"],"downwards":[]}
{"upwards":["Structure (mathematical logic)","Other generalizations","Structures that are proper classes"],"downwards":[]}
{"upwards":["Interpretation (logic)"],"downwards":["Formal languages","General properties of truth-functional interpretations","Interpretation of a theory","Interpretations for propositional logic","First-order logic","Higher-order predicate logics","Non-classical interpretations","Intended interpretations","Other concepts of interpretation"]}
{"upwards":["Interpretation (logic)","Formal languages"],"downwards":["Example","Logical constants"]}
{"upwards":["Interpretation (logic)","Formal languages","Example"],"downwards":[]}
{"upwards":["Interpretation (logic)","Formal languages","Logical constants"],"downwards":[]}
{"upwards":["Interpretation (logic)","General properties of truth-functional interpretations"],"downwards":["Logical connectives"]}
{"upwards":["Interpretation (logic)","General properties of truth-functional interpretations","Logical connectives"],"downwards":[]}
{"upwards":["Interpretation (logic)","Interpretation of a theory"],"downwards":[]}
{"upwards":["Interpretation (logic)","Interpretations for propositional logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic"],"downwards":["Formal languages for first-order logic","Interpretations of a first-order language","Example of a first-order interpretation","Non-empty domain requirement","Interpreting equality","Many-sorted first-order logic"]}
{"upwards":["Interpretation (logic)","First-order logic","Formal languages for first-order logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Interpretations of a first-order language"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Example of a first-order interpretation"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Non-empty domain requirement"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Interpreting equality"],"downwards":[]}
{"upwards":["Interpretation (logic)","First-order logic","Many-sorted first-order logic"],"downwards":[]}
{"upwards":["Interpretation (logic)","Higher-order predicate logics"],"downwards":[]}
{"upwards":["Interpretation (logic)","Non-classical interpretations"],"downwards":[]}
{"upwards":["Interpretation (logic)","Intended interpretations"],"downwards":["Example"]}
{"upwards":["Interpretation (logic)","Intended interpretations","Example"],"downwards":[]}
{"upwards":["Interpretation (logic)","Other concepts of interpretation"],"downwards":[]}
{"upwards":["List of statements independent of ZFC"],"downwards":["Axiomatic set theory","Set theory of the real line","Order theory","Abstract algebra","Number theory","Measure theory","Topology","Functional analysis","Model theory","Computability theory","References","External links"]}
{"upwards":["List of statements independent of ZFC","Axiomatic set theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Set theory of the real line"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Order theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Abstract algebra"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Number theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Measure theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Topology"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Functional analysis"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Model theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","Computability theory"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","References"],"downwards":[]}
{"upwards":["List of statements independent of ZFC","External links"],"downwards":[]}
{"upwards":["Interpretation (model theory)"],"downwards":["Definition","Bi-interpretability","Example","References"]}
{"upwards":["Interpretation (model theory)","Definition"],"downwards":[]}
{"upwards":["Interpretation (model theory)","Bi-interpretability"],"downwards":[]}
{"upwards":["Interpretation (model theory)","Example"],"downwards":[]}
{"upwards":["Interpretation (model theory)","References"],"downwards":[]}
{"upwards":["Proof of impossibility"],"downwards":["Types of proof","Types of disproof","Pythagorean proof of the existence of irrational numbers","Impossible constructions sought by the ancient Greeks","Euclid's parallel postulate","Fermat's Last Theorem","Richard's paradox","Incompleteness of axiomatic systems: G\u00f6del's proof","Halting problem: Turing's first proof","String compressibility: Chaitin's proof","Integer solutions of Diophantine equations: Hilbert's tenth problem","In social science","In natural science"]}
{"upwards":["Proof of impossibility","Types of proof"],"downwards":["By contradiction"]}
{"upwards":["Proof of impossibility","Types of proof","By contradiction"],"downwards":["By descent"]}
{"upwards":["Proof of impossibility","Types of proof","By contradiction","By descent"],"downwards":[]}
{"upwards":["Proof of impossibility","Types of disproof"],"downwards":[]}
{"upwards":["Proof of impossibility","Pythagorean proof of the existence of irrational numbers"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks"],"downwards":["Angle trisection and doubling the cube","Squaring the circle","Constructing an equilateral n-gon"]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Angle trisection and doubling the cube"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Squaring the circle"],"downwards":[]}
{"upwards":["Proof of impossibility","Impossible constructions sought by the ancient Greeks","Constructing an equilateral n-gon"],"downwards":[]}
{"upwards":["Proof of impossibility","Euclid's parallel postulate"],"downwards":[]}
{"upwards":["Proof of impossibility","Fermat's Last Theorem"],"downwards":[]}
{"upwards":["Proof of impossibility","Richard's paradox"],"downwards":[]}
{"upwards":["Proof of impossibility","Incompleteness of axiomatic systems: G\u00f6del's proof"],"downwards":[]}
{"upwards":["Proof of impossibility","Halting problem: Turing's first proof"],"downwards":[]}
{"upwards":["Proof of impossibility","String compressibility: Chaitin's proof"],"downwards":[]}
{"upwards":["Proof of impossibility","Integer solutions of Diophantine equations: Hilbert's tenth problem"],"downwards":[]}
{"upwards":["Proof of impossibility","In social science"],"downwards":[]}
{"upwards":["Proof of impossibility","In natural science"],"downwards":[]}
{"upwards":["Finite model theory"],"downwards":["Axiomatisability","Zero-one laws","Descriptive complexity theory","Applications","History","Citations","References","Further reading","External links"]}
{"upwards":["Finite model theory","Axiomatisability"],"downwards":["Characterisation of a single structure","Characterisation of a class of structures"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure"],"downwards":["Problem","Approach","Extension to a fixed number of structures","Extension to an infinite structure"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Problem"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Approach"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Extension to a fixed number of structures"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a single structure","Extension to an infinite structure"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures"],"downwards":["Problem","Approach","Example"]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Problem"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Approach"],"downwards":[]}
{"upwards":["Finite model theory","Axiomatisability","Characterisation of a class of structures","Example"],"downwards":[]}
{"upwards":["Finite model theory","Zero-one laws"],"downwards":[]}
{"upwards":["Finite model theory","Descriptive complexity theory"],"downwards":[]}
{"upwards":["Finite model theory","Applications"],"downwards":["Database theory","Querying and search"]}
{"upwards":["Finite model theory","Applications","Database theory"],"downwards":[]}
{"upwards":["Finite model theory","Applications","Querying and search"],"downwards":[]}
{"upwards":["Finite model theory","History"],"downwards":[]}
{"upwards":["Finite model theory","Citations"],"downwards":[]}
{"upwards":["Finite model theory","References"],"downwards":[]}
{"upwards":["Finite model theory","Further reading"],"downwards":[]}
{"upwards":["Finite model theory","External links"],"downwards":[]}
{"upwards":["Elementary equivalence"],"downwards":["Elementarily equivalent structures","Elementary substructures and elementary extensions","Tarski\u2013Vaught test","Elementary embeddings","References"]}
{"upwards":["Elementary equivalence","Elementarily equivalent structures"],"downwards":[]}
{"upwards":["Elementary equivalence","Elementary substructures and elementary extensions"],"downwards":[]}
{"upwards":["Elementary equivalence","Tarski\u2013Vaught test"],"downwards":[]}
{"upwards":["Elementary equivalence","Elementary embeddings"],"downwards":[]}
{"upwards":["Elementary equivalence","References"],"downwards":[]}
{"upwards":["Saturated model"],"downwards":["Definition","Motivation","Examples","Relationship to prime models","Notes","References"]}
{"upwards":["Saturated model","Definition"],"downwards":[]}
{"upwards":["Saturated model","Motivation"],"downwards":[]}
{"upwards":["Saturated model","Examples"],"downwards":[]}
{"upwards":["Saturated model","Relationship to prime models"],"downwards":[]}
{"upwards":["Saturated model","Notes"],"downwards":[]}
{"upwards":["Saturated model","References"],"downwards":[]}
{"upwards":["Non-standard model"],"downwards":["Existence","Importance"]}
{"upwards":["Non-standard model","Existence"],"downwards":[]}
{"upwards":["Non-standard model","Importance"],"downwards":[]}
{"upwards":["Categorical theory"],"downwards":["History and motivation","Examples","Properties"]}
{"upwards":["Categorical theory","History and motivation"],"downwards":[]}
{"upwards":["Categorical theory","Examples"],"downwards":[]}
{"upwards":["Categorical theory","Properties"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic"],"downwards":["Existence","Structure of countable non-standard models","References"]}
{"upwards":["Non-standard model of arithmetic","Existence"],"downwards":["From the compactness theorem","From the incompleteness theorems","From an ultraproduct"]}
{"upwards":["Non-standard model of arithmetic","Existence","From the compactness theorem"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Existence","From the incompleteness theorems"],"downwards":["Arithmetic unsoundness for models with ~G true"]}
{"upwards":["Non-standard model of arithmetic","Existence","From the incompleteness theorems","Arithmetic unsoundness for models with ~G true"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Existence","From an ultraproduct"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","Structure of countable non-standard models"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","References"],"downwards":["Citations","Sources"]}
{"upwards":["Non-standard model of arithmetic","References","Citations"],"downwards":[]}
{"upwards":["Non-standard model of arithmetic","References","Sources"],"downwards":[]}
{"upwards":["Model complete theory"],"downwards":["Model companion and model completion","Examples","Non-examples","Sufficient condition for completeness of model-complete theories","Notes","References"]}
{"upwards":["Model complete theory","Model companion and model completion"],"downwards":[]}
{"upwards":["Model complete theory","Examples"],"downwards":[]}
{"upwards":["Model complete theory","Non-examples"],"downwards":[]}
{"upwards":["Model complete theory","Sufficient condition for completeness of model-complete theories"],"downwards":[]}
{"upwards":["Model complete theory","Notes"],"downwards":[]}
{"upwards":["Model complete theory","References"],"downwards":[]}
{"upwards":["Diagram (mathematical logic)"],"downwards":["Definition"]}
{"upwards":["Diagram (mathematical logic)","Definition"],"downwards":[]}
{"upwards":["Elementary diagram"],"downwards":["Definition"]}
{"upwards":["Elementary diagram","Definition"],"downwards":[]}
{"upwards":["Substructure (mathematics)"],"downwards":["Definition","Example","As subobjects","Submodel"]}
{"upwards":["Substructure (mathematics)","Definition"],"downwards":[]}
{"upwards":["Substructure (mathematics)","Example"],"downwards":[]}
{"upwards":["Substructure (mathematics)","As subobjects"],"downwards":[]}
{"upwards":["Substructure (mathematics)","Submodel"],"downwards":[]}
{"upwards":["Spectrum of a theory"],"downwards":["Early results","List of possible spectra of a countable theory"]}
{"upwards":["Spectrum of a theory","Early results"],"downwards":[]}
{"upwards":["Spectrum of a theory","List of possible spectra of a countable theory"],"downwards":[]}
{"upwards":["Semantics of logic"],"downwards":["Overview"]}
{"upwards":["Semantics of logic","Overview"],"downwards":[]}
{"upwards":["Strength (mathematical logic)"],"downwards":[]}
{"upwards":["T-schema"],"downwards":["The inductive definition","Natural languages"]}
{"upwards":["T-schema","The inductive definition"],"downwards":[]}
{"upwards":["T-schema","Natural languages"],"downwards":[]}
{"upwards":["Truth"],"downwards":["Definition and etymology","Major theories","Formal theories","Folk beliefs","Notable views"]}
{"upwards":["Truth","Definition and etymology"],"downwards":[]}
{"upwards":["Truth","Major theories"],"downwards":["Substantive","Minimalist","Pluralist"]}
{"upwards":["Truth","Major theories","Substantive"],"downwards":["Correspondence","Coherence","Pragmatic","Constructivist","Consensus"]}
{"upwards":["Truth","Major theories","Substantive","Correspondence"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Coherence"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Pragmatic"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Constructivist"],"downwards":[]}
{"upwards":["Truth","Major theories","Substantive","Consensus"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist"],"downwards":["Deflationary","Performative","Redundancy and related","Philosophical skepticism"]}
{"upwards":["Truth","Major theories","Minimalist","Deflationary"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Performative"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Redundancy and related"],"downwards":[]}
{"upwards":["Truth","Major theories","Minimalist","Philosophical skepticism"],"downwards":[]}
{"upwards":["Truth","Major theories","Pluralist"],"downwards":[]}
{"upwards":["Truth","Formal theories"],"downwards":["Logic","Mathematics","Tarski's semantics","Kripke's semantics"]}
{"upwards":["Truth","Formal theories","Logic"],"downwards":[]}
{"upwards":["Truth","Formal theories","Mathematics"],"downwards":[]}
{"upwards":["Truth","Formal theories","Tarski's semantics"],"downwards":[]}
{"upwards":["Truth","Formal theories","Kripke's semantics"],"downwards":[]}
{"upwards":["Truth","Folk beliefs"],"downwards":[]}
{"upwards":["Truth","Notable views"],"downwards":["Ancient Greek philosophy","Medieval philosophy","Modern philosophy","Theological views"]}
{"upwards":["Truth","Notable views","Ancient Greek philosophy"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy"],"downwards":["Avicenna (980\u20131037)","Aquinas (1225\u20131274)","Changing concepts of truth in the Middle Ages"]}
{"upwards":["Truth","Notable views","Medieval philosophy","Avicenna (980\u20131037)"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy","Aquinas (1225\u20131274)"],"downwards":[]}
{"upwards":["Truth","Notable views","Medieval philosophy","Changing concepts of truth in the Middle Ages"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy"],"downwards":["Kant (1724\u20131804)","Hegel (1770\u20131831)","Schopenhauer (1788\u20131860)","Kierkegaard (1813\u20131855)","Nietzsche (1844\u20131900)","Heidegger (1889\u20131976)","Sartre (1905\u20131980)","Camus (1913\u20131960)","Whitehead (1861\u20131947)","Peirce (1839\u20131914)","Nishida (1870\u20131945)","Fromm (1900\u20131980)","Foucault (1926\u20131984)","Baudrillard (1929\u20132007)","Other contemporary positions"]}
{"upwards":["Truth","Notable views","Modern philosophy","Kant (1724\u20131804)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Hegel (1770\u20131831)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Schopenhauer (1788\u20131860)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Kierkegaard (1813\u20131855)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Nietzsche (1844\u20131900)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Heidegger (1889\u20131976)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Sartre (1905\u20131980)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Camus (1913\u20131960)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Whitehead (1861\u20131947)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Peirce (1839\u20131914)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Nishida (1870\u20131945)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Fromm (1900\u20131980)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Foucault (1926\u20131984)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Baudrillard (1929\u20132007)"],"downwards":[]}
{"upwards":["Truth","Notable views","Modern philosophy","Other contemporary positions"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views"],"downwards":["Hinduism","Buddhism","Christianity"]}
{"upwards":["Truth","Notable views","Theological views","Hinduism"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views","Buddhism"],"downwards":[]}
{"upwards":["Truth","Notable views","Theological views","Christianity"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Reverse mathematics"],"downwards":["General principles","The big five subsystems of second-order arithmetic","Additional systems","\u03c9-models and \u03b2-models"]}
{"upwards":["Reverse mathematics","General principles"],"downwards":["Use of second-order arithmetic","Use of higher-order arithmetic"]}
{"upwards":["Reverse mathematics","General principles","Use of second-order arithmetic"],"downwards":[]}
{"upwards":["Reverse mathematics","General principles","Use of higher-order arithmetic"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic"],"downwards":["The base system RCA0","Weak K\u0151nig's lemma WKL0","Arithmetical comprehension ACA0","Arithmetical transfinite recursion ATR0","\u03a011 comprehension \u03a011-CA0"]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","The base system RCA0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Weak K\u0151nig's lemma WKL0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Arithmetical comprehension ACA0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","Arithmetical transfinite recursion ATR0"],"downwards":[]}
{"upwards":["Reverse mathematics","The big five subsystems of second-order arithmetic","\u03a011 comprehension \u03a011-CA0"],"downwards":[]}
{"upwards":["Reverse mathematics","Additional systems"],"downwards":["Stronger Systems"]}
{"upwards":["Reverse mathematics","Additional systems","Stronger Systems"],"downwards":[]}
{"upwards":["Reverse mathematics","\u03c9-models and \u03b2-models"],"downwards":[]}
{"upwards":["Satisfiability"],"downwards":["Reduction of validity to satisfiability","Propositional satisfiability for classical logic","Satisfiability in first-order logic","Satisfiability in model theory","Finite satisfiability","Numerical constraints"]}
{"upwards":["Satisfiability","Reduction of validity to satisfiability"],"downwards":[]}
{"upwards":["Satisfiability","Propositional satisfiability for classical logic"],"downwards":[]}
{"upwards":["Satisfiability","Satisfiability in first-order logic"],"downwards":[]}
{"upwards":["Satisfiability","Satisfiability in model theory"],"downwards":[]}
{"upwards":["Satisfiability","Finite satisfiability"],"downwards":[]}
{"upwards":["Satisfiability","Numerical constraints"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth"],"downwards":["Origin","Tarski's theory of truth","Kripke's theory of truth"]}
{"upwards":["Semantic theory of truth","Origin"],"downwards":[]}
{"upwards":["Semantic theory of truth","Tarski's theory of truth"],"downwards":[]}
{"upwards":["Semantic theory of truth","Kripke's theory of truth"],"downwards":[]}
{"upwards":["Type theory"],"downwards":["History","Introduction","Differences from set theory","Technical details","Interpretations","List of type theories","Applications"]}
{"upwards":["Type theory","History"],"downwards":[]}
{"upwards":["Type theory","Introduction"],"downwards":["Basics","Functions","Common \"by rule\" types and terms"]}
{"upwards":["Type theory","Introduction","Basics"],"downwards":["Terms and types","Computation"]}
{"upwards":["Type theory","Introduction","Basics","Terms and types"],"downwards":[]}
{"upwards":["Type theory","Introduction","Basics","Computation"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions"],"downwards":["Lambda terms","Dependent typing","Universes"]}
{"upwards":["Type theory","Introduction","Functions","Lambda terms"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions","Dependent typing"],"downwards":[]}
{"upwards":["Type theory","Introduction","Functions","Universes"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms"],"downwards":["Empty type","Unit type","Boolean type","Product type","Sum type","Natural numbers","Identity type","Inductive types"]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Empty type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Unit type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Boolean type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Product type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Sum type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Natural numbers"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Identity type"],"downwards":[]}
{"upwards":["Type theory","Introduction","Common \"by rule\" types and terms","Inductive types"],"downwards":[]}
{"upwards":["Type theory","Differences from set theory"],"downwards":[]}
{"upwards":["Type theory","Technical details"],"downwards":["Terms","Judgements","Rules","Properties of type theories","Axioms","Decision problems","Research areas"]}
{"upwards":["Type theory","Technical details","Terms"],"downwards":[]}
{"upwards":["Type theory","Technical details","Judgements"],"downwards":[]}
{"upwards":["Type theory","Technical details","Rules"],"downwards":[]}
{"upwards":["Type theory","Technical details","Properties of type theories"],"downwards":[]}
{"upwards":["Type theory","Technical details","Axioms"],"downwards":[]}
{"upwards":["Type theory","Technical details","Decision problems"],"downwards":["Type inhabitation","Type inference"]}
{"upwards":["Type theory","Technical details","Decision problems","Type inhabitation"],"downwards":[]}
{"upwards":["Type theory","Technical details","Decision problems","Type inference"],"downwards":[]}
{"upwards":["Type theory","Technical details","Research areas"],"downwards":[]}
{"upwards":["Type theory","Interpretations"],"downwards":["Types are propositions; terms are proofs","Curry-Howard correspondence","Intuitionistic logic","Constructive mathematics","Category theory"]}
{"upwards":["Type theory","Interpretations","Types are propositions; terms are proofs"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Curry-Howard correspondence"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Intuitionistic logic"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Constructive mathematics"],"downwards":[]}
{"upwards":["Type theory","Interpretations","Category theory"],"downwards":[]}
{"upwards":["Type theory","List of type theories"],"downwards":["Major","Minor","Active research"]}
{"upwards":["Type theory","List of type theories","Major"],"downwards":[]}
{"upwards":["Type theory","List of type theories","Minor"],"downwards":[]}
{"upwards":["Type theory","List of type theories","Active research"],"downwards":[]}
{"upwards":["Type theory","Applications"],"downwards":["Mathematical foundations","Proof assistants","Programming languages","Linguistics","Social sciences"]}
{"upwards":["Type theory","Applications","Mathematical foundations"],"downwards":[]}
{"upwards":["Type theory","Applications","Proof assistants"],"downwards":[]}
{"upwards":["Type theory","Applications","Programming languages"],"downwards":[]}
{"upwards":["Type theory","Applications","Linguistics"],"downwards":[]}
{"upwards":["Type theory","Applications","Social sciences"],"downwards":[]}
{"upwards":["Transfer principle"],"downwards":["History","Transfer principle for the hyperreals","Generalizations of the concept of number","Differences between R and *R","Constructions of the hyperreals","Statement","Three examples"]}
{"upwards":["Transfer principle","History"],"downwards":[]}
{"upwards":["Transfer principle","Transfer principle for the hyperreals"],"downwards":["Example"]}
{"upwards":["Transfer principle","Transfer principle for the hyperreals","Example"],"downwards":[]}
{"upwards":["Transfer principle","Generalizations of the concept of number"],"downwards":[]}
{"upwards":["Transfer principle","Differences between R and *R"],"downwards":[]}
{"upwards":["Transfer principle","Constructions of the hyperreals"],"downwards":[]}
{"upwards":["Transfer principle","Statement"],"downwards":[]}
{"upwards":["Transfer principle","Three examples"],"downwards":[]}
{"upwards":["Truth predicate"],"downwards":["Languages which allow a truth predicate"]}
{"upwards":["Truth predicate","Languages which allow a truth predicate"],"downwards":[]}
{"upwards":["Type (model theory)"],"downwards":["Formal definition","Examples of types","Stone spaces","Omitting types theorem","References"]}
{"upwards":["Type (model theory)","Formal definition"],"downwards":[]}
{"upwards":["Type (model theory)","Examples of types"],"downwards":[]}
{"upwards":["Type (model theory)","Stone spaces"],"downwards":[]}
{"upwards":["Type (model theory)","Omitting types theorem"],"downwards":[]}
{"upwards":["Type (model theory)","References"],"downwards":[]}
{"upwards":["Truth value"],"downwards":["Computing","Classical logic","Intuitionistic and constructive logic","Multi-valued logic","Algebraic semantics","In other theories"]}
{"upwards":["Truth value","Computing"],"downwards":[]}
{"upwards":["Truth value","Classical logic"],"downwards":[]}
{"upwards":["Truth value","Intuitionistic and constructive logic"],"downwards":[]}
{"upwards":["Truth value","Multi-valued logic"],"downwards":[]}
{"upwards":["Truth value","Algebraic semantics"],"downwards":[]}
{"upwards":["Truth value","In other theories"],"downwards":[]}
{"upwards":["Ultraproduct"],"downwards":["Definition","Examples","\u0141o\u015b's theorem","Direct limits of ultrapowers (ultralimits)","Ultraproduct monad"]}
{"upwards":["Ultraproduct","Definition"],"downwards":["Ultrapower"]}
{"upwards":["Ultraproduct","Definition","Ultrapower"],"downwards":[]}
{"upwards":["Ultraproduct","Examples"],"downwards":[]}
{"upwards":["Ultraproduct","\u0141o\u015b's theorem"],"downwards":["Examples"]}
{"upwards":["Ultraproduct","\u0141o\u015b's theorem","Examples"],"downwards":[]}
{"upwards":["Ultraproduct","Direct limits of ultrapowers (ultralimits)"],"downwards":[]}
{"upwards":["Ultraproduct","Ultraproduct monad"],"downwards":[]}
{"upwards":["Computability theory"],"downwards":["Introduction","Turing computability","Areas of research","Relationships between definability, proof and computability","Name","Professional organizations"]}
{"upwards":["Computability theory","Introduction"],"downwards":[]}
{"upwards":["Computability theory","Turing computability"],"downwards":[]}
{"upwards":["Computability theory","Areas of research"],"downwards":["Relative computability and the Turing degrees","Other reducibilities","Rice's theorem and the arithmetical hierarchy","Reverse mathematics","Numberings","The priority method","The lattice of computably enumerable sets","Automorphism problems","Kolmogorov complexity","Frequency computation","Inductive inference","Generalizations of Turing computability","Continuous computability theory"]}
{"upwards":["Computability theory","Areas of research","Relative computability and the Turing degrees"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Other reducibilities"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Rice's theorem and the arithmetical hierarchy"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Reverse mathematics"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Numberings"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","The priority method"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","The lattice of computably enumerable sets"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Automorphism problems"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Kolmogorov complexity"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Frequency computation"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Inductive inference"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Generalizations of Turing computability"],"downwards":[]}
{"upwards":["Computability theory","Areas of research","Continuous computability theory"],"downwards":[]}
{"upwards":["Computability theory","Relationships between definability, proof and computability"],"downwards":[]}
{"upwards":["Computability theory","Name"],"downwards":[]}
{"upwards":["Computability theory","Professional organizations"],"downwards":[]}
{"upwards":["Church encoding"],"downwards":["Use","Church numerals","Church Booleans","Predicates","Church pairs","List encodings"]}
{"upwards":["Church encoding","Use"],"downwards":[]}
{"upwards":["Church encoding","Church numerals"],"downwards":["Calculation with Church numerals","Table of functions on Church numerals","Derivation of predecessor function","Division","Signed numbers","Plus and minus","Multiply and  divide","Rational and real numbers","Translation with other representations"]}
{"upwards":["Church encoding","Church numerals","Calculation with Church numerals"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Table of functions on Church numerals"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Derivation of predecessor function"],"downwards":["Another way of defining pred"]}
{"upwards":["Church encoding","Church numerals","Derivation of predecessor function","Another way of defining pred"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Division"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Signed numbers"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Plus and minus"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Multiply and  divide"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Rational and real numbers"],"downwards":[]}
{"upwards":["Church encoding","Church numerals","Translation with other representations"],"downwards":[]}
{"upwards":["Church encoding","Church Booleans"],"downwards":[]}
{"upwards":["Church encoding","Predicates"],"downwards":[]}
{"upwards":["Church encoding","Church pairs"],"downwards":[]}
{"upwards":["Church encoding","List encodings"],"downwards":["Two pairs as a list node","One pair as a list node","Represent the list using right fold","Represent the list using Scott encoding"]}
{"upwards":["Church encoding","List encodings","Two pairs as a list node"],"downwards":[]}
{"upwards":["Church encoding","List encodings","One pair as a list node"],"downwards":[]}
{"upwards":["Church encoding","List encodings","Represent the list using right fold"],"downwards":[]}
{"upwards":["Church encoding","List encodings","Represent the list using Scott encoding"],"downwards":[]}
{"upwards":["Logical connective"],"downwards":["Overview","Natural language","Properties","Order of precedence","Computer science","Table and Hasse diagram"]}
{"upwards":["Logical connective","Overview"],"downwards":["List of common logical connectives","History of notations","Redundancy"]}
{"upwards":["Logical connective","Overview","List of common logical connectives"],"downwards":[]}
{"upwards":["Logical connective","Overview","History of notations"],"downwards":[]}
{"upwards":["Logical connective","Overview","Redundancy"],"downwards":[]}
{"upwards":["Logical connective","Natural language"],"downwards":[]}
{"upwards":["Logical connective","Properties"],"downwards":[]}
{"upwards":["Logical connective","Order of precedence"],"downwards":[]}
{"upwards":["Logical connective","Computer science"],"downwards":[]}
{"upwards":["Logical connective","Table and Hasse diagram"],"downwards":[]}
{"upwards":["Equiconsistency"],"downwards":["Consistency","Consistency strength"]}
{"upwards":["Equiconsistency","Consistency"],"downwards":[]}
{"upwards":["Equiconsistency","Consistency strength"],"downwards":[]}
{"upwards":["Syllogism"],"downwards":["Early history","Modern history","Basic structure","Types","Terms in syllogism","Existential import","Syllogistic fallacies","Other types of syllogism"]}
{"upwards":["Syllogism","Early history"],"downwards":["Aristotle","Medieval syllogism"]}
{"upwards":["Syllogism","Early history","Aristotle"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism"],"downwards":["Boethius","Peter Abelard","Jean Buridan"]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Boethius"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Peter Abelard"],"downwards":[]}
{"upwards":["Syllogism","Early history","Medieval syllogism","Jean Buridan"],"downwards":[]}
{"upwards":["Syllogism","Modern history"],"downwards":["Boole's acceptance of Aristotle"]}
{"upwards":["Syllogism","Modern history","Boole's acceptance of Aristotle"],"downwards":[]}
{"upwards":["Syllogism","Basic structure"],"downwards":["Polysyllogism"]}
{"upwards":["Syllogism","Basic structure","Polysyllogism"],"downwards":[]}
{"upwards":["Syllogism","Types"],"downwards":["Examples","Table of all syllogisms"]}
{"upwards":["Syllogism","Types","Examples"],"downwards":["Barbara (AAA-1)","Celarent (EAE-1)","Darii (AII-1)","Ferio (EIO-1)","Baroco (AOO-2)","Bocardo (OAO-3)","Barbari (AAI-1)","Celaront (EAO-1)","Camestros (AEO-2)","Felapton (EAO-3)","Darapti (AAI-3)"]}
{"upwards":["Syllogism","Types","Examples","Barbara (AAA-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Celarent (EAE-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Darii (AII-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Ferio (EIO-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Baroco (AOO-2)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Bocardo (OAO-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Barbari (AAI-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Celaront (EAO-1)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Camestros (AEO-2)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Felapton (EAO-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Examples","Darapti (AAI-3)"],"downwards":[]}
{"upwards":["Syllogism","Types","Table of all syllogisms"],"downwards":[]}
{"upwards":["Syllogism","Terms in syllogism"],"downwards":[]}
{"upwards":["Syllogism","Existential import"],"downwards":[]}
{"upwards":["Syllogism","Syllogistic fallacies"],"downwards":[]}
{"upwards":["Syllogism","Other types of syllogism"],"downwards":[]}
{"upwards":["Many-valued logic"],"downwards":["History","Examples","Relation to classical logic","Functional completeness of many-valued logics","Applications","Research venues"]}
{"upwards":["Many-valued logic","History"],"downwards":[]}
{"upwards":["Many-valued logic","Examples"],"downwards":["Kleene (strong) K3 and Priest logic P3","Bochvar's internal three-valued logic","Belnap logic (B4)","G\u00f6del logics Gk and G\u221e","\u0141ukasiewicz logics Lv and L\u221e","Product logic \u03a0","Post logics Pm","Rose logics"]}
{"upwards":["Many-valued logic","Examples","Kleene (strong) K3 and Priest logic P3"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Bochvar's internal three-valued logic"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Belnap logic (B4)"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","G\u00f6del logics Gk and G\u221e"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","\u0141ukasiewicz logics Lv and L\u221e"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Product logic \u03a0"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Post logics Pm"],"downwards":[]}
{"upwards":["Many-valued logic","Examples","Rose logics"],"downwards":[]}
{"upwards":["Many-valued logic","Relation to classical logic"],"downwards":["Suszko's thesis"]}
{"upwards":["Many-valued logic","Relation to classical logic","Suszko's thesis"],"downwards":[]}
{"upwards":["Many-valued logic","Functional completeness of many-valued logics"],"downwards":[]}
{"upwards":["Many-valued logic","Applications"],"downwards":[]}
{"upwards":["Many-valued logic","Research venues"],"downwards":[]}
{"upwards":["Paradoxes of set theory"],"downwards":["Basics","Paradoxes of the infinite sets","Paradoxes of the Supertask","Paradoxes of proof and definability"]}
{"upwards":["Paradoxes of set theory","Basics"],"downwards":["Cardinal numbers","Ordinal numbers","Power sets"]}
{"upwards":["Paradoxes of set theory","Basics","Cardinal numbers"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Basics","Ordinal numbers"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Basics","Power sets"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets"],"downwards":["Paradoxes of enumeration","Je le vois, mais je ne crois pas","Paradoxes of well-ordering"]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Paradoxes of enumeration"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Je le vois, mais je ne crois pas"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the infinite sets","Paradoxes of well-ordering"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask"],"downwards":["The diary of Tristram Shandy","The Ross-Littlewood paradox"]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask","The diary of Tristram Shandy"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of the Supertask","The Ross-Littlewood paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability"],"downwards":["Early paradoxes: the set of all sets","Paradoxes by change of language","Paradox of L\u00f6wenheim and Skolem"]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Early paradoxes: the set of all sets"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language"],"downwards":["K\u00f6nig's paradox","Richard's paradox"]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language","K\u00f6nig's paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradoxes by change of language","Richard's paradox"],"downwards":[]}
{"upwards":["Paradoxes of set theory","Paradoxes of proof and definability","Paradox of L\u00f6wenheim and Skolem"],"downwards":[]}
{"upwards":["Validity (logic)"],"downwards":["Arguments","Valid formula","Statements","Soundness","Satisfiability","Preservation"]}
{"upwards":["Validity (logic)","Arguments"],"downwards":[]}
{"upwards":["Validity (logic)","Valid formula"],"downwards":[]}
{"upwards":["Validity (logic)","Statements"],"downwards":[]}
{"upwards":["Validity (logic)","Soundness"],"downwards":[]}
{"upwards":["Validity (logic)","Satisfiability"],"downwards":[]}
{"upwards":["Validity (logic)","Preservation"],"downwards":[]}
{"upwards":["Three-valued logic"],"downwards":["Pre-discovery","Representation of values","Logics","Applications"]}
{"upwards":["Three-valued logic","Pre-discovery"],"downwards":[]}
{"upwards":["Three-valued logic","Representation of values"],"downwards":[]}
{"upwards":["Three-valued logic","Logics"],"downwards":["Kleene and Priest logics","\u0141ukasiewicz logic","HT logic","Bochvar logic","Ternary Post logic","Modular algebras"]}
{"upwards":["Three-valued logic","Logics","Kleene and Priest logics"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","\u0141ukasiewicz logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","HT logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","Bochvar logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","Ternary Post logic"],"downwards":[]}
{"upwards":["Three-valued logic","Logics","Modular algebras"],"downwards":[]}
{"upwards":["Three-valued logic","Applications"],"downwards":["SQL"]}
{"upwards":["Three-valued logic","Applications","SQL"],"downwards":[]}
{"upwards":["Argument"],"downwards":["Etymology","Formal and informal","Standard logical account of argument types","Defeasible arguments and argumentation schemes","By analogy","Other kinds","Explanations","Fallacies and non-arguments","Elliptical or ethymematic arguments","Argument mining"]}
{"upwards":["Argument","Etymology"],"downwards":[]}
{"upwards":["Argument","Formal and informal"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types"],"downwards":["Deductive arguments","Inductive arguments"]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments"],"downwards":["Validity","Soundness"]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments","Validity"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types","Deductive arguments","Soundness"],"downwards":[]}
{"upwards":["Argument","Standard logical account of argument types","Inductive arguments"],"downwards":[]}
{"upwards":["Argument","Defeasible arguments and argumentation schemes"],"downwards":[]}
{"upwards":["Argument","By analogy"],"downwards":[]}
{"upwards":["Argument","Other kinds"],"downwards":["World-disclosing"]}
{"upwards":["Argument","Other kinds","World-disclosing"],"downwards":[]}
{"upwards":["Argument","Explanations"],"downwards":[]}
{"upwards":["Argument","Fallacies and non-arguments"],"downwards":[]}
{"upwards":["Argument","Elliptical or ethymematic arguments"],"downwards":[]}
{"upwards":["Argument","Argument mining"],"downwards":[]}
{"upwards":["Boolean function"],"downwards":["Examples","Representation","Analysis","Real polynomial form","Applications"]}
{"upwards":["Boolean function","Examples"],"downwards":[]}
{"upwards":["Boolean function","Representation"],"downwards":[]}
{"upwards":["Boolean function","Analysis"],"downwards":["Properties","Derived functions","Cryptographic analysis"]}
{"upwards":["Boolean function","Analysis","Properties"],"downwards":[]}
{"upwards":["Boolean function","Analysis","Derived functions"],"downwards":[]}
{"upwards":["Boolean function","Analysis","Cryptographic analysis"],"downwards":["Linear approximation table"]}
{"upwards":["Boolean function","Analysis","Cryptographic analysis","Linear approximation table"],"downwards":[]}
{"upwards":["Boolean function","Real polynomial form"],"downwards":["On the unit hypercube","On the symmetric hypercube"]}
{"upwards":["Boolean function","Real polynomial form","On the unit hypercube"],"downwards":[]}
{"upwards":["Boolean function","Real polynomial form","On the symmetric hypercube"],"downwards":[]}
{"upwards":["Boolean function","Applications"],"downwards":[]}
{"upwards":["Soundness"],"downwards":["Definition","Use in mathematical logic"]}
{"upwards":["Soundness","Definition"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic"],"downwards":["Logical systems","Relation to completeness"]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems"],"downwards":["Soundness","Strong soundness","Arithmetic soundness"]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Strong soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Logical systems","Arithmetic soundness"],"downwards":[]}
{"upwards":["Soundness","Use in mathematical logic","Relation to completeness"],"downwards":[]}
{"upwards":["Boolean algebra"],"downwards":["History","Values","Operations","Laws","Diagrammatic representations","Boolean algebras","Axiomatizing Boolean algebra","Propositional logic","Applications"]}
{"upwards":["Boolean algebra","History"],"downwards":[]}
{"upwards":["Boolean algebra","Values"],"downwards":[]}
{"upwards":["Boolean algebra","Operations"],"downwards":["Basic operations","Secondary operations"]}
{"upwards":["Boolean algebra","Operations","Basic operations"],"downwards":[]}
{"upwards":["Boolean algebra","Operations","Secondary operations"],"downwards":[]}
{"upwards":["Boolean algebra","Laws"],"downwards":["Monotone laws","Nonmonotone laws","Completeness","Duality principle"]}
{"upwards":["Boolean algebra","Laws","Monotone laws"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Nonmonotone laws"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Completeness"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Duality principle"],"downwards":[]}
{"upwards":["Boolean algebra","Diagrammatic representations"],"downwards":["Venn diagrams","Digital logic gates"]}
{"upwards":["Boolean algebra","Diagrammatic representations","Venn diagrams"],"downwards":[]}
{"upwards":["Boolean algebra","Diagrammatic representations","Digital logic gates"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras"],"downwards":["Concrete Boolean algebras","Subsets as bit vectors","The prototypical Boolean algebra","Boolean algebras: the definition","Representable Boolean algebras"]}
{"upwards":["Boolean algebra","Boolean algebras","Concrete Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Subsets as bit vectors"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","The prototypical Boolean algebra"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Boolean algebras: the definition"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Representable Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebra","Axiomatizing Boolean algebra"],"downwards":[]}
{"upwards":["Boolean algebra","Propositional logic"],"downwards":["Applications","Deductive systems for propositional logic"]}
{"upwards":["Boolean algebra","Propositional logic","Applications"],"downwards":[]}
{"upwards":["Boolean algebra","Propositional logic","Deductive systems for propositional logic"],"downwards":["Sequent calculus"]}
{"upwards":["Boolean algebra","Propositional logic","Deductive systems for propositional logic","Sequent calculus"],"downwards":[]}
{"upwards":["Boolean algebra","Applications"],"downwards":["Computers","Two-valued logic","Boolean operations"]}
{"upwards":["Boolean algebra","Applications","Computers"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Two-valued logic"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations"],"downwards":["Natural language","Digital logic","Naive set theory","Video cards","Modeling and CAD","Boolean searches"]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Natural language"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Digital logic"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Naive set theory"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Video cards"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Modeling and CAD"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Boolean searches"],"downwards":[]}
{"upwards":["Truth table"],"downwards":["Unary operations","Binary operations","Size of truth tables","Applications","History"]}
{"upwards":["Truth table","Unary operations"],"downwards":["Logical true","Logical false","Logical identity","Logical negation"]}
{"upwards":["Truth table","Unary operations","Logical true"],"downwards":[]}
{"upwards":["Truth table","Unary operations","Logical false"],"downwards":[]}
{"upwards":["Truth table","Unary operations","Logical identity"],"downwards":[]}
{"upwards":["Truth table","Unary operations","Logical negation"],"downwards":[]}
{"upwards":["Truth table","Binary operations"],"downwards":["Truth table for all binary logical operators","Logical conjunction (AND)","Logical disjunction (OR)","Logical implication","Logical equality","Exclusive disjunction","Logical NAND","Logical NOR"]}
{"upwards":["Truth table","Binary operations","Truth table for all binary logical operators"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical conjunction (AND)"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical disjunction (OR)"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical implication"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical equality"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Exclusive disjunction"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical NAND"],"downwards":[]}
{"upwards":["Truth table","Binary operations","Logical NOR"],"downwards":[]}
{"upwards":["Truth table","Size of truth tables"],"downwards":[]}
{"upwards":["Truth table","Applications"],"downwards":["Truth table for most commonly used logical operators","Condensed truth tables for binary operators","Truth tables in digital logic","Applications of truth tables in digital electronics"]}
{"upwards":["Truth table","Applications","Truth table for most commonly used logical operators"],"downwards":[]}
{"upwards":["Truth table","Applications","Condensed truth tables for binary operators"],"downwards":[]}
{"upwards":["Truth table","Applications","Truth tables in digital logic"],"downwards":[]}
{"upwards":["Truth table","Applications","Applications of truth tables in digital electronics"],"downwards":[]}
{"upwards":["Truth table","History"],"downwards":[]}
{"upwards":["Universal set"],"downwards":["Reasons for nonexistence","Theories of universality"]}
{"upwards":["Universal set","Reasons for nonexistence"],"downwards":["Regularity","Russell's paradox","Cantor's theorem"]}
{"upwards":["Universal set","Reasons for nonexistence","Regularity"],"downwards":[]}
{"upwards":["Universal set","Reasons for nonexistence","Russell's paradox"],"downwards":[]}
{"upwards":["Universal set","Reasons for nonexistence","Cantor's theorem"],"downwards":[]}
{"upwards":["Universal set","Theories of universality"],"downwards":["Restricted comprehension","Universal objects that are not sets"]}
{"upwards":["Universal set","Theories of universality","Restricted comprehension"],"downwards":[]}
{"upwards":["Universal set","Theories of universality","Universal objects that are not sets"],"downwards":[]}
{"upwards":["Infinite-valued logic"],"downwards":["History","Examples"]}
{"upwards":["Infinite-valued logic","History"],"downwards":[]}
{"upwards":["Infinite-valued logic","Examples"],"downwards":[]}
{"upwards":["Predicate (mathematical logic)"],"downwards":["Predicates in different systems"]}
{"upwards":["Predicate (mathematical logic)","Predicates in different systems"],"downwards":[]}
{"upwards":["First-order logic"],"downwards":["Introduction","Syntax","Semantics","Deductive systems","Equality and its axioms","Metalogical properties","Limitations","Restrictions, extensions, and variations","Automated theorem proving and formal methods"]}
{"upwards":["First-order logic","Introduction"],"downwards":[]}
{"upwards":["First-order logic","Syntax"],"downwards":["Alphabet","Formation rules","Free and bound variables","Example: ordered abelian groups"]}
{"upwards":["First-order logic","Syntax","Alphabet"],"downwards":["Logical symbols","Non-logical symbols"]}
{"upwards":["First-order logic","Syntax","Alphabet","Logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Alphabet","Non-logical symbols"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules"],"downwards":["Terms","Formulas","Notational conventions"]}
{"upwards":["First-order logic","Syntax","Formation rules","Terms"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Formulas"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Formation rules","Notational conventions"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Free and bound variables"],"downwards":[]}
{"upwards":["First-order logic","Syntax","Example: ordered abelian groups"],"downwards":[]}
{"upwards":["First-order logic","Semantics"],"downwards":["First-order structures","Evaluation of truth values","Validity, satisfiability, and logical consequence","Algebraizations","First-order theories, models, and elementary classes","Empty domains"]}
{"upwards":["First-order logic","Semantics","First-order structures"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Evaluation of truth values"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Validity, satisfiability, and logical consequence"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Algebraizations"],"downwards":[]}
{"upwards":["First-order logic","Semantics","First-order theories, models, and elementary classes"],"downwards":[]}
{"upwards":["First-order logic","Semantics","Empty domains"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems"],"downwards":["Rules of inference","Hilbert-style systems and natural deduction","Sequent calculus","Tableaux method","Resolution","Provable identities"]}
{"upwards":["First-order logic","Deductive systems","Rules of inference"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Hilbert-style systems and natural deduction"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Sequent calculus"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Tableaux method"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Resolution"],"downwards":[]}
{"upwards":["First-order logic","Deductive systems","Provable identities"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms"],"downwards":["First-order logic without equality","Defining equality within a theory"]}
{"upwards":["First-order logic","Equality and its axioms","First-order logic without equality"],"downwards":[]}
{"upwards":["First-order logic","Equality and its axioms","Defining equality within a theory"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties"],"downwards":["Completeness and undecidability","The L\u00f6wenheim\u2013Skolem theorem","The compactness theorem","Lindstr\u00f6m's theorem"]}
{"upwards":["First-order logic","Metalogical properties","Completeness and undecidability"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The L\u00f6wenheim\u2013Skolem theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","The compactness theorem"],"downwards":[]}
{"upwards":["First-order logic","Metalogical properties","Lindstr\u00f6m's theorem"],"downwards":[]}
{"upwards":["First-order logic","Limitations"],"downwards":["Expressiveness","Formalizing natural languages"]}
{"upwards":["First-order logic","Limitations","Expressiveness"],"downwards":[]}
{"upwards":["First-order logic","Limitations","Formalizing natural languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations"],"downwards":["Restricted languages","Many-sorted logic","Additional quantifiers","Infinitary logics","Non-classical and modal logics","Fixpoint logic","Higher-order logics"]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Restricted languages"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Many-sorted logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Additional quantifiers"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Infinitary logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Non-classical and modal logics"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Fixpoint logic"],"downwards":[]}
{"upwards":["First-order logic","Restrictions, extensions, and variations","Higher-order logics"],"downwards":[]}
{"upwards":["First-order logic","Automated theorem proving and formal methods"],"downwards":[]}
{"upwards":["List of first-order theories"],"downwards":["Preliminaries","Pure identity theories","Unary relations","Equivalence relations","Orders","Lattices","Graphs","Boolean algebras","Groups","Rings and fields","Geometry","Differential algebra","Addition","Arithmetic","Second order arithmetic","Set theories"]}
{"upwards":["List of first-order theories","Preliminaries"],"downwards":[]}
{"upwards":["List of first-order theories","Pure identity theories"],"downwards":[]}
{"upwards":["List of first-order theories","Unary relations"],"downwards":[]}
{"upwards":["List of first-order theories","Equivalence relations"],"downwards":[]}
{"upwards":["List of first-order theories","Orders"],"downwards":[]}
{"upwards":["List of first-order theories","Lattices"],"downwards":[]}
{"upwards":["List of first-order theories","Graphs"],"downwards":[]}
{"upwards":["List of first-order theories","Boolean algebras"],"downwards":[]}
{"upwards":["List of first-order theories","Groups"],"downwards":[]}
{"upwards":["List of first-order theories","Rings and fields"],"downwards":[]}
{"upwards":["List of first-order theories","Geometry"],"downwards":[]}
{"upwards":["List of first-order theories","Differential algebra"],"downwards":[]}
{"upwards":["List of first-order theories","Addition"],"downwards":[]}
{"upwards":["List of first-order theories","Arithmetic"],"downwards":[]}
{"upwards":["List of first-order theories","Second order arithmetic"],"downwards":[]}
{"upwards":["List of first-order theories","Set theories"],"downwards":[]}
{"upwards":["Second-order logic"],"downwards":["Examples","Syntax and fragments","Semantics","Expressive power","Deductive systems","Non-reducibility to first-order logic","Metalogical results","History and disputed value","Relation to computational complexity"]}
{"upwards":["Second-order logic","Examples"],"downwards":[]}
{"upwards":["Second-order logic","Syntax and fragments"],"downwards":[]}
{"upwards":["Second-order logic","Semantics"],"downwards":[]}
{"upwards":["Second-order logic","Expressive power"],"downwards":[]}
{"upwards":["Second-order logic","Deductive systems"],"downwards":[]}
{"upwards":["Second-order logic","Non-reducibility to first-order logic"],"downwards":[]}
{"upwards":["Second-order logic","Metalogical results"],"downwards":[]}
{"upwards":["Second-order logic","History and disputed value"],"downwards":[]}
{"upwards":["Second-order logic","Relation to computational complexity"],"downwards":[]}
{"upwards":["Higher-order logic"],"downwards":["Quantification scope","Semantics","Properties"]}
{"upwards":["Higher-order logic","Quantification scope"],"downwards":[]}
{"upwards":["Higher-order logic","Semantics"],"downwards":[]}
{"upwards":["Higher-order logic","Properties"],"downwards":[]}
{"upwards":["Forcing (mathematics)"],"downwards":["Intuition","Forcing conditions and forcing posets","Generic filters","P-names and interpretations","Forcing","Consistency","Cohen forcing","The countable chain condition","Easton forcing","Random reals","Boolean-valued models","Meta-mathematical explanation","Logical explanation"]}
{"upwards":["Forcing (mathematics)","Intuition"],"downwards":["The role of the model"]}
{"upwards":["Forcing (mathematics)","Intuition","The role of the model"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Forcing conditions and forcing posets"],"downwards":["Examples"]}
{"upwards":["Forcing (mathematics)","Forcing conditions and forcing posets","Examples"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Generic filters"],"downwards":[]}
{"upwards":["Forcing (mathematics)","P-names and interpretations"],"downwards":["Rigorous definitions"]}
{"upwards":["Forcing (mathematics)","P-names and interpretations","Rigorous definitions"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Forcing"],"downwards":["Internal definition"]}
{"upwards":["Forcing (mathematics)","Forcing","Internal definition"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Consistency"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Cohen forcing"],"downwards":[]}
{"upwards":["Forcing (mathematics)","The countable chain condition"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Easton forcing"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Random reals"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Boolean-valued models"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Meta-mathematical explanation"],"downwards":[]}
{"upwards":["Forcing (mathematics)","Logical explanation"],"downwards":[]}
{"upwards":["Propositional formula"],"downwards":["Propositions","An algebra of propositions, the propositional calculus","Propositional connectives","More complex formulas","Inductive definition","Parsing formulas","Well-formed formulas (wffs)","Reduced sets of connectives","Normal forms","Impredicative propositions","Propositional formula with \"feedback\"","Historical development","Footnotes","Citations","References","External links"]}
{"upwards":["Propositional formula","Propositions"],"downwards":["Relationship between propositional and predicate formulas","Identity"]}
{"upwards":["Propositional formula","Propositions","Relationship between propositional and predicate formulas"],"downwards":[]}
{"upwards":["Propositional formula","Propositions","Identity"],"downwards":[]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus"],"downwards":["Usefulness of propositional formulas","Propositional variables","Truth-value assignments, formula evaluations"]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus","Usefulness of propositional formulas"],"downwards":[]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus","Propositional variables"],"downwards":[]}
{"upwards":["Propositional formula","An algebra of propositions, the propositional calculus","Truth-value assignments, formula evaluations"],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives"],"downwards":["Connectives of rhetoric, philosophy and mathematics","Engineering connectives","CASE connective: IF ... THEN ... ELSE ...","IDENTITY and evaluation"]}
{"upwards":["Propositional formula","Propositional connectives","Connectives of rhetoric, philosophy and mathematics"],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives","Engineering connectives"],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives","CASE connective: IF ... THEN ... ELSE ..."],"downwards":[]}
{"upwards":["Propositional formula","Propositional connectives","IDENTITY and evaluation"],"downwards":[]}
{"upwards":["Propositional formula","More complex formulas"],"downwards":["Definitions","Axiom and definition schemas","Substitution versus replacement"]}
{"upwards":["Propositional formula","More complex formulas","Definitions"],"downwards":[]}
{"upwards":["Propositional formula","More complex formulas","Axiom and definition schemas"],"downwards":[]}
{"upwards":["Propositional formula","More complex formulas","Substitution versus replacement"],"downwards":[]}
{"upwards":["Propositional formula","Inductive definition"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas"],"downwards":["Connective seniority (symbol rank)","Commutative and associative laws","Distributive laws","De Morgan's laws","Laws of absorption","Laws of evaluation: Identity, nullity, and complement","Double negative (involution)"]}
{"upwards":["Propositional formula","Parsing formulas","Connective seniority (symbol rank)"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Commutative and associative laws"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Distributive laws"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","De Morgan's laws"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Laws of absorption"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Laws of evaluation: Identity, nullity, and complement"],"downwards":[]}
{"upwards":["Propositional formula","Parsing formulas","Double negative (involution)"],"downwards":[]}
{"upwards":["Propositional formula","Well-formed formulas (wffs)"],"downwards":["Well-formed formulas versus valid formulas in inferences"]}
{"upwards":["Propositional formula","Well-formed formulas (wffs)","Well-formed formulas versus valid formulas in inferences"],"downwards":[]}
{"upwards":["Propositional formula","Reduced sets of connectives"],"downwards":["The stroke (NAND)","IF ... THEN ... ELSE"]}
{"upwards":["Propositional formula","Reduced sets of connectives","The stroke (NAND)"],"downwards":[]}
{"upwards":["Propositional formula","Reduced sets of connectives","IF ... THEN ... ELSE"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms"],"downwards":["Reduction to normal form","Reduction by use of the map method (Veitch, Karnaugh)"]}
{"upwards":["Propositional formula","Normal forms","Reduction to normal form"],"downwards":["Literal, term and alterm","Minterms"]}
{"upwards":["Propositional formula","Normal forms","Reduction to normal form","Literal, term and alterm"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction to normal form","Minterms"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)"],"downwards":["Produce the formula's truth table","Create the formula's Karnaugh map","Reduce minterms","Verify reduction with a truth table"]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Produce the formula's truth table"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Create the formula's Karnaugh map"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Reduce minterms"],"downwards":[]}
{"upwards":["Propositional formula","Normal forms","Reduction by use of the map method (Veitch, Karnaugh)","Verify reduction with a truth table"],"downwards":[]}
{"upwards":["Propositional formula","Impredicative propositions"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\""],"downwards":["Oscillation","Memory"]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Oscillation"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory"],"downwards":["Once-flip memory","Flip-flop memory","Clocked flip-flop memory"]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory","Once-flip memory"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory","Flip-flop memory"],"downwards":[]}
{"upwards":["Propositional formula","Propositional formula with \"feedback\"","Memory","Clocked flip-flop memory"],"downwards":[]}
{"upwards":["Propositional formula","Historical development"],"downwards":[]}
{"upwards":["Propositional formula","Footnotes"],"downwards":[]}
{"upwards":["Propositional formula","Citations"],"downwards":[]}
{"upwards":["Propositional formula","References"],"downwards":[]}
{"upwards":["Propositional formula","External links"],"downwards":[]}
{"upwards":["Extensionality"],"downwards":["Example","In mathematics"]}
{"upwards":["Extensionality","Example"],"downwards":[]}
{"upwards":["Extensionality","In mathematics"],"downwards":[]}
{"upwards":["Finite-valued logic"],"downwards":["History","Examples"]}
{"upwards":["Finite-valued logic","History"],"downwards":[]}
{"upwards":["Finite-valued logic","Examples"],"downwards":[]}
{"upwards":["Ordered pair"],"downwards":["Generalities","Informal and formal definitions","Defining the ordered pair using set theory","Category theory"]}
{"upwards":["Ordered pair","Generalities"],"downwards":[]}
{"upwards":["Ordered pair","Informal and formal definitions"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory"],"downwards":["Wiener's definition","Hausdorff's definition","Kuratowski's definition","Quine\u2013Rosser definition","Cantor\u2013Frege definition","Morse definition","Axiomatic definition"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Wiener's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Hausdorff's definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition"],"downwards":["Variants","Proving that definitions satisfy the characteristic property"]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Variants"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Kuratowski's definition","Proving that definitions satisfy the characteristic property"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Quine\u2013Rosser definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Cantor\u2013Frege definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Morse definition"],"downwards":[]}
{"upwards":["Ordered pair","Defining the ordered pair using set theory","Axiomatic definition"],"downwards":[]}
{"upwards":["Ordered pair","Category theory"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory"],"downwards":["Axioms","Implementation in the Mizar system","Implementation in Metamath"]}
{"upwards":["Tarski\u2013Grothendieck set theory","Axioms"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory","Implementation in the Mizar system"],"downwards":[]}
{"upwards":["Tarski\u2013Grothendieck set theory","Implementation in Metamath"],"downwards":[]}
{"upwards":["Quantifier (logic)"],"downwards":["Relations to logical conjunction and disjunction","Algebraic approaches to quantification","Notation","Order of quantifiers (nesting)","Equivalent expressions","Range of quantification","Formal semantics","Paucal, multal and other degree quantifiers","Other quantifiers","History"]}
{"upwards":["Quantifier (logic)","Relations to logical conjunction and disjunction"],"downwards":["Infinite domain of discourse"]}
{"upwards":["Quantifier (logic)","Relations to logical conjunction and disjunction","Infinite domain of discourse"],"downwards":[]}
{"upwards":["Quantifier (logic)","Algebraic approaches to quantification"],"downwards":[]}
{"upwards":["Quantifier (logic)","Notation"],"downwards":[]}
{"upwards":["Quantifier (logic)","Order of quantifiers (nesting)"],"downwards":[]}
{"upwards":["Quantifier (logic)","Equivalent expressions"],"downwards":["Quantifier elimination"]}
{"upwards":["Quantifier (logic)","Equivalent expressions","Quantifier elimination"],"downwards":[]}
{"upwards":["Quantifier (logic)","Range of quantification"],"downwards":[]}
{"upwards":["Quantifier (logic)","Formal semantics"],"downwards":[]}
{"upwards":["Quantifier (logic)","Paucal, multal and other degree quantifiers"],"downwards":[]}
{"upwards":["Quantifier (logic)","Other quantifiers"],"downwards":[]}
{"upwards":["Quantifier (logic)","History"],"downwards":[]}
{"upwards":["Ordinal number"],"downwards":["Ordinals extend the natural numbers","Definitions","Transfinite sequence","Transfinite induction","Arithmetic of ordinals","Ordinals and cardinals","Some \"large\" countable ordinals","Topology and ordinals","History"]}
{"upwards":["Ordinal number","Ordinals extend the natural numbers"],"downwards":[]}
{"upwards":["Ordinal number","Definitions"],"downwards":["Well-ordered sets","Definition of an ordinal as an equivalence class","Von Neumann definition of ordinals","Other definitions"]}
{"upwards":["Ordinal number","Definitions","Well-ordered sets"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Definition of an ordinal as an equivalence class"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Von Neumann definition of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Definitions","Other definitions"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite sequence"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction"],"downwards":["Transfinite recursion","Successor and limit ordinals","Indexing classes of ordinals","Closed unbounded sets and classes"]}
{"upwards":["Ordinal number","Transfinite induction","Transfinite recursion"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Successor and limit ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Indexing classes of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Transfinite induction","Closed unbounded sets and classes"],"downwards":[]}
{"upwards":["Ordinal number","Arithmetic of ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals"],"downwards":["Initial ordinal of a cardinal","Cofinality"]}
{"upwards":["Ordinal number","Ordinals and cardinals","Initial ordinal of a cardinal"],"downwards":[]}
{"upwards":["Ordinal number","Ordinals and cardinals","Cofinality"],"downwards":[]}
{"upwards":["Ordinal number","Some \"large\" countable ordinals"],"downwards":[]}
{"upwards":["Ordinal number","Topology and ordinals"],"downwards":[]}
{"upwards":["Ordinal number","History"],"downwards":[]}
{"upwards":["Hereditary set"],"downwards":["Examples","In formulations of set theory","Assumptions"]}
{"upwards":["Hereditary set","Examples"],"downwards":[]}
{"upwards":["Hereditary set","In formulations of set theory"],"downwards":[]}
{"upwards":["Hereditary set","Assumptions"],"downwards":[]}
{"upwards":["Free logic"],"downwards":["Explanation","Interpretation"]}
{"upwards":["Free logic","Explanation"],"downwards":[]}
{"upwards":["Free logic","Interpretation"],"downwards":[]}
{"upwards":["Monadic second-order logic"],"downwards":["Variants","Computational complexity of evaluation","Decidability and complexity of satisfiability"]}
{"upwards":["Monadic second-order logic","Variants"],"downwards":[]}
{"upwards":["Monadic second-order logic","Computational complexity of evaluation"],"downwards":[]}
{"upwards":["Monadic second-order logic","Decidability and complexity of satisfiability"],"downwards":["Use of satisfiability of MSO on trees in verification"]}
{"upwards":["Monadic second-order logic","Decidability and complexity of satisfiability","Use of satisfiability of MSO on trees in verification"],"downwards":[]}
{"upwards":["Monadic predicate calculus"],"downwards":["Expressiveness","Relationship with term logic","Variants","Footnotes"]}
{"upwards":["Monadic predicate calculus","Expressiveness"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Relationship with term logic"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Variants"],"downwards":[]}
{"upwards":["Monadic predicate calculus","Footnotes"],"downwards":[]}
{"upwards":["Equivalence relation"],"downwards":["Notation","Definition","Examples","Connections to other relations","Well-definedness under an equivalence relation","Related important definitions","Fundamental theorem of equivalence relations","Comparing equivalence relations","Generating equivalence relations","Algebraic structure","Equivalence relations and mathematical logic"]}
{"upwards":["Equivalence relation","Notation"],"downwards":[]}
{"upwards":["Equivalence relation","Definition"],"downwards":["Alternative definition using relational algebra"]}
{"upwards":["Equivalence relation","Definition","Alternative definition using relational algebra"],"downwards":[]}
{"upwards":["Equivalence relation","Examples"],"downwards":["Simple example","Equivalence relations","Relations that are not equivalences"]}
{"upwards":["Equivalence relation","Examples","Simple example"],"downwards":[]}
{"upwards":["Equivalence relation","Examples","Equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Examples","Relations that are not equivalences"],"downwards":[]}
{"upwards":["Equivalence relation","Connections to other relations"],"downwards":[]}
{"upwards":["Equivalence relation","Well-definedness under an equivalence relation"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions"],"downwards":["Equivalence class","Quotient set","Projection","Equivalence kernel","Partition"]}
{"upwards":["Equivalence relation","Related important definitions","Equivalence class"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Quotient set"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Projection"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Equivalence kernel"],"downwards":[]}
{"upwards":["Equivalence relation","Related important definitions","Partition"],"downwards":["Counting partitions"]}
{"upwards":["Equivalence relation","Related important definitions","Partition","Counting partitions"],"downwards":[]}
{"upwards":["Equivalence relation","Fundamental theorem of equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Comparing equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Generating equivalence relations"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure"],"downwards":["Group theory","Categories and groupoids","Lattices"]}
{"upwards":["Equivalence relation","Algebraic structure","Group theory"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure","Categories and groupoids"],"downwards":[]}
{"upwards":["Equivalence relation","Algebraic structure","Lattices"],"downwards":[]}
{"upwards":["Equivalence relation","Equivalence relations and mathematical logic"],"downwards":[]}
{"upwards":["Urelement"],"downwards":["Theory","Urelements in set theory","Quine atoms","References","External links"]}
{"upwards":["Urelement","Theory"],"downwards":[]}
{"upwards":["Urelement","Urelements in set theory"],"downwards":[]}
{"upwards":["Urelement","Quine atoms"],"downwards":[]}
{"upwards":["Urelement","References"],"downwards":[]}
{"upwards":["Urelement","External links"],"downwards":[]}
{"upwards":["New Foundations"],"downwards":["The Type Theory TST","Quinean set theory","Consequences","The consistency problem and related partial results","Models of NFU","Strong axioms of infinity","The system ML (Mathematical Logic)"]}
{"upwards":["New Foundations","The Type Theory TST"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory"],"downwards":["Axioms and stratification","Finite axiomatization","Ordered pairs","Natural numbers and the axiom of infinity"]}
{"upwards":["New Foundations","Quinean set theory","Axioms and stratification"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Finite axiomatization"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Ordered pairs"],"downwards":[]}
{"upwards":["New Foundations","Quinean set theory","Natural numbers and the axiom of infinity"],"downwards":[]}
{"upwards":["New Foundations","Consequences"],"downwards":["Admissibility of useful large sets","Resolution of set-theoretic paradoxes","Cartesian closure"]}
{"upwards":["New Foundations","Consequences","Admissibility of useful large sets"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes"],"downwards":["Cantor's paradox and Cantorian sets","Burali-Forti paradox and the T operation"]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes","Cantor's paradox and Cantorian sets"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Resolution of set-theoretic paradoxes","Burali-Forti paradox and the T operation"],"downwards":[]}
{"upwards":["New Foundations","Consequences","Cartesian closure"],"downwards":[]}
{"upwards":["New Foundations","The consistency problem and related partial results"],"downwards":[]}
{"upwards":["New Foundations","Models of NFU"],"downwards":["Self-sufficiency of mathematical foundations in NFU","Facts about the automorphism j"]}
{"upwards":["New Foundations","Models of NFU","Self-sufficiency of mathematical foundations in NFU"],"downwards":[]}
{"upwards":["New Foundations","Models of NFU","Facts about the automorphism j"],"downwards":[]}
{"upwards":["New Foundations","Strong axioms of infinity"],"downwards":[]}
{"upwards":["New Foundations","The system ML (Mathematical Logic)"],"downwards":[]}
{"upwards":["Ackermann set theory"],"downwards":["The language","The axioms","Relation to Zermelo\u2013Fraenkel set theory","AST and category theory"]}
{"upwards":["Ackermann set theory","The language"],"downwards":[]}
{"upwards":["Ackermann set theory","The axioms"],"downwards":[]}
{"upwards":["Ackermann set theory","Relation to Zermelo\u2013Fraenkel set theory"],"downwards":[]}
{"upwards":["Ackermann set theory","AST and category theory"],"downwards":[]}
{"upwards":["Surjective function"],"downwards":["Definition","Examples","Properties","Space of surjections","Gallery"]}
{"upwards":["Surjective function","Definition"],"downwards":[]}
{"upwards":["Surjective function","Examples"],"downwards":[]}
{"upwards":["Surjective function","Properties"],"downwards":["Surjections as right invertible functions","Surjections as epimorphisms","Surjections as binary relations","Cardinality of the domain of a surjection","Composition and decomposition","Induced surjection and induced bijection"]}
{"upwards":["Surjective function","Properties","Surjections as right invertible functions"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as epimorphisms"],"downwards":[]}
{"upwards":["Surjective function","Properties","Surjections as binary relations"],"downwards":[]}
{"upwards":["Surjective function","Properties","Cardinality of the domain of a surjection"],"downwards":[]}
{"upwards":["Surjective function","Properties","Composition and decomposition"],"downwards":[]}
{"upwards":["Surjective function","Properties","Induced surjection and induced bijection"],"downwards":[]}
{"upwards":["Surjective function","Space of surjections"],"downwards":[]}
{"upwards":["Surjective function","Gallery"],"downwards":[]}
{"upwards":["Inaccessible cardinal"],"downwards":["Models and consistency","Existence of a proper class of inaccessibles","\u03b1-inaccessible cardinals and hyper-inaccessible cardinals","Two model-theoretic characterisations of inaccessibility"]}
{"upwards":["Inaccessible cardinal","Models and consistency"],"downwards":[]}
{"upwards":["Inaccessible cardinal","Existence of a proper class of inaccessibles"],"downwards":[]}
{"upwards":["Inaccessible cardinal","\u03b1-inaccessible cardinals and hyper-inaccessible cardinals"],"downwards":[]}
{"upwards":["Inaccessible cardinal","Two model-theoretic characterisations of inaccessibility"],"downwards":[]}
{"upwards":["G\u00f6del numbering"],"downwards":["Simplified overview","G\u00f6del's encoding","Lack of uniqueness","Application to formal arithmetic","Generalizations","G\u00f6del  sets"]}
{"upwards":["G\u00f6del numbering","Simplified overview"],"downwards":[]}
{"upwards":["G\u00f6del numbering","G\u00f6del's encoding"],"downwards":["Example"]}
{"upwards":["G\u00f6del numbering","G\u00f6del's encoding","Example"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Lack of uniqueness"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Application to formal arithmetic"],"downwards":["Recursion","Expressing statements and proofs by numbers"]}
{"upwards":["G\u00f6del numbering","Application to formal arithmetic","Recursion"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Application to formal arithmetic","Expressing statements and proofs by numbers"],"downwards":[]}
{"upwards":["G\u00f6del numbering","Generalizations"],"downwards":[]}
{"upwards":["G\u00f6del numbering","G\u00f6del  sets"],"downwards":[]}
{"upwards":["Finite set"],"downwards":["Definition and terminology","Basic properties","Necessary and sufficient conditions for finiteness","Foundational issues","Set-theoretic definitions of finiteness"]}
{"upwards":["Finite set","Definition and terminology"],"downwards":[]}
{"upwards":["Finite set","Basic properties"],"downwards":[]}
{"upwards":["Finite set","Necessary and sufficient conditions for finiteness"],"downwards":[]}
{"upwards":["Finite set","Foundational issues"],"downwards":[]}
{"upwards":["Finite set","Set-theoretic definitions of finiteness"],"downwards":["Other concepts of finiteness"]}
{"upwards":["Finite set","Set-theoretic definitions of finiteness","Other concepts of finiteness"],"downwards":[]}
{"upwards":["Constructible universe"],"downwards":["What L is","Additional facts about the sets L\u03b1","L is a standard inner model of ZFC","L is absolute and minimal","L can be well-ordered","L has a reflection principle","The generalized continuum hypothesis holds in L","Constructible sets are definable from the ordinals","Relative constructibility"]}
{"upwards":["Constructible universe","What L is"],"downwards":[]}
{"upwards":["Constructible universe","Additional facts about the sets L\u03b1"],"downwards":[]}
{"upwards":["Constructible universe","L is a standard inner model of ZFC"],"downwards":[]}
{"upwards":["Constructible universe","L is absolute and minimal"],"downwards":["L and large cardinals"]}
{"upwards":["Constructible universe","L is absolute and minimal","L and large cardinals"],"downwards":[]}
{"upwards":["Constructible universe","L can be well-ordered"],"downwards":[]}
{"upwards":["Constructible universe","L has a reflection principle"],"downwards":[]}
{"upwards":["Constructible universe","The generalized continuum hypothesis holds in L"],"downwards":[]}
{"upwards":["Constructible universe","Constructible sets are definable from the ordinals"],"downwards":[]}
{"upwards":["Constructible universe","Relative constructibility"],"downwards":[]}
{"upwards":["Von Neumann universe"],"downwards":["Definition","Applications and interpretations","History","Philosophical perspectives"]}
{"upwards":["Von Neumann universe","Definition"],"downwards":["Finite and low cardinality stages of the hierarchy"]}
{"upwards":["Von Neumann universe","Definition","Finite and low cardinality stages of the hierarchy"],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations"],"downwards":["Applications of V as models for set theories","Interpretation of V as the \"set of all sets\"","V and the axiom of regularity","The existential status of V"]}
{"upwards":["Von Neumann universe","Applications and interpretations","Applications of V as models for set theories"],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations","Interpretation of V as the \"set of all sets\""],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations","V and the axiom of regularity"],"downwards":[]}
{"upwards":["Von Neumann universe","Applications and interpretations","The existential status of V"],"downwards":[]}
{"upwards":["Von Neumann universe","History"],"downwards":[]}
{"upwards":["Von Neumann universe","Philosophical perspectives"],"downwards":[]}
{"upwards":["Aleph number"],"downwards":["Aleph-nought","Aleph-one","Continuum hypothesis","Aleph-omega","Aleph-\u03b1 for general \u03b1","Fixed points of omega","Role of axiom of choice"]}
{"upwards":["Aleph number","Aleph-nought"],"downwards":[]}
{"upwards":["Aleph number","Aleph-one"],"downwards":[]}
{"upwards":["Aleph number","Continuum hypothesis"],"downwards":[]}
{"upwards":["Aleph number","Aleph-omega"],"downwards":[]}
{"upwards":["Aleph number","Aleph-\u03b1 for general \u03b1"],"downwards":[]}
{"upwards":["Aleph number","Fixed points of omega"],"downwards":[]}
{"upwards":["Aleph number","Role of axiom of choice"],"downwards":[]}
{"upwards":["Inhabited set"],"downwards":["Comparison with nonempty sets","Example"]}
{"upwards":["Inhabited set","Comparison with nonempty sets"],"downwards":[]}
{"upwards":["Inhabited set","Example"],"downwards":[]}
{"upwards":["Computable set"],"downwards":["Formal definition","Examples and non-examples","Properties"]}
{"upwards":["Computable set","Formal definition"],"downwards":[]}
{"upwards":["Computable set","Examples and non-examples"],"downwards":[]}
{"upwards":["Computable set","Properties"],"downwards":[]}
{"upwards":["Enumeration"],"downwards":["Combinatorics","Set theory","Computability and complexity theory"]}
{"upwards":["Enumeration","Combinatorics"],"downwards":[]}
{"upwards":["Enumeration","Set theory"],"downwards":["Listing","Countable vs. uncountable","Ordinals","Comparison of cardinalities"]}
{"upwards":["Enumeration","Set theory","Listing"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Countable vs. uncountable"],"downwards":["Examples","Properties"]}
{"upwards":["Enumeration","Set theory","Countable vs. uncountable","Examples"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Countable vs. uncountable","Properties"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Ordinals"],"downwards":[]}
{"upwards":["Enumeration","Set theory","Comparison of cardinalities"],"downwards":[]}
{"upwards":["Enumeration","Computability and complexity theory"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem"],"downwards":["Proof","History","Prerequisites"]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Proof"],"downwards":["Another proof","Examples"]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Proof","Another proof"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Proof","Examples"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","History"],"downwards":[]}
{"upwards":["Schr\u00f6der\u2013Bernstein theorem","Prerequisites"],"downwards":[]}
{"upwards":["Injective function"],"downwards":["Definition","Examples","Injections can be undone","Injections may be made invertible","Other properties","Proving that functions are injective","Gallery"]}
{"upwards":["Injective function","Definition"],"downwards":[]}
{"upwards":["Injective function","Examples"],"downwards":[]}
{"upwards":["Injective function","Injections can be undone"],"downwards":[]}
{"upwards":["Injective function","Injections may be made invertible"],"downwards":[]}
{"upwards":["Injective function","Other properties"],"downwards":[]}
{"upwards":["Injective function","Proving that functions are injective"],"downwards":[]}
{"upwards":["Injective function","Gallery"],"downwards":[]}
{"upwards":["Large cardinal"],"downwards":["Partial definition","Hierarchy of consistency strength","Motivations and epistemic status"]}
{"upwards":["Large cardinal","Partial definition"],"downwards":[]}
{"upwards":["Large cardinal","Hierarchy of consistency strength"],"downwards":[]}
{"upwards":["Large cardinal","Motivations and epistemic status"],"downwards":[]}
{"upwards":["Grothendieck universe"],"downwards":["Properties","Grothendieck universes and inaccessible cardinals"]}
{"upwards":["Grothendieck universe","Properties"],"downwards":[]}
{"upwards":["Grothendieck universe","Grothendieck universes and inaccessible cardinals"],"downwards":[]}
{"upwards":["Isomorphism"],"downwards":["Examples","Applications","Category theoretic view","Relation with equality"]}
{"upwards":["Isomorphism","Examples"],"downwards":["Logarithm and exponential","Integers modulo 6","Relation-preserving isomorphism"]}
{"upwards":["Isomorphism","Examples","Logarithm and exponential"],"downwards":[]}
{"upwards":["Isomorphism","Examples","Integers modulo 6"],"downwards":[]}
{"upwards":["Isomorphism","Examples","Relation-preserving isomorphism"],"downwards":[]}
{"upwards":["Isomorphism","Applications"],"downwards":[]}
{"upwards":["Isomorphism","Category theoretic view"],"downwards":["Isomorphism vs. bijective morphism"]}
{"upwards":["Isomorphism","Category theoretic view","Isomorphism vs. bijective morphism"],"downwards":[]}
{"upwards":["Isomorphism","Relation with equality"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory"],"downwards":["Classes in set theory","Axiomatization of NBG","History","NBG, ZFC, and MK","Category theory","Notes","References","Bibliography","External links"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory"],"downwards":["The uses of classes","Axiom schema versus class existence theorem"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory","The uses of classes"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Classes in set theory","Axiom schema versus class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG"],"downwards":["Classes and sets","Definitions and axioms of extensionality and pairing","Class existence axioms and axiom of regularity","Class existence theorem","Extending the class existence theorem","Set axioms","Axiom of global choice"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Classes and sets"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Definitions and axioms of extensionality and pairing"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Class existence axioms and axiom of regularity"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Extending the class existence theorem"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Set axioms"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Axiomatization of NBG","Axiom of global choice"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History"],"downwards":["Von Neumann's 1925 axiom system","Von Neumann's 1929 axiom system","Bernays' axiom system","G\u00f6del's axiom system (NBG)"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Von Neumann's 1925 axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Von Neumann's 1929 axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","Bernays' axiom system"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","History","G\u00f6del's axiom system (NBG)"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","NBG, ZFC, and MK"],"downwards":["Models"]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","NBG, ZFC, and MK","Models"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Category theory"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Notes"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","References"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","Bibliography"],"downwards":[]}
{"upwards":["Von Neumann\u2013Bernays\u2013G\u00f6del set theory","External links"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)"],"downwards":["Definitions","Generalization to ultra prefilters","Characterizations","Examples, properties, and sufficient conditions","The ultrafilter lemma","Completeness","Ordering on ultrafilters","Ultrafilters on \u2118(\u03c9)"]}
{"upwards":["Ultrafilter (set theory)","Definitions"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Generalization to ultra prefilters"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Characterizations"],"downwards":["Grills and filter-grills","Free or principal"]}
{"upwards":["Ultrafilter (set theory)","Characterizations","Grills and filter-grills"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Characterizations","Free or principal"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Examples, properties, and sufficient conditions"],"downwards":["Monad structure"]}
{"upwards":["Ultrafilter (set theory)","Examples, properties, and sufficient conditions","Monad structure"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma"],"downwards":["Relationships to other statements under ZF"]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF"],"downwards":["Statements that cannot be deduced","Equivalent statements","Weaker statements"]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Statements that cannot be deduced"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Equivalent statements"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","The ultrafilter lemma","Relationships to other statements under ZF","Weaker statements"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Completeness"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Ordering on ultrafilters"],"downwards":[]}
{"upwards":["Ultrafilter (set theory)","Ultrafilters on \u2118(\u03c9)"],"downwards":[]}
{"upwards":["Image (mathematics)"],"downwards":["Definition","Inverse image","Notation for image and inverse image","Examples","Properties"]}
{"upwards":["Image (mathematics)","Definition"],"downwards":["Image of an element","Image of a subset","Image of a function","Generalization to binary relations"]}
{"upwards":["Image (mathematics)","Definition","Image of an element"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a subset"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Image of a function"],"downwards":[]}
{"upwards":["Image (mathematics)","Definition","Generalization to binary relations"],"downwards":[]}
{"upwards":["Image (mathematics)","Inverse image"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image"],"downwards":["Arrow notation","Star notation","Other terminology"]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Arrow notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Star notation"],"downwards":[]}
{"upwards":["Image (mathematics)","Notation for image and inverse image","Other terminology"],"downwards":[]}
{"upwards":["Image (mathematics)","Examples"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties"],"downwards":["General","Multiple functions","Multiple subsets of domain or codomain"]}
{"upwards":["Image (mathematics)","Properties","General"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple functions"],"downwards":[]}
{"upwards":["Image (mathematics)","Properties","Multiple subsets of domain or codomain"],"downwards":[]}
{"upwards":["Transitive set"],"downwards":["Examples","Properties","Transitive closure","Transitive models of set theory"]}
{"upwards":["Transitive set","Examples"],"downwards":[]}
{"upwards":["Transitive set","Properties"],"downwards":[]}
{"upwards":["Transitive set","Transitive closure"],"downwards":[]}
{"upwards":["Transitive set","Transitive models of set theory"],"downwards":[]}
{"upwards":["Formal system"],"downwards":["Background","History","Examples","Variants"]}
{"upwards":["Formal system","Background"],"downwards":["Recursive","Inference and entailment"]}
{"upwards":["Formal system","Background","Recursive"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment"],"downwards":["Formal language","Deductive system","Logical system"]}
{"upwards":["Formal system","Background","Inference and entailment","Formal language"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Deductive system"],"downwards":[]}
{"upwards":["Formal system","Background","Inference and entailment","Logical system"],"downwards":[]}
{"upwards":["Formal system","History"],"downwards":["Formalism","QED manifesto"]}
{"upwards":["Formal system","History","Formalism"],"downwards":["Hilbert's program"]}
{"upwards":["Formal system","History","Formalism","Hilbert's program"],"downwards":[]}
{"upwards":["Formal system","History","QED manifesto"],"downwards":[]}
{"upwards":["Formal system","Examples"],"downwards":[]}
{"upwards":["Formal system","Variants"],"downwards":["Proof system"]}
{"upwards":["Formal system","Variants","Proof system"],"downwards":[]}
{"upwards":["General set theory"],"downwards":["Ontology","Axioms","Discussion","History","Footnotes","References","External links"]}
{"upwards":["General set theory","Ontology"],"downwards":[]}
{"upwards":["General set theory","Axioms"],"downwards":[]}
{"upwards":["General set theory","Discussion"],"downwards":["Metamathematics","Peano arithmetic","Infinite sets"]}
{"upwards":["General set theory","Discussion","Metamathematics"],"downwards":[]}
{"upwards":["General set theory","Discussion","Peano arithmetic"],"downwards":[]}
{"upwards":["General set theory","Discussion","Infinite sets"],"downwards":[]}
{"upwards":["General set theory","History"],"downwards":[]}
{"upwards":["General set theory","Footnotes"],"downwards":[]}
{"upwards":["General set theory","References"],"downwards":[]}
{"upwards":["General set theory","External links"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory"],"downwards":["MK axioms and ontology","Discussion","The axioms in Kelley's General Topology","Notes","References","External links"]}
{"upwards":["Morse\u2013Kelley set theory","MK axioms and ontology"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Discussion"],"downwards":["Model theory","History"]}
{"upwards":["Morse\u2013Kelley set theory","Discussion","Model theory"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Discussion","History"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","The axioms in Kelley's General Topology"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","Notes"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","References"],"downwards":[]}
{"upwards":["Morse\u2013Kelley set theory","External links"],"downwards":[]}
{"upwards":["Operation (mathematics)"],"downwards":["Types of operation","Definition"]}
{"upwards":["Operation (mathematics)","Types of operation"],"downwards":[]}
{"upwards":["Operation (mathematics)","Definition"],"downwards":[]}
{"upwards":["Binary operation"],"downwards":["Terminology","Properties and examples","Notation","Binary operations as ternary relations","External binary operations"]}
{"upwards":["Binary operation","Terminology"],"downwards":[]}
{"upwards":["Binary operation","Properties and examples"],"downwards":[]}
{"upwards":["Binary operation","Notation"],"downwards":[]}
{"upwards":["Binary operation","Binary operations as ternary relations"],"downwards":[]}
{"upwards":["Binary operation","External binary operations"],"downwards":[]}
{"upwards":["NP (complexity)"],"downwards":["Formal definition","Background","Properties","Why some NP problems are hard to solve","Equivalence of definitions","Relationship to other classes","Other characterizations","Examples"]}
{"upwards":["NP (complexity)","Formal definition"],"downwards":[]}
{"upwards":["NP (complexity)","Background"],"downwards":["Verifier-based definition","Machine-definition"]}
{"upwards":["NP (complexity)","Background","Verifier-based definition"],"downwards":[]}
{"upwards":["NP (complexity)","Background","Machine-definition"],"downwards":[]}
{"upwards":["NP (complexity)","Properties"],"downwards":[]}
{"upwards":["NP (complexity)","Why some NP problems are hard to solve"],"downwards":[]}
{"upwards":["NP (complexity)","Equivalence of definitions"],"downwards":[]}
{"upwards":["NP (complexity)","Relationship to other classes"],"downwards":[]}
{"upwards":["NP (complexity)","Other characterizations"],"downwards":[]}
{"upwards":["NP (complexity)","Examples"],"downwards":["P","Integer factorization","NP-complete problems"]}
{"upwards":["NP (complexity)","Examples","P"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","Integer factorization"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","NP-complete problems"],"downwards":["Boolean satisfiability","Travelling salesman","Subgraph isomorphism"]}
{"upwards":["NP (complexity)","Examples","NP-complete problems","Boolean satisfiability"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","NP-complete problems","Travelling salesman"],"downwards":[]}
{"upwards":["NP (complexity)","Examples","NP-complete problems","Subgraph isomorphism"],"downwards":[]}
{"upwards":["P (complexity)"],"downwards":["Definition","Notable problems in P","Relationships to other classes","Properties","Pure existence proofs of polynomial-time algorithms","Alternative characterizations","History","Notes","References","External links"]}
{"upwards":["P (complexity)","Definition"],"downwards":[]}
{"upwards":["P (complexity)","Notable problems in P"],"downwards":[]}
{"upwards":["P (complexity)","Relationships to other classes"],"downwards":[]}
{"upwards":["P (complexity)","Properties"],"downwards":[]}
{"upwards":["P (complexity)","Pure existence proofs of polynomial-time algorithms"],"downwards":[]}
{"upwards":["P (complexity)","Alternative characterizations"],"downwards":[]}
{"upwards":["P (complexity)","History"],"downwards":[]}
{"upwards":["P (complexity)","Notes"],"downwards":[]}
{"upwards":["P (complexity)","References"],"downwards":[]}
{"upwards":["P (complexity)","External links"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory"],"downwards":["Axioms","Related definitions","Theorems","Metalogic"]}
{"upwards":["Kripke\u2013Platek set theory","Axioms"],"downwards":["Empty set","Comparison with Zermelo-Fraenkel set theory"]}
{"upwards":["Kripke\u2013Platek set theory","Axioms","Empty set"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Axioms","Comparison with Zermelo-Fraenkel set theory"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Related definitions"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Theorems"],"downwards":["Admissible sets","Cartesian products exist"]}
{"upwards":["Kripke\u2013Platek set theory","Theorems","Admissible sets"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Theorems","Cartesian products exist"],"downwards":[]}
{"upwards":["Kripke\u2013Platek set theory","Metalogic"],"downwards":[]}
{"upwards":["Computable set"],"downwards":["Formal definition","Examples and non-examples","Properties"]}
{"upwards":["Computable set","Formal definition"],"downwards":[]}
{"upwards":["Computable set","Examples and non-examples"],"downwards":[]}
{"upwards":["Computable set","Properties"],"downwards":[]}
{"upwards":["Computable function"],"downwards":["Definition","Characteristics of computable functions","Computable sets and relations","Formal languages","Examples","Church\u2013Turing thesis","Provability","Uncomputable functions and unsolvable problems","Extensions of computability"]}
{"upwards":["Computable function","Definition"],"downwards":[]}
{"upwards":["Computable function","Characteristics of computable functions"],"downwards":[]}
{"upwards":["Computable function","Computable sets and relations"],"downwards":[]}
{"upwards":["Computable function","Formal languages"],"downwards":[]}
{"upwards":["Computable function","Examples"],"downwards":[]}
{"upwards":["Computable function","Church\u2013Turing thesis"],"downwards":[]}
{"upwards":["Computable function","Provability"],"downwards":["Relation to recursively defined functions","Total functions that are not provably total"]}
{"upwards":["Computable function","Provability","Relation to recursively defined functions"],"downwards":[]}
{"upwards":["Computable function","Provability","Total functions that are not provably total"],"downwards":[]}
{"upwards":["Computable function","Uncomputable functions and unsolvable problems"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability"],"downwards":["Relative computability","Higher recursion theory","Hyper-computation"]}
{"upwards":["Computable function","Extensions of computability","Relative computability"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability","Higher recursion theory"],"downwards":[]}
{"upwards":["Computable function","Extensions of computability","Hyper-computation"],"downwards":[]}
{"upwards":["Undecidable problem"],"downwards":["Background","Example: the halting problem in computability theory","Relationship with G\u00f6del's incompleteness theorem","Examples of undecidable problems","Examples of undecidable statements"]}
{"upwards":["Undecidable problem","Background"],"downwards":[]}
{"upwards":["Undecidable problem","Example: the halting problem in computability theory"],"downwards":[]}
{"upwards":["Undecidable problem","Relationship with G\u00f6del's incompleteness theorem"],"downwards":[]}
{"upwards":["Undecidable problem","Examples of undecidable problems"],"downwards":[]}
{"upwards":["Undecidable problem","Examples of undecidable statements"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis"],"downwards":["Statement in Church's and Turing's words","History","Success of the thesis","Informal usage in proofs","Variations","Philosophical implications","Non-computable functions"]}
{"upwards":["Church\u2013Turing thesis","Statement in Church's and Turing's words"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History"],"downwards":["Circa 1930\u20131952","Later developments","The thesis as a definition"]}
{"upwards":["Church\u2013Turing thesis","History","Circa 1930\u20131952"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History","Later developments"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","History","The thesis as a definition"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Success of the thesis"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Informal usage in proofs"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Variations"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Philosophical implications"],"downwards":[]}
{"upwards":["Church\u2013Turing thesis","Non-computable functions"],"downwards":[]}
{"upwards":["Computably enumerable set"],"downwards":["Formal definition","Equivalent formulations","Examples","Properties","Remarks"]}
{"upwards":["Computably enumerable set","Formal definition"],"downwards":[]}
{"upwards":["Computably enumerable set","Equivalent formulations"],"downwards":[]}
{"upwards":["Computably enumerable set","Examples"],"downwards":[]}
{"upwards":["Computably enumerable set","Properties"],"downwards":[]}
{"upwards":["Computably enumerable set","Remarks"],"downwards":[]}
{"upwards":["Decision problem"],"downwards":["Definition","Examples","Decidability","Complete problems","Function problems","Optimization problems"]}
{"upwards":["Decision problem","Definition"],"downwards":[]}
{"upwards":["Decision problem","Examples"],"downwards":[]}
{"upwards":["Decision problem","Decidability"],"downwards":[]}
{"upwards":["Decision problem","Complete problems"],"downwards":[]}
{"upwards":["Decision problem","Function problems"],"downwards":[]}
{"upwards":["Decision problem","Optimization problems"],"downwards":[]}
{"upwards":["Decidability (logic)"],"downwards":["Decidability of a logical system","Decidability of a theory","Some decidable theories","Some undecidable theories","Semidecidability","Relationship with completeness","Relationship to computability","In context of games"]}
{"upwards":["Decidability (logic)","Decidability of a logical system"],"downwards":[]}
{"upwards":["Decidability (logic)","Decidability of a theory"],"downwards":[]}
{"upwards":["Decidability (logic)","Some decidable theories"],"downwards":[]}
{"upwards":["Decidability (logic)","Some undecidable theories"],"downwards":[]}
{"upwards":["Decidability (logic)","Semidecidability"],"downwards":[]}
{"upwards":["Decidability (logic)","Relationship with completeness"],"downwards":[]}
{"upwards":["Decidability (logic)","Relationship to computability"],"downwards":[]}
{"upwards":["Decidability (logic)","In context of games"],"downwards":[]}
{"upwards":["Abstract logic"],"downwards":[]}
{"upwards":["Concrete category"],"downwards":["Definition","Remarks","Further examples","Counter-examples","Implicit structure of concrete categories","Relative concreteness","Notes","References"]}
{"upwards":["Concrete category","Definition"],"downwards":[]}
{"upwards":["Concrete category","Remarks"],"downwards":[]}
{"upwards":["Concrete category","Further examples"],"downwards":[]}
{"upwards":["Concrete category","Counter-examples"],"downwards":[]}
{"upwards":["Concrete category","Implicit structure of concrete categories"],"downwards":[]}
{"upwards":["Concrete category","Relative concreteness"],"downwards":[]}
{"upwards":["Concrete category","Notes"],"downwards":[]}
{"upwards":["Concrete category","References"],"downwards":[]}
{"upwards":["Primitive recursive function"],"downwards":["Definition","Examples","Use in first-order Peano arithmetic","Elimination of parameters","Relationship to recursive functions","Limitations","Variants","Finitism and consistency results","History"]}
{"upwards":["Primitive recursive function","Definition"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples"],"downwards":["Addition","Doubling","Multiplication","Predecessor","Truncated subtraction","Converting predicates to numeric functions","Predicate \"Is zero\"","Predicate \"Less or equal\"","Predicate \"Greater or equal\"","If-then-else","Junctors","Equality predicate","Other operations on natural numbers","Operations on integers and rational numbers","Some common primitive recursive functions"]}
{"upwards":["Primitive recursive function","Examples","Addition"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Doubling"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Multiplication"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predecessor"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Truncated subtraction"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Converting predicates to numeric functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predicate \"Is zero\""],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predicate \"Less or equal\""],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Predicate \"Greater or equal\""],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","If-then-else"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Junctors"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Equality predicate"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Other operations on natural numbers"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Operations on integers and rational numbers"],"downwards":[]}
{"upwards":["Primitive recursive function","Examples","Some common primitive recursive functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Use in first-order Peano arithmetic"],"downwards":[]}
{"upwards":["Primitive recursive function","Elimination of parameters"],"downwards":[]}
{"upwards":["Primitive recursive function","Relationship to recursive functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Limitations"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants"],"downwards":["Constant functions","Weak primitive recursion","Iterative functions","Additional primitive recursive forms","Computer language definition"]}
{"upwards":["Primitive recursive function","Variants","Constant functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Weak primitive recursion"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Iterative functions"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Additional primitive recursive forms"],"downwards":[]}
{"upwards":["Primitive recursive function","Variants","Computer language definition"],"downwards":[]}
{"upwards":["Primitive recursive function","Finitism and consistency results"],"downwards":[]}
{"upwards":["Primitive recursive function","History"],"downwards":[]}
{"upwards":["Recursion"],"downwards":["Formal definitions","Informal definition","In language","In mathematics","In computer science","In biology","In business","In art","In culture"]}
{"upwards":["Recursion","Formal definitions"],"downwards":[]}
{"upwards":["Recursion","Informal definition"],"downwards":[]}
{"upwards":["Recursion","In language"],"downwards":["Recursive humor"]}
{"upwards":["Recursion","In language","Recursive humor"],"downwards":[]}
{"upwards":["Recursion","In mathematics"],"downwards":["Recursively defined sets","Finite subdivision rules","Functional recursion","Proofs involving recursive definitions","Recursive optimization","The recursion theorem"]}
{"upwards":["Recursion","In mathematics","Recursively defined sets"],"downwards":["Example: the natural numbers","Example: Proof procedure"]}
{"upwards":["Recursion","In mathematics","Recursively defined sets","Example: the natural numbers"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Recursively defined sets","Example: Proof procedure"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Finite subdivision rules"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Functional recursion"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Proofs involving recursive definitions"],"downwards":[]}
{"upwards":["Recursion","In mathematics","Recursive optimization"],"downwards":[]}
{"upwards":["Recursion","In mathematics","The recursion theorem"],"downwards":["Proof of uniqueness"]}
{"upwards":["Recursion","In mathematics","The recursion theorem","Proof of uniqueness"],"downwards":[]}
{"upwards":["Recursion","In computer science"],"downwards":[]}
{"upwards":["Recursion","In biology"],"downwards":[]}
{"upwards":["Recursion","In business"],"downwards":[]}
{"upwards":["Recursion","In art"],"downwards":[]}
{"upwards":["Recursion","In culture"],"downwards":[]}
{"upwards":["Turing machine"],"downwards":["Overview","Description","Formal definition","Additional details required to visualize or implement Turing machines","Equivalent models","Choice c-machines, oracle o-machines","Universal Turing machines","Comparison with real machines","History"]}
{"upwards":["Turing machine","Overview"],"downwards":["Physical description"]}
{"upwards":["Turing machine","Overview","Physical description"],"downwards":[]}
{"upwards":["Turing machine","Description"],"downwards":[]}
{"upwards":["Turing machine","Formal definition"],"downwards":[]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines"],"downwards":["Alternative definitions","The \"state\"","\"State\" diagrams"]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines","Alternative definitions"],"downwards":[]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines","The \"state\""],"downwards":[]}
{"upwards":["Turing machine","Additional details required to visualize or implement Turing machines","\"State\" diagrams"],"downwards":[]}
{"upwards":["Turing machine","Equivalent models"],"downwards":[]}
{"upwards":["Turing machine","Choice c-machines, oracle o-machines"],"downwards":[]}
{"upwards":["Turing machine","Universal Turing machines"],"downwards":[]}
{"upwards":["Turing machine","Comparison with real machines"],"downwards":["Limitations"]}
{"upwards":["Turing machine","Comparison with real machines","Limitations"],"downwards":["Computational complexity theory","Interaction"]}
{"upwards":["Turing machine","Comparison with real machines","Limitations","Computational complexity theory"],"downwards":[]}
{"upwards":["Turing machine","Comparison with real machines","Limitations","Interaction"],"downwards":[]}
{"upwards":["Turing machine","History"],"downwards":["Historical background: computational machinery","The Entscheidungsproblem (the \"decision problem\"): Hilbert's tenth question of 1900","Alan Turing's a-machine","1937\u20131970: The \"digital computer\", the birth of \"computer science\"","1970\u2013present: as a model of computation"]}
{"upwards":["Turing machine","History","Historical background: computational machinery"],"downwards":[]}
{"upwards":["Turing machine","History","The Entscheidungsproblem (the \"decision problem\"): Hilbert's tenth question of 1900"],"downwards":[]}
{"upwards":["Turing machine","History","Alan Turing's a-machine"],"downwards":[]}
{"upwards":["Turing machine","History","1937\u20131970: The \"digital computer\", the birth of \"computer science\""],"downwards":[]}
{"upwards":["Turing machine","History","1970\u2013present: as a model of computation"],"downwards":[]}
{"upwards":["Category theory"],"downwards":["Categories, objects, and morphisms","Functors","Natural transformations","Other concepts","Historical notes"]}
{"upwards":["Category theory","Categories, objects, and morphisms"],"downwards":["Categories","Morphisms"]}
{"upwards":["Category theory","Categories, objects, and morphisms","Categories"],"downwards":[]}
{"upwards":["Category theory","Categories, objects, and morphisms","Morphisms"],"downwards":[]}
{"upwards":["Category theory","Functors"],"downwards":[]}
{"upwards":["Category theory","Natural transformations"],"downwards":[]}
{"upwards":["Category theory","Other concepts"],"downwards":["Universal constructions, limits, and colimits","Equivalent categories","Further concepts and results","Higher-dimensional categories"]}
{"upwards":["Category theory","Other concepts","Universal constructions, limits, and colimits"],"downwards":[]}
{"upwards":["Category theory","Other concepts","Equivalent categories"],"downwards":[]}
{"upwards":["Category theory","Other concepts","Further concepts and results"],"downwards":[]}
{"upwards":["Category theory","Other concepts","Higher-dimensional categories"],"downwards":[]}
{"upwards":["Category theory","Historical notes"],"downwards":[]}
{"upwards":["Kolmogorov complexity"],"downwards":["Definition","Invariance theorem","History and context","Basic results","Compression","Chaitin's incompleteness theorem","Minimum message length","Kolmogorov randomness","Relation to entropy","Conditional versions"]}
{"upwards":["Kolmogorov complexity","Definition"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Invariance theorem"],"downwards":["Informal treatment","A more formal treatment"]}
{"upwards":["Kolmogorov complexity","Invariance theorem","Informal treatment"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Invariance theorem","A more formal treatment"],"downwards":[]}
{"upwards":["Kolmogorov complexity","History and context"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Basic results"],"downwards":["Uncomputability of Kolmogorov complexity","Chain rule for Kolmogorov complexity"]}
{"upwards":["Kolmogorov complexity","Basic results","Uncomputability of Kolmogorov complexity"],"downwards":["A naive attempt at a program to compute K","Formal proof of uncomputability of K"]}
{"upwards":["Kolmogorov complexity","Basic results","Uncomputability of Kolmogorov complexity","A naive attempt at a program to compute K"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Basic results","Uncomputability of Kolmogorov complexity","Formal proof of uncomputability of K"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Basic results","Chain rule for Kolmogorov complexity"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Compression"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Chaitin's incompleteness theorem"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Minimum message length"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Kolmogorov randomness"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Relation to entropy"],"downwards":[]}
{"upwards":["Kolmogorov complexity","Conditional versions"],"downwards":[]}
{"upwards":["Lambda calculus"],"downwards":["Explanation and applications","History","Informal description","Formal definition","Reduction","Normal forms and confluence","Encoding datatypes","Additional programming techniques","Typed lambda calculus","Reduction strategies","Computability","Complexity","Lambda calculus and programming languages","Semantics","Variations and extensions"]}
{"upwards":["Lambda calculus","Explanation and applications"],"downwards":[]}
{"upwards":["Lambda calculus","History"],"downwards":["Origin of the \u03bb symbol"]}
{"upwards":["Lambda calculus","History","Origin of the \u03bb symbol"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description"],"downwards":["Motivation","The lambda calculus"]}
{"upwards":["Lambda calculus","Informal description","Motivation"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus"],"downwards":["Lambda terms","Functions that operate on functions","Alpha equivalence","Free variables","Capture-avoiding substitutions","\u03b2-reduction"]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Lambda terms"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Functions that operate on functions"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Alpha equivalence"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Free variables"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","Capture-avoiding substitutions"],"downwards":[]}
{"upwards":["Lambda calculus","Informal description","The lambda calculus","\u03b2-reduction"],"downwards":[]}
{"upwards":["Lambda calculus","Formal definition"],"downwards":["Definition","Notation","Free and bound variables"]}
{"upwards":["Lambda calculus","Formal definition","Definition"],"downwards":[]}
{"upwards":["Lambda calculus","Formal definition","Notation"],"downwards":[]}
{"upwards":["Lambda calculus","Formal definition","Free and bound variables"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction"],"downwards":["\u03b1-conversion","\u03b2-reduction","\u03b7-reduction"]}
{"upwards":["Lambda calculus","Reduction","\u03b1-conversion"],"downwards":["Substitution"]}
{"upwards":["Lambda calculus","Reduction","\u03b1-conversion","Substitution"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction","\u03b2-reduction"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction","\u03b7-reduction"],"downwards":[]}
{"upwards":["Lambda calculus","Normal forms and confluence"],"downwards":[]}
{"upwards":["Lambda calculus","Encoding datatypes"],"downwards":["Arithmetic in lambda calculus","Logic and predicates","Pairs"]}
{"upwards":["Lambda calculus","Encoding datatypes","Arithmetic in lambda calculus"],"downwards":[]}
{"upwards":["Lambda calculus","Encoding datatypes","Logic and predicates"],"downwards":[]}
{"upwards":["Lambda calculus","Encoding datatypes","Pairs"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques"],"downwards":["Named constants","Recursion and fixed points","Standard terms","Abstraction elimination"]}
{"upwards":["Lambda calculus","Additional programming techniques","Named constants"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques","Recursion and fixed points"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques","Standard terms"],"downwards":[]}
{"upwards":["Lambda calculus","Additional programming techniques","Abstraction elimination"],"downwards":[]}
{"upwards":["Lambda calculus","Typed lambda calculus"],"downwards":[]}
{"upwards":["Lambda calculus","Reduction strategies"],"downwards":[]}
{"upwards":["Lambda calculus","Computability"],"downwards":[]}
{"upwards":["Lambda calculus","Complexity"],"downwards":[]}
{"upwards":["Lambda calculus","Lambda calculus and programming languages"],"downwards":["Anonymous functions","Parallelism and concurrency"]}
{"upwards":["Lambda calculus","Lambda calculus and programming languages","Anonymous functions"],"downwards":[]}
{"upwards":["Lambda calculus","Lambda calculus and programming languages","Parallelism and concurrency"],"downwards":[]}
{"upwards":["Lambda calculus","Semantics"],"downwards":[]}
{"upwards":["Lambda calculus","Variations and extensions"],"downwards":[]}
{"upwards":["Mathematical logic"],"downwards":["Subfields and scope","History","Formal logical systems","Set theory","Model theory","Recursion theory","Proof theory and constructive mathematics","Applications","Connections with computer science","Foundations of mathematics"]}
{"upwards":["Mathematical logic","Subfields and scope"],"downwards":[]}
{"upwards":["Mathematical logic","History"],"downwards":["Early history","19th century","20th century"]}
{"upwards":["Mathematical logic","History","Early history"],"downwards":[]}
{"upwards":["Mathematical logic","History","19th century"],"downwards":["Foundational theories"]}
{"upwards":["Mathematical logic","History","19th century","Foundational theories"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century"],"downwards":["Set theory and paradoxes","Symbolic logic","Beginnings of the other branches"]}
{"upwards":["Mathematical logic","History","20th century","Set theory and paradoxes"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Symbolic logic"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Beginnings of the other branches"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems"],"downwards":["First-order logic","Other classical logics","Nonclassical and modal logic","Algebraic logic"]}
{"upwards":["Mathematical logic","Formal logical systems","First-order logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Other classical logics"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Nonclassical and modal logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Algebraic logic"],"downwards":[]}
{"upwards":["Mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Mathematical logic","Model theory"],"downwards":[]}
{"upwards":["Mathematical logic","Recursion theory"],"downwards":["Algorithmically unsolvable problems"]}
{"upwards":["Mathematical logic","Recursion theory","Algorithmically unsolvable problems"],"downwards":[]}
{"upwards":["Mathematical logic","Proof theory and constructive mathematics"],"downwards":[]}
{"upwards":["Mathematical logic","Applications"],"downwards":[]}
{"upwards":["Mathematical logic","Connections with computer science"],"downwards":[]}
{"upwards":["Mathematical logic","Foundations of mathematics"],"downwards":[]}
{"upwards":["History of logic"],"downwards":["Logic in the East","Logic in the West","Medieval logic","Traditional logic","Rise of modern logic","Modern logic"]}
{"upwards":["History of logic","Logic in the East"],"downwards":["Logic in India","Hindu logic","Origin","Before Gautama","Dattatreya","Medhatithi Gautama","Panini","Nyaya-Vaisheshika","Jain Logic","Buddhist logic","Nagarjuna","Dignaga","Syllogism and influence","Logic in China"]}
{"upwards":["History of logic","Logic in the East","Logic in India"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Hindu logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Origin"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Before Gautama"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Dattatreya"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Medhatithi Gautama"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Panini"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Nyaya-Vaisheshika"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Jain Logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Buddhist logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Nagarjuna"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Dignaga"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Syllogism and influence"],"downwards":[]}
{"upwards":["History of logic","Logic in the East","Logic in China"],"downwards":[]}
{"upwards":["History of logic","Logic in the West"],"downwards":["Prehistory of logic","Ancient Greece before Aristotle","Aristotle","Stoics"]}
{"upwards":["History of logic","Logic in the West","Prehistory of logic"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle"],"downwards":["Thales","Pythagoras","Heraclitus and Parmenides","Plato"]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Thales"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Pythagoras"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Heraclitus and Parmenides"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Ancient Greece before Aristotle","Plato"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Aristotle"],"downwards":["The Organon"]}
{"upwards":["History of logic","Logic in the West","Aristotle","The Organon"],"downwards":[]}
{"upwards":["History of logic","Logic in the West","Stoics"],"downwards":[]}
{"upwards":["History of logic","Medieval logic"],"downwards":["Logic in the Middle East","Logic in medieval Europe"]}
{"upwards":["History of logic","Medieval logic","Logic in the Middle East"],"downwards":[]}
{"upwards":["History of logic","Medieval logic","Logic in medieval Europe"],"downwards":[]}
{"upwards":["History of logic","Traditional logic"],"downwards":["The textbook tradition","Logic in Hegel's philosophy","Logic and psychology"]}
{"upwards":["History of logic","Traditional logic","The textbook tradition"],"downwards":[]}
{"upwards":["History of logic","Traditional logic","Logic in Hegel's philosophy"],"downwards":[]}
{"upwards":["History of logic","Traditional logic","Logic and psychology"],"downwards":[]}
{"upwards":["History of logic","Rise of modern logic"],"downwards":[]}
{"upwards":["History of logic","Modern logic"],"downwards":["Embryonic period","Algebraic period","Logicist period","Metamathematical period","Logic after WWII"]}
{"upwards":["History of logic","Modern logic","Embryonic period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Algebraic period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Logicist period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Metamathematical period"],"downwards":[]}
{"upwards":["History of logic","Modern logic","Logic after WWII"],"downwards":[]}
{"upwards":["Category (mathematics)"],"downwards":["Definition","Small and large categories","Examples","Construction of new categories","Types of morphisms","Types of categories"]}
{"upwards":["Category (mathematics)","Definition"],"downwards":[]}
{"upwards":["Category (mathematics)","Small and large categories"],"downwards":[]}
{"upwards":["Category (mathematics)","Examples"],"downwards":[]}
{"upwards":["Category (mathematics)","Construction of new categories"],"downwards":["Dual category","Product categories"]}
{"upwards":["Category (mathematics)","Construction of new categories","Dual category"],"downwards":[]}
{"upwards":["Category (mathematics)","Construction of new categories","Product categories"],"downwards":[]}
{"upwards":["Category (mathematics)","Types of morphisms"],"downwards":[]}
{"upwards":["Category (mathematics)","Types of categories"],"downwards":[]}
{"upwards":["P versus NP problem"],"downwards":["Example","History","Context","NP-completeness","Harder problems","Problems in NP not known to be in P or NP-complete","Does P mean \"easy\"?","Reasons to believe P \u2260 NP or P = NP","Consequences of solution","Results about difficulty of proof","Logical characterizations","Polynomial-time algorithms","Formal definitions","Claimed solutions","Popular culture"]}
{"upwards":["P versus NP problem","Example"],"downwards":[]}
{"upwards":["P versus NP problem","History"],"downwards":[]}
{"upwards":["P versus NP problem","Context"],"downwards":[]}
{"upwards":["P versus NP problem","NP-completeness"],"downwards":[]}
{"upwards":["P versus NP problem","Harder problems"],"downwards":[]}
{"upwards":["P versus NP problem","Problems in NP not known to be in P or NP-complete"],"downwards":[]}
{"upwards":["P versus NP problem","Does P mean \"easy\"?"],"downwards":[]}
{"upwards":["P versus NP problem","Reasons to believe P \u2260 NP or P = NP"],"downwards":[]}
{"upwards":["P versus NP problem","Consequences of solution"],"downwards":["P = NP","P \u2260 NP"]}
{"upwards":["P versus NP problem","Consequences of solution","P = NP"],"downwards":[]}
{"upwards":["P versus NP problem","Consequences of solution","P \u2260 NP"],"downwards":[]}
{"upwards":["P versus NP problem","Results about difficulty of proof"],"downwards":[]}
{"upwards":["P versus NP problem","Logical characterizations"],"downwards":[]}
{"upwards":["P versus NP problem","Polynomial-time algorithms"],"downwards":[]}
{"upwards":["P versus NP problem","Formal definitions"],"downwards":["P and NP","NP-completeness"]}
{"upwards":["P versus NP problem","Formal definitions","P and NP"],"downwards":["Example"]}
{"upwards":["P versus NP problem","Formal definitions","P and NP","Example"],"downwards":[]}
{"upwards":["P versus NP problem","Formal definitions","NP-completeness"],"downwards":[]}
{"upwards":["P versus NP problem","Claimed solutions"],"downwards":[]}
{"upwards":["P versus NP problem","Popular culture"],"downwards":[]}
{"upwards":["Axiom of dependent choice"],"downwards":["Formal statement","Use","Equivalent statements","Relation with other axioms","Notes","References"]}
{"upwards":["Axiom of dependent choice","Formal statement"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Use"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Equivalent statements"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Relation with other axioms"],"downwards":[]}
{"upwards":["Axiom of dependent choice","Notes"],"downwards":[]}
{"upwards":["Axiom of dependent choice","References"],"downwards":[]}
{"upwards":["Axiom of countable choice"],"downwards":["Overview","Use","References"]}
{"upwards":["Axiom of countable choice","Overview"],"downwards":[]}
{"upwards":["Axiom of countable choice","Use"],"downwards":[]}
{"upwards":["Axiom of countable choice","References"],"downwards":[]}
{"upwards":["Logicism"],"downwards":["Overview","Origin of the name 'logicism'","Intent, or goal, of logicism","Epistemology, ontology and logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","The unit class, impredicativity, and the vicious circle principle","Neo-logicism"]}
{"upwards":["Logicism","Overview"],"downwards":[]}
{"upwards":["Logicism","Origin of the name 'logicism'"],"downwards":[]}
{"upwards":["Logicism","Intent, or goal, of logicism"],"downwards":["History"]}
{"upwards":["Logicism","Intent, or goal, of logicism","History"],"downwards":[]}
{"upwards":["Logicism","Epistemology, ontology and logicism"],"downwards":[]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia"],"downwards":["Preliminaries","The definition of the natural numbers","Criticism"]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","Preliminaries"],"downwards":[]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","The definition of the natural numbers"],"downwards":[]}
{"upwards":["Logicism","An example of a logicist construction of the natural numbers: Russell's construction in the Principia","Criticism"],"downwards":[]}
{"upwards":["Logicism","The unit class, impredicativity, and the vicious circle principle"],"downwards":["A solution to impredicativity: a hierarchy of types","G\u00f6del's criticism and suggestions"]}
{"upwards":["Logicism","The unit class, impredicativity, and the vicious circle principle","A solution to impredicativity: a hierarchy of types"],"downwards":[]}
{"upwards":["Logicism","The unit class, impredicativity, and the vicious circle principle","G\u00f6del's criticism and suggestions"],"downwards":[]}
{"upwards":["Logicism","Neo-logicism"],"downwards":[]}
{"upwards":["Axiom of adjunction"],"downwards":["Interpretability of arithmetic","References"]}
{"upwards":["Axiom of adjunction","Interpretability of arithmetic"],"downwards":[]}
{"upwards":["Axiom of adjunction","References"],"downwards":[]}
{"upwards":["Timeline of mathematical logic"],"downwards":["19th century","20th century"]}
{"upwards":["Timeline of mathematical logic","19th century"],"downwards":[]}
{"upwards":["Timeline of mathematical logic","20th century"],"downwards":["1950-1999"]}
{"upwards":["Timeline of mathematical logic","20th century","1950-1999"],"downwards":[]}
{"upwards":["Axiom of global choice"],"downwards":["Statement","References"]}
{"upwards":["Axiom of global choice","Statement"],"downwards":[]}
{"upwards":["Axiom of global choice","References"],"downwards":[]}
{"upwards":["Philosophy of mathematics"],"downwards":["History","Major themes","Contemporary schools of thought","Arguments","Aesthetics","Journals"]}
{"upwards":["Philosophy of mathematics","History"],"downwards":["Contemporary philosophy"]}
{"upwards":["Philosophy of mathematics","History","Contemporary philosophy"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Major themes"],"downwards":["Mathematical realism","Mathematical anti-realism"]}
{"upwards":["Philosophy of mathematics","Major themes","Mathematical realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Major themes","Mathematical anti-realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought"],"downwards":["Artistic","Platonism","Mathematicism","Logicism","Formalism","Conventionalism","Intuitionism","Structuralism","Embodied mind theories","Fictionalism","Social constructivism","Beyond the traditional schools"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Artistic"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Platonism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Mathematicism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Logicism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Formalism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Conventionalism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Intuitionism"],"downwards":["Constructivism","Finitism"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Intuitionism","Constructivism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Intuitionism","Finitism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Structuralism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories"],"downwards":["Aristotelian realism","Psychologism","Empiricism"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories","Aristotelian realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories","Psychologism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Embodied mind theories","Empiricism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Fictionalism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Social constructivism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools"],"downwards":["Unreasonable effectiveness","Popper's two senses of number statements","Philosophy of language"]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools","Unreasonable effectiveness"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools","Popper's two senses of number statements"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Contemporary schools of thought","Beyond the traditional schools","Philosophy of language"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Arguments"],"downwards":["Indispensability argument for realism","Epistemic argument against realism"]}
{"upwards":["Philosophy of mathematics","Arguments","Indispensability argument for realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Arguments","Epistemic argument against realism"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Aesthetics"],"downwards":[]}
{"upwards":["Philosophy of mathematics","Journals"],"downwards":[]}
{"upwards":["Supertask"],"downwards":["History","Prominent supertasks"]}
{"upwards":["Supertask","History"],"downwards":["Zeno","Thomson","Benacerraf","Modern literature"]}
{"upwards":["Supertask","History","Zeno"],"downwards":["Motion","Achilles and the tortoise"]}
{"upwards":["Supertask","History","Zeno","Motion"],"downwards":[]}
{"upwards":["Supertask","History","Zeno","Achilles and the tortoise"],"downwards":[]}
{"upwards":["Supertask","History","Thomson"],"downwards":[]}
{"upwards":["Supertask","History","Benacerraf"],"downwards":[]}
{"upwards":["Supertask","History","Modern literature"],"downwards":["Philosophy of mathematics","Physical possibility","Super Turing machines"]}
{"upwards":["Supertask","History","Modern literature","Philosophy of mathematics"],"downwards":[]}
{"upwards":["Supertask","History","Modern literature","Physical possibility"],"downwards":[]}
{"upwards":["Supertask","History","Modern literature","Super Turing machines"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks"],"downwards":["Ross\u2013Littlewood paradox","Benardete's paradox","Grim Reaper paradox","Davies' super-machine"]}
{"upwards":["Supertask","Prominent supertasks","Ross\u2013Littlewood paradox"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks","Benardete's paradox"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks","Grim Reaper paradox"],"downwards":[]}
{"upwards":["Supertask","Prominent supertasks","Davies' super-machine"],"downwards":[]}
{"upwards":["Axiom of determinacy"],"downwards":["Types of game that are determined","Incompatibility of the axiom of determinacy with the axiom of choice","Infinitary logic and the axiom of determinacy","Large cardinals and the axiom of determinacy","Projective ordinals"]}
{"upwards":["Axiom of determinacy","Types of game that are determined"],"downwards":[]}
{"upwards":["Axiom of determinacy","Incompatibility of the axiom of determinacy with the axiom of choice"],"downwards":[]}
{"upwards":["Axiom of determinacy","Infinitary logic and the axiom of determinacy"],"downwards":[]}
{"upwards":["Axiom of determinacy","Large cardinals and the axiom of determinacy"],"downwards":[]}
{"upwards":["Axiom of determinacy","Projective ordinals"],"downwards":[]}
{"upwards":["Axiom of constructibility"],"downwards":["Implications","In arithmetic","Significance","Statements true in L","References","External links"]}
{"upwards":["Axiom of constructibility","Implications"],"downwards":[]}
{"upwards":["Axiom of constructibility","In arithmetic"],"downwards":[]}
{"upwards":["Axiom of constructibility","Significance"],"downwards":[]}
{"upwards":["Axiom of constructibility","Statements true in L"],"downwards":[]}
{"upwards":["Axiom of constructibility","References"],"downwards":[]}
{"upwards":["Axiom of constructibility","External links"],"downwards":[]}
{"upwards":["Axiom of extensionality"],"downwards":["Formal statement","Interpretation","In predicate logic without equality","In set theory with ur-elements"]}
{"upwards":["Axiom of extensionality","Formal statement"],"downwards":[]}
{"upwards":["Axiom of extensionality","Interpretation"],"downwards":[]}
{"upwards":["Axiom of extensionality","In predicate logic without equality"],"downwards":[]}
{"upwards":["Axiom of extensionality","In set theory with ur-elements"],"downwards":[]}
{"upwards":["Axiom of infinity"],"downwards":["Formal statement","Interpretation and consequences","Extracting the natural numbers from the infinite set","An apparently weaker version","Independence"]}
{"upwards":["Axiom of infinity","Formal statement"],"downwards":[]}
{"upwards":["Axiom of infinity","Interpretation and consequences"],"downwards":[]}
{"upwards":["Axiom of infinity","Extracting the natural numbers from the infinite set"],"downwards":["Alternative method"]}
{"upwards":["Axiom of infinity","Extracting the natural numbers from the infinite set","Alternative method"],"downwards":[]}
{"upwards":["Axiom of infinity","An apparently weaker version"],"downwards":[]}
{"upwards":["Axiom of infinity","Independence"],"downwards":[]}
{"upwards":["Axiom of limitation of size"],"downwards":["Formal statement","Implications of the axiom","Zermelo's models and the axiom of limitation of size","Limitation of size doctrine","History","Notes","References","Bibliography"]}
{"upwards":["Axiom of limitation of size","Formal statement"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Implications of the axiom"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Zermelo's models and the axiom of limitation of size"],"downwards":["The model V\u03c9","The models V\u03ba where \u03ba is a strongly inaccessible cardinal"]}
{"upwards":["Axiom of limitation of size","Zermelo's models and the axiom of limitation of size","The model V\u03c9"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Zermelo's models and the axiom of limitation of size","The models V\u03ba where \u03ba is a strongly inaccessible cardinal"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Limitation of size doctrine"],"downwards":[]}
{"upwards":["Axiom of limitation of size","History"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Notes"],"downwards":[]}
{"upwards":["Axiom of limitation of size","References"],"downwards":[]}
{"upwards":["Axiom of limitation of size","Bibliography"],"downwards":[]}
{"upwards":["Axiom of pairing"],"downwards":["Formal statement","Consequences","Alternatives","References"]}
{"upwards":["Axiom of pairing","Formal statement"],"downwards":[]}
{"upwards":["Axiom of pairing","Consequences"],"downwards":[]}
{"upwards":["Axiom of pairing","Alternatives"],"downwards":["Non-independence","Weaker","Stronger"]}
{"upwards":["Axiom of pairing","Alternatives","Non-independence"],"downwards":[]}
{"upwards":["Axiom of pairing","Alternatives","Weaker"],"downwards":[]}
{"upwards":["Axiom of pairing","Alternatives","Stronger"],"downwards":[]}
{"upwards":["Axiom of pairing","References"],"downwards":[]}
{"upwards":["Axiom of power set"],"downwards":["Consequences","Limitations","References"]}
{"upwards":["Axiom of power set","Consequences"],"downwards":[]}
{"upwards":["Axiom of power set","Limitations"],"downwards":[]}
{"upwards":["Axiom of power set","References"],"downwards":[]}
{"upwards":["Axiom of regularity"],"downwards":["Elementary implications of regularity","The axiom of dependent choice and no infinite descending sequence of sets implies regularity","Regularity and the rest of ZF(C) axioms","Regularity and Russell's paradox","Regularity, the cumulative hierarchy, and types","History","Regularity in the presence of urelements"]}
{"upwards":["Axiom of regularity","Elementary implications of regularity"],"downwards":["No set is an element of itself","No infinite descending sequence of sets exists","Simpler set-theoretic definition of the ordered pair","Every set has an ordinal rank","For every two sets, only one can be an element of the other"]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","No set is an element of itself"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","No infinite descending sequence of sets exists"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","Simpler set-theoretic definition of the ordered pair"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","Every set has an ordinal rank"],"downwards":[]}
{"upwards":["Axiom of regularity","Elementary implications of regularity","For every two sets, only one can be an element of the other"],"downwards":[]}
{"upwards":["Axiom of regularity","The axiom of dependent choice and no infinite descending sequence of sets implies regularity"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity and the rest of ZF(C) axioms"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity and Russell's paradox"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity, the cumulative hierarchy, and types"],"downwards":[]}
{"upwards":["Axiom of regularity","History"],"downwards":[]}
{"upwards":["Axiom of regularity","Regularity in the presence of urelements"],"downwards":[]}
{"upwards":["Constructive set theory"],"downwards":["Introduction","Subtheories of ZF","Sorted theories"]}
{"upwards":["Constructive set theory","Introduction"],"downwards":["Constructive outlook","History and overview"]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook"],"downwards":["Preliminary on the use of intuitionistic logic","Imposed restrictions on a set theory","Metalogic","Anti-classical principles"]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Preliminary on the use of intuitionistic logic"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Imposed restrictions on a set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","Constructive outlook","Anti-classical principles"],"downwards":[]}
{"upwards":["Constructive set theory","Introduction","History and overview"],"downwards":["Models"]}
{"upwards":["Constructive set theory","Introduction","History and overview","Models"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF"],"downwards":["Notation","Equality","Merging sets","Set existence","BCST","ECST","Functions","Choice functions","Arithmetic","Exponentiation","Analysis","Non-constructive principles","Induction","Strong Collection","Constructive Zermelo\u2013Fraenkel","Metalogic of CZF","Intuitionistic Zermelo\u2013Fraenkel","Intuitionistic Z","Intuitionistic KP"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation"],"downwards":["Language","Variables","Classes"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Language"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Variables"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Notation","Classes"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Equality"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Merging sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Set existence"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST"],"downwards":["Separation","No universal set","Predicativity","Replacement"]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Separation"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","No universal set"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Predicativity"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","BCST","Replacement"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST"],"downwards":["Weaker formulations of infinity","Number bounds"]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST","Weaker formulations of infinity"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","ECST","Number bounds"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions"],"downwards":["Finiteness","General note on programs and functions","Total functional relations","Characteristic functions","Computable sets","Boundedness criteria"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Finiteness"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","General note on programs and functions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Total functional relations"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Characteristic functions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Computable sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Functions","Boundedness criteria"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions"],"downwards":["Diaconescu's theorem","Analysis of Diaconescu's theorem","Regularity implies PEM"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Diaconescu's theorem"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Analysis of Diaconescu's theorem"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Choice functions","Regularity implies PEM"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic"],"downwards":["Recursion","Moderate induction in ECST","Induction without infinite sets"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Recursion"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Moderate induction in ECST"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Arithmetic","Induction without infinite sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation"],"downwards":["On countable sets","The class of subsets of a set","Metalogic","Category and type theoretic notions"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","On countable sets"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","The class of subsets of a set"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Exponentiation","Category and type theoretic notions"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis"],"downwards":["Cauchy sequences","Cauchy reals","Towards the Dedekind reals","Constructive schools"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Cauchy sequences"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Cauchy reals"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Towards the Dedekind reals"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Analysis","Constructive schools"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Non-constructive principles"],"downwards":["Infinite trees"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Non-constructive principles","Infinite trees"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction"],"downwards":["Mathematical induction","Set Induction","Metalogic","Relation to ZF"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Mathematical induction"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Set Induction"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Induction","Relation to ZF"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Strong Collection"],"downwards":["Metalogic"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Strong Collection","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel"],"downwards":["Binary refinement","Subset Collection","Fullness"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Binary refinement"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Subset Collection"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Constructive Zermelo\u2013Fraenkel","Fullness"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF"],"downwards":["Unprovable claims","Ordinal analysis","Models","Indecomposability","Breaking with ZF"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Unprovable claims"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Ordinal analysis"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Models"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Indecomposability"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Metalogic of CZF","Breaking with ZF"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Zermelo\u2013Fraenkel"],"downwards":["Metalogic"]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Zermelo\u2013Fraenkel","Metalogic"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic Z"],"downwards":[]}
{"upwards":["Constructive set theory","Subtheories of ZF","Intuitionistic KP"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories"],"downwards":["Constructive set theory","Bishop style set theory","Category theories"]}
{"upwards":["Constructive set theory","Sorted theories","Constructive set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories","Bishop style set theory"],"downwards":[]}
{"upwards":["Constructive set theory","Sorted theories","Category theories"],"downwards":[]}
{"upwards":["List of set identities and relations"],"downwards":["Notation","One subset involved","Two sets involved","Three sets involved","Cartesian products \u2a2f of finitely many sets","Arbitrary families of sets","Functions and sets","Sequences and collections of families of sets"]}
{"upwards":["List of set identities and relations","Notation"],"downwards":["Elementary set operations"]}
{"upwards":["List of set identities and relations","Notation","Elementary set operations"],"downwards":[]}
{"upwards":["List of set identities and relations","One subset involved"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved"],"downwards":["Formulas for binary set operations \u22c2, \u22c3, \\, and \u2206","De Morgan's laws","Commutativity","Other identities involving two sets","Subsets \u2286 and supersets \u2287"]}
{"upwards":["List of set identities and relations","Two sets involved","Formulas for binary set operations \u22c2, \u22c3, \\, and \u2206"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","De Morgan's laws"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Commutativity"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Other identities involving two sets"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Subsets \u2286 and supersets \u2287"],"downwards":["Set equality","Meets, Joins, and lattice properties"]}
{"upwards":["List of set identities and relations","Two sets involved","Subsets \u2286 and supersets \u2287","Set equality"],"downwards":[]}
{"upwards":["List of set identities and relations","Two sets involved","Subsets \u2286 and supersets \u2287","Meets, Joins, and lattice properties"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved"],"downwards":["Associativity","Distributivity","Two set subtractions","One set subtraction","Three operations on three sets","Other simplifications"]}
{"upwards":["List of set identities and relations","Three sets involved","Associativity"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Distributivity"],"downwards":["Distributivity and symmetric difference \u2206","Distributivity and set subtraction \\"]}
{"upwards":["List of set identities and relations","Three sets involved","Distributivity","Distributivity and symmetric difference \u2206"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Distributivity","Distributivity and set subtraction \\"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Two set subtractions"],"downwards":["(L\\M)\\R","L\\(M\\R)"]}
{"upwards":["List of set identities and relations","Three sets involved","Two set subtractions","(L\\M)\\R"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Two set subtractions","L\\(M\\R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction"],"downwards":["(L\\M) \u204e R","L\\(M \u204e R)","(L \u204e M)\\R","L \u204e (M\\R)"]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","(L\\M) \u204e R"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","L\\(M \u204e R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","(L \u204e M)\\R"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","One set subtraction","L \u204e (M\\R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets"],"downwards":["(L \u2022 M) \u204e (M \u2022 R)","(L \u2022 M) \u204e (R\\M)","(L\\M) \u204e (L\\R)"]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets","(L \u2022 M) \u204e (M \u2022 R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets","(L \u2022 M) \u204e (R\\M)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Three operations on three sets","(L\\M) \u204e (L\\R)"],"downwards":[]}
{"upwards":["List of set identities and relations","Three sets involved","Other simplifications"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets"],"downwards":["Binary \u22c2 of finite \u2a2f","Binary \u22c3 of finite \u2a2f","Difference \\ of finite \u2a2f","Finite \u2a2f of differences \\","Symmetric difference \u2206 and finite \u2a2f"]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Binary \u22c2 of finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Binary \u22c3 of finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Difference \\ of finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Finite \u2a2f of differences \\"],"downwards":[]}
{"upwards":["List of set identities and relations","Cartesian products \u2a2f of finitely many sets","Symmetric difference \u2206 and finite \u2a2f"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets"],"downwards":["Definitions","Distributing unions and intersections","Distributing subtraction over \u22c3 and \u22c2","Commutativity and associativity of \u22c3 and \u22c2","Cartesian products \u03a0 of arbitrarily many sets"]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Definitions"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections"],"downwards":["Binary \u22c2 of arbitrary \u22c3's","Binary \u22c3 of arbitrary \u22c2's","Arbitrary \u22c2's and arbitrary \u22c3's"]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections","Binary \u22c2 of arbitrary \u22c3's"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections","Binary \u22c3 of arbitrary \u22c2's"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing unions and intersections","Arbitrary \u22c2's and arbitrary \u22c3's"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Distributing subtraction over \u22c3 and \u22c2"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Commutativity and associativity of \u22c3 and \u22c2"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets"],"downwards":["Intersections \u22c2 of \u03a0","Unions \u22c3 of \u03a0","Difference \\ of \u03a0","Symmetric difference \u2206 of \u03a0"]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Intersections \u22c2 of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Unions \u22c3 of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Difference \\ of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Arbitrary families of sets","Cartesian products \u03a0 of arbitrarily many sets","Symmetric difference \u2206 of \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets"],"downwards":["Definitions","(Pre)Images of arbitrary unions \u22c3's and intersections \u22c2's","(Pre)Images of binary set operations","(Pre)Images of set operations on (pre)images","(Pre)Images and Cartesian products \u03a0","(Pre)Image of a single set"]}
{"upwards":["List of set identities and relations","Functions and sets","Definitions"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of arbitrary unions \u22c3's and intersections \u22c2's"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations"],"downwards":["Counter-examples: images of operations not distributing","Conditions guaranteeing that images distribute over set operations","Exact formulas\/equalities for images of set operations"]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations","Counter-examples: images of operations not distributing"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations","Conditions guaranteeing that images distribute over set operations"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of binary set operations","Exact formulas\/equalities for images of set operations"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images of set operations on (pre)images"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Images and Cartesian products \u03a0"],"downwards":[]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Image of a single set"],"downwards":["Containments \u2286 and intersections \u22c2 of images and preimages"]}
{"upwards":["List of set identities and relations","Functions and sets","(Pre)Image of a single set","Containments \u2286 and intersections \u22c2 of images and preimages"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets"],"downwards":["Definitions","Power set","Sequences of sets"]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Definitions"],"downwards":["Definitions of categories of families of sets","Elementwise operations on families"]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Definitions","Definitions of categories of families of sets"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Definitions","Elementwise operations on families"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Power set"],"downwards":[]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Sequences of sets"],"downwards":["Partitions"]}
{"upwards":["List of set identities and relations","Sequences and collections of families of sets","Sequences of sets","Partitions"],"downwards":[]}
{"upwards":["Axiom of union"],"downwards":["Formal statement","Relation to Pairing","Relation to Replacement","Relation to Separation","Relation to Intersection","References","Further reading"]}
{"upwards":["Axiom of union","Formal statement"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Pairing"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Replacement"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Separation"],"downwards":[]}
{"upwards":["Axiom of union","Relation to Intersection"],"downwards":[]}
{"upwards":["Axiom of union","References"],"downwards":[]}
{"upwards":["Axiom of union","Further reading"],"downwards":[]}
{"upwards":["Disjoint union"],"downwards":["Example","Set theory definition","Category theory point of view"]}
{"upwards":["Disjoint union","Example"],"downwards":[]}
{"upwards":["Disjoint union","Set theory definition"],"downwards":[]}
{"upwards":["Disjoint union","Category theory point of view"],"downwards":[]}
{"upwards":["Almost"],"downwards":[]}
{"upwards":["Axiom schema of replacement"],"downwards":["Statement","Applications","Relation to other axiom schemas","History","References"]}
{"upwards":["Axiom schema of replacement","Statement"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Applications"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas"],"downwards":["Collection","Separation"]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas","Collection"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas","Separation"],"downwards":[]}
{"upwards":["Axiom schema of replacement","History"],"downwards":[]}
{"upwards":["Axiom schema of replacement","References"],"downwards":[]}
{"upwards":["Axiom schema of specification"],"downwards":["Statement","Relation to the axiom schema of replacement","Unrestricted comprehension","In NBG class theory","In higher-order settings","In Quine's New Foundations","References"]}
{"upwards":["Axiom schema of specification","Statement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Relation to the axiom schema of replacement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Unrestricted comprehension"],"downwards":[]}
{"upwards":["Axiom schema of specification","In NBG class theory"],"downwards":[]}
{"upwards":["Axiom schema of specification","In higher-order settings"],"downwards":[]}
{"upwards":["Axiom schema of specification","In Quine's New Foundations"],"downwards":[]}
{"upwards":["Axiom schema of specification","References"],"downwards":["Citations"]}
{"upwards":["Axiom schema of specification","References","Citations"],"downwards":[]}
{"upwards":["Cardinal number"],"downwards":["History","Motivation","Formal definition","Cardinal arithmetic","The continuum hypothesis"]}
{"upwards":["Cardinal number","History"],"downwards":[]}
{"upwards":["Cardinal number","Motivation"],"downwards":[]}
{"upwards":["Cardinal number","Formal definition"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic"],"downwards":["Successor cardinal","Cardinal addition","Cardinal multiplication","Cardinal exponentiation"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Successor cardinal"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal addition"],"downwards":["Subtraction"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal addition","Subtraction"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal multiplication"],"downwards":["Division"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal multiplication","Division"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal exponentiation"],"downwards":["Roots","Logarithms"]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal exponentiation","Roots"],"downwards":[]}
{"upwards":["Cardinal number","Cardinal arithmetic","Cardinal exponentiation","Logarithms"],"downwards":[]}
{"upwards":["Cardinal number","The continuum hypothesis"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
{"upwards":["Subset","Definition"],"downwards":[]}
{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["Zermelo set theory"],"downwards":["The axioms of Zermelo set theory","Connection with standard set theory","Mac Lane set theory","The aim of Zermelo's paper","The axiom of separation","Cantor's theorem"]}
{"upwards":["Zermelo set theory","The axioms of Zermelo set theory"],"downwards":[]}
{"upwards":["Zermelo set theory","Connection with standard set theory"],"downwards":[]}
{"upwards":["Zermelo set theory","Mac Lane set theory"],"downwards":[]}
{"upwards":["Zermelo set theory","The aim of Zermelo's paper"],"downwards":[]}
{"upwards":["Zermelo set theory","The axiom of separation"],"downwards":[]}
{"upwards":["Zermelo set theory","Cantor's theorem"],"downwards":[]}
{"upwards":["Transfinite induction"],"downwards":["Induction by cases","Transfinite recursion","Relationship to the axiom of choice"]}
{"upwards":["Transfinite induction","Induction by cases"],"downwards":[]}
{"upwards":["Transfinite induction","Transfinite recursion"],"downwards":[]}
{"upwards":["Transfinite induction","Relationship to the axiom of choice"],"downwards":[]}
{"upwards":["Filter (set theory)"],"downwards":["Preliminaries, notation, and basic notions","Filters and prefilters","Set theoretic properties and constructions","Filters and nets"]}
{"upwards":["Filter (set theory)","Preliminaries, notation, and basic notions"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters"],"downwards":["Basic examples","Ultrafilters","Kernels","Finer\/coarser, subordination, and meshing"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Basic examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Ultrafilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels"],"downwards":["Classifying families by their kernels","Characterizing fixed ultra prefilters"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Classifying families by their kernels"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Characterizing fixed ultra prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing"],"downwards":["Equivalent families of sets"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing","Equivalent families of sets"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions"],"downwards":["Trace and meshing","Images and preimages under functions","Products of prefilters","Set subtraction and some examples"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Trace and meshing"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions"],"downwards":["Subordination is preserved by images and preimages"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions","Subordination is preserved by images and preimages"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Products of prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Set subtraction and some examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets"],"downwards":["Nets to prefilters","Prefilters to nets","Subordinate filters and subnets"]}
{"upwards":["Filter (set theory)","Filters and nets","Nets to prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets"],"downwards":["Partially ordered net"]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets","Partially ordered net"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets"],"downwards":["Non\u2013equivalence of subnets and subordinate filters"]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets","Non\u2013equivalence of subnets and subordinate filters"],"downwards":[]}
{"upwards":["Filter (set theory)"],"downwards":["Preliminaries, notation, and basic notions","Filters and prefilters","Set theoretic properties and constructions","Filters and nets"]}
{"upwards":["Filter (set theory)","Preliminaries, notation, and basic notions"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters"],"downwards":["Basic examples","Ultrafilters","Kernels","Finer\/coarser, subordination, and meshing"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Basic examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Ultrafilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels"],"downwards":["Classifying families by their kernels","Characterizing fixed ultra prefilters"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Classifying families by their kernels"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Characterizing fixed ultra prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing"],"downwards":["Equivalent families of sets"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing","Equivalent families of sets"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions"],"downwards":["Trace and meshing","Images and preimages under functions","Products of prefilters","Set subtraction and some examples"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Trace and meshing"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions"],"downwards":["Subordination is preserved by images and preimages"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions","Subordination is preserved by images and preimages"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Products of prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Set subtraction and some examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets"],"downwards":["Nets to prefilters","Prefilters to nets","Subordinate filters and subnets"]}
{"upwards":["Filter (set theory)","Filters and nets","Nets to prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets"],"downwards":["Partially ordered net"]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets","Partially ordered net"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets"],"downwards":["Non\u2013equivalence of subnets and subordinate filters"]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets","Non\u2013equivalence of subnets and subordinate filters"],"downwards":[]}
{"upwards":["Burali-Forti paradox"],"downwards":["Stated in terms of von Neumann ordinals","Stated more generally","Resolutions of the paradox"]}
{"upwards":["Burali-Forti paradox","Stated in terms of von Neumann ordinals"],"downwards":[]}
{"upwards":["Burali-Forti paradox","Stated more generally"],"downwards":[]}
{"upwards":["Burali-Forti paradox","Resolutions of the paradox"],"downwards":[]}
{"upwards":["Dedekind-infinite set"],"downwards":["Comparison with the usual definition of infinite set","Dedekind-infinite sets in ZF","History","Relation to the axiom of choice","Proof of equivalence to infinity, assuming axiom of countable choice","Generalizations","Notes","References"]}
{"upwards":["Dedekind-infinite set","Comparison with the usual definition of infinite set"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Dedekind-infinite sets in ZF"],"downwards":[]}
{"upwards":["Dedekind-infinite set","History"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Relation to the axiom of choice"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Proof of equivalence to infinity, assuming axiom of countable choice"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Generalizations"],"downwards":[]}
{"upwards":["Dedekind-infinite set","Notes"],"downwards":[]}
{"upwards":["Dedekind-infinite set","References"],"downwards":[]}
{"upwards":["Amorphous set"],"downwards":["Existence","Additional properties","Variations","References"]}
{"upwards":["Amorphous set","Existence"],"downwards":[]}
{"upwards":["Amorphous set","Additional properties"],"downwards":[]}
{"upwards":["Amorphous set","Variations"],"downwards":[]}
{"upwards":["Amorphous set","References"],"downwards":[]}
{"upwards":["Abraham Fraenkel"],"downwards":["Biography","Mathematician","Awards","Published works"]}
{"upwards":["Abraham Fraenkel","Biography"],"downwards":[]}
{"upwards":["Abraham Fraenkel","Mathematician"],"downwards":[]}
{"upwards":["Abraham Fraenkel","Awards"],"downwards":[]}
{"upwards":["Abraham Fraenkel","Published works"],"downwards":[]}
{"upwards":["Mathematics"],"downwards":["Etymology","Areas of mathematics","History","Symbolic notation and terminology","Relationship with sciences","Relationship with astrology and esotericism","Philosophy","Training and practice","Cultural impact","Awards and prize problems"]}
{"upwards":["Mathematics","Etymology"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics"],"downwards":["Number theory","Geometry","Algebra","Calculus and analysis","Discrete mathematics","Mathematical logic and set theory","Statistics and other decision sciences","Computational mathematics"]}
{"upwards":["Mathematics","Areas of mathematics","Number theory"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Geometry"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Algebra"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Calculus and analysis"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Discrete mathematics"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Mathematical logic and set theory"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Statistics and other decision sciences"],"downwards":[]}
{"upwards":["Mathematics","Areas of mathematics","Computational mathematics"],"downwards":[]}
{"upwards":["Mathematics","History"],"downwards":["Ancient","Medieval and later"]}
{"upwards":["Mathematics","History","Ancient"],"downwards":[]}
{"upwards":["Mathematics","History","Medieval and later"],"downwards":[]}
{"upwards":["Mathematics","Symbolic notation and terminology"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences"],"downwards":["Pure and applied mathematics","Unreasonable effectiveness","Specific sciences"]}
{"upwards":["Mathematics","Relationship with sciences","Pure and applied mathematics"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Unreasonable effectiveness"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences"],"downwards":["Physics","Computing","Biology and chemistry","Earth sciences","Social sciences"]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Physics"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Computing"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Biology and chemistry"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Earth sciences"],"downwards":[]}
{"upwards":["Mathematics","Relationship with sciences","Specific sciences","Social sciences"],"downwards":[]}
{"upwards":["Mathematics","Relationship with astrology and esotericism"],"downwards":[]}
{"upwards":["Mathematics","Philosophy"],"downwards":["Reality","Proposed definitions","Rigor"]}
{"upwards":["Mathematics","Philosophy","Reality"],"downwards":[]}
{"upwards":["Mathematics","Philosophy","Proposed definitions"],"downwards":[]}
{"upwards":["Mathematics","Philosophy","Rigor"],"downwards":[]}
{"upwards":["Mathematics","Training and practice"],"downwards":["Education","Psychology (aesthetic, creativity and intuition)"]}
{"upwards":["Mathematics","Training and practice","Education"],"downwards":[]}
{"upwards":["Mathematics","Training and practice","Psychology (aesthetic, creativity and intuition)"],"downwards":[]}
{"upwards":["Mathematics","Cultural impact"],"downwards":["Artistic expression","Popularization"]}
{"upwards":["Mathematics","Cultural impact","Artistic expression"],"downwards":[]}
{"upwards":["Mathematics","Cultural impact","Popularization"],"downwards":[]}
{"upwards":["Mathematics","Awards and prize problems"],"downwards":[]}
{"upwards":["Thomas Jech"],"downwards":["Life","Work","Bibliography","References","External links"]}
{"upwards":["Thomas Jech","Life"],"downwards":[]}
{"upwards":["Thomas Jech","Work"],"downwards":[]}
{"upwards":["Thomas Jech","Bibliography"],"downwards":[]}
{"upwards":["Thomas Jech","References"],"downwards":[]}
{"upwards":["Thomas Jech","External links"],"downwards":[]}
{"upwards":["Willard Van Orman Quine"],"downwards":["Biography","Work","In popular culture","Bibliography","Filmography"]}
{"upwards":["Willard Van Orman Quine","Biography"],"downwards":["Education","World War II","Personal","Harvard"]}
{"upwards":["Willard Van Orman Quine","Biography","Education"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","World War II"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","Personal"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Biography","Harvard"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work"],"downwards":["Logic","Set theory","Metaphysics","Epistemology"]}
{"upwards":["Willard Van Orman Quine","Work","Logic"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Set theory"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics"],"downwards":["Rejection of the analytic\u2013synthetic distinction","Confirmation holism and ontological relativity","Existence and its contrary","Ontological commitment","Indispensability argument for mathematical realism"]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Rejection of the analytic\u2013synthetic distinction"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Confirmation holism and ontological relativity"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Existence and its contrary"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Ontological commitment"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Metaphysics","Indispensability argument for mathematical realism"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Work","Epistemology"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","In popular culture"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Bibliography"],"downwards":["Selected books","Important articles"]}
{"upwards":["Willard Van Orman Quine","Bibliography","Selected books"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Bibliography","Important articles"],"downwards":[]}
{"upwards":["Willard Van Orman Quine","Filmography"],"downwards":[]}
{"upwards":["Thoralf Skolem"],"downwards":["Life","Mathematics","Completeness"]}
{"upwards":["Thoralf Skolem","Life"],"downwards":[]}
{"upwards":["Thoralf Skolem","Mathematics"],"downwards":[]}
{"upwards":["Thoralf Skolem","Completeness"],"downwards":[]}
{"upwards":["Ernst Zermelo"],"downwards":["Life","Research in set theory","Zermelo's navigation problem","Publications"]}
{"upwards":["Ernst Zermelo","Life"],"downwards":[]}
{"upwards":["Ernst Zermelo","Research in set theory"],"downwards":[]}
{"upwards":["Ernst Zermelo","Zermelo's navigation problem"],"downwards":[]}
{"upwards":["Ernst Zermelo","Publications"],"downwards":[]}
{"upwards":["Subset"],"downwards":["Definition","Properties","\u2282 and \u2283 symbols","Examples of subsets","Other properties of inclusion"]}
{"upwards":["Subset","Definition"],"downwards":[]}
{"upwards":["Subset","Properties"],"downwards":[]}
{"upwards":["Subset","\u2282 and \u2283 symbols"],"downwards":[]}
{"upwards":["Subset","Examples of subsets"],"downwards":[]}
{"upwards":["Subset","Other properties of inclusion"],"downwards":[]}
{"upwards":["Richard Dedekind"],"downwards":["Life","Work","Bibliography"]}
{"upwards":["Richard Dedekind","Life"],"downwards":[]}
{"upwards":["Richard Dedekind","Work"],"downwards":[]}
{"upwards":["Richard Dedekind","Bibliography"],"downwards":[]}
{"upwards":["Free content"],"downwards":["Definition","Legal matters","Usage","Open content"]}
{"upwards":["Free content","Definition"],"downwards":[]}
{"upwards":["Free content","Legal matters"],"downwards":["Copyright","Public domain","Copyleft"]}
{"upwards":["Free content","Legal matters","Copyright"],"downwards":[]}
{"upwards":["Free content","Legal matters","Public domain"],"downwards":[]}
{"upwards":["Free content","Legal matters","Copyleft"],"downwards":[]}
{"upwards":["Free content","Usage"],"downwards":["Media","Software","Engineering and technology","Academia","Legislation"]}
{"upwards":["Free content","Usage","Media"],"downwards":[]}
{"upwards":["Free content","Usage","Software"],"downwards":[]}
{"upwards":["Free content","Usage","Engineering and technology"],"downwards":[]}
{"upwards":["Free content","Usage","Academia"],"downwards":[]}
{"upwards":["Free content","Usage","Legislation"],"downwards":[]}
{"upwards":["Free content","Open content"],"downwards":["History","\"Open content\" definition","Open access","Open content and education","Textbooks","Licenses"]}
{"upwards":["Free content","Open content","History"],"downwards":[]}
{"upwards":["Free content","Open content","\"Open content\" definition"],"downwards":[]}
{"upwards":["Free content","Open content","Open access"],"downwards":[]}
{"upwards":["Free content","Open content","Open content and education"],"downwards":[]}
{"upwards":["Free content","Open content","Textbooks"],"downwards":[]}
{"upwards":["Free content","Open content","Licenses"],"downwards":[]}
{"upwards":["Paul Bernays"],"downwards":["Biography","Mathematical work","Publications","Notes","References","External links"]}
{"upwards":["Paul Bernays","Biography"],"downwards":[]}
{"upwards":["Paul Bernays","Mathematical work"],"downwards":[]}
{"upwards":["Paul Bernays","Publications"],"downwards":[]}
{"upwards":["Paul Bernays","Notes"],"downwards":[]}
{"upwards":["Paul Bernays","References"],"downwards":[]}
{"upwards":["Paul Bernays","External links"],"downwards":[]}
{"upwards":["John von Neumann"],"downwards":["Life and education","Career and private life","Mathematics","Physics","Economics","Computer science","Defense work","Personality","Recognition","Legacy","Selected works"]}
{"upwards":["John von Neumann","Life and education"],"downwards":["Family background","Child prodigy","University studies"]}
{"upwards":["John von Neumann","Life and education","Family background"],"downwards":[]}
{"upwards":["John von Neumann","Life and education","Child prodigy"],"downwards":[]}
{"upwards":["John von Neumann","Life and education","University studies"],"downwards":[]}
{"upwards":["John von Neumann","Career and private life"],"downwards":["Illness and death"]}
{"upwards":["John von Neumann","Career and private life","Illness and death"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics"],"downwards":["Set theory","Proof theory","Ergodic theory","Measure theory","Topological groups","Functional analysis","Operator algebras","Lattice theory","Mathematical statistics","Other work"]}
{"upwards":["John von Neumann","Mathematics","Set theory"],"downwards":["Von Neumann paradox"]}
{"upwards":["John von Neumann","Mathematics","Set theory","Von Neumann paradox"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Proof theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Ergodic theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Measure theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Topological groups"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Functional analysis"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Operator algebras"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Lattice theory"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Mathematical statistics"],"downwards":[]}
{"upwards":["John von Neumann","Mathematics","Other work"],"downwards":[]}
{"upwards":["John von Neumann","Physics"],"downwards":["Quantum mechanics","Fluid dynamics","Other work"]}
{"upwards":["John von Neumann","Physics","Quantum mechanics"],"downwards":["Von Neumann entropy","Quantum mutual information","Density matrix","Von Neumann measurement scheme","Quantum logic"]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Von Neumann entropy"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Quantum mutual information"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Density matrix"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Von Neumann measurement scheme"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Quantum mechanics","Quantum logic"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Fluid dynamics"],"downwards":[]}
{"upwards":["John von Neumann","Physics","Other work"],"downwards":[]}
{"upwards":["John von Neumann","Economics"],"downwards":["Game theory","Mathematical economics","Linear programming"]}
{"upwards":["John von Neumann","Economics","Game theory"],"downwards":[]}
{"upwards":["John von Neumann","Economics","Mathematical economics"],"downwards":[]}
{"upwards":["John von Neumann","Economics","Linear programming"],"downwards":[]}
{"upwards":["John von Neumann","Computer science"],"downwards":["Cellular automata, DNA and the universal constructor","Scientific computing and numerical analysis","Weather systems and global warming","Technological singularity hypothesis"]}
{"upwards":["John von Neumann","Computer science","Cellular automata, DNA and the universal constructor"],"downwards":[]}
{"upwards":["John von Neumann","Computer science","Scientific computing and numerical analysis"],"downwards":[]}
{"upwards":["John von Neumann","Computer science","Weather systems and global warming"],"downwards":[]}
{"upwards":["John von Neumann","Computer science","Technological singularity hypothesis"],"downwards":[]}
{"upwards":["John von Neumann","Defense work"],"downwards":["Manhattan Project","Post war","Atomic Energy Commission","Mutual assured destruction","Consultancies"]}
{"upwards":["John von Neumann","Defense work","Manhattan Project"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Post war"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Atomic Energy Commission"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Mutual assured destruction"],"downwards":[]}
{"upwards":["John von Neumann","Defense work","Consultancies"],"downwards":[]}
{"upwards":["John von Neumann","Personality"],"downwards":["Mathematical style"]}
{"upwards":["John von Neumann","Personality","Mathematical style"],"downwards":[]}
{"upwards":["John von Neumann","Recognition"],"downwards":["Cognitive abilities","Eidetic memory"]}
{"upwards":["John von Neumann","Recognition","Cognitive abilities"],"downwards":[]}
{"upwards":["John von Neumann","Recognition","Eidetic memory"],"downwards":[]}
{"upwards":["John von Neumann","Legacy"],"downwards":["Mastery of mathematics","Honors and awards"]}
{"upwards":["John von Neumann","Legacy","Mastery of mathematics"],"downwards":[]}
{"upwards":["John von Neumann","Legacy","Honors and awards"],"downwards":[]}
{"upwards":["John von Neumann","Selected works"],"downwards":["Books authored \/ coauthored","Scholarly articles","Popular articles","Collected works"]}
{"upwards":["John von Neumann","Selected works","Books authored \/ coauthored"],"downwards":[]}
{"upwards":["John von Neumann","Selected works","Scholarly articles"],"downwards":[]}
{"upwards":["John von Neumann","Selected works","Popular articles"],"downwards":[]}
{"upwards":["John von Neumann","Selected works","Collected works"],"downwards":[]}
{"upwards":["Kurt G\u00f6del"],"downwards":["Early life and education","Career","Awards and honours","Later life and death","Religious views","Legacy","Bibliography"]}
{"upwards":["Kurt G\u00f6del","Early life and education"],"downwards":["Childhood","Studies in Vienna"]}
{"upwards":["Kurt G\u00f6del","Early life and education","Childhood"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Early life and education","Studies in Vienna"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Career"],"downwards":["Incompleteness theorems","Mid-1930s: further work and U.S. visits","Princeton, Einstein, U.S. citizenship"]}
{"upwards":["Kurt G\u00f6del","Career","Incompleteness theorems"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Career","Mid-1930s: further work and U.S. visits"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Career","Princeton, Einstein, U.S. citizenship"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Awards and honours"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Later life and death"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Religious views"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Legacy"],"downwards":[]}
{"upwards":["Kurt G\u00f6del","Bibliography"],"downwards":["Important publications"]}
{"upwards":["Kurt G\u00f6del","Bibliography","Important publications"],"downwards":[]}
{"upwards":["Encyclopedia"],"downwards":["Etymology","Characteristics","Pre-modern encyclopedias","Printed encyclopedias","Digital encyclopedias"]}
{"upwards":["Encyclopedia","Etymology"],"downwards":[]}
{"upwards":["Encyclopedia","Characteristics"],"downwards":["Four major elements","Encyclopedic dictionaries","Differences between encyclopedias and dictionaries"]}
{"upwards":["Encyclopedia","Characteristics","Four major elements"],"downwards":[]}
{"upwards":["Encyclopedia","Characteristics","Encyclopedic dictionaries"],"downwards":[]}
{"upwards":["Encyclopedia","Characteristics","Differences between encyclopedias and dictionaries"],"downwards":[]}
{"upwards":["Encyclopedia","Pre-modern encyclopedias"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias"],"downwards":["Encyclop\u00e9die","Encyclop\u00e6dia Britannica","Brockhaus Enzyklop\u00e4die","Encyclopedias in the US"]}
{"upwards":["Encyclopedia","Printed encyclopedias","Encyclop\u00e9die"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias","Encyclop\u00e6dia Britannica"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias","Brockhaus Enzyklop\u00e4die"],"downwards":[]}
{"upwards":["Encyclopedia","Printed encyclopedias","Encyclopedias in the US"],"downwards":[]}
{"upwards":["Encyclopedia","Digital encyclopedias"],"downwards":["Physical media","Online"]}
{"upwards":["Encyclopedia","Digital encyclopedias","Physical media"],"downwards":[]}
{"upwards":["Encyclopedia","Digital encyclopedias","Online"],"downwards":["Free encyclopedias"]}
{"upwards":["Encyclopedia","Digital encyclopedias","Online","Free encyclopedias"],"downwards":[]}
{"upwards":["List of states of Mexico"],"downwards":["List"]}
{"upwards":["List of states of Mexico","List"],"downwards":[]}
{"upwards":["Tropical cyclone scales"],"downwards":["Background","Atlantic, Eastern and Central Pacific","Western Pacific","North Indian Ocean","South-West Indian Ocean","Australia and Fiji","Alternative scales"]}
{"upwards":["Tropical cyclone scales","Background"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Atlantic, Eastern and Central Pacific"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Western Pacific"],"downwards":[]}
{"upwards":["Tropical cyclone scales","North Indian Ocean"],"downwards":[]}
{"upwards":["Tropical cyclone scales","South-West Indian Ocean"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Australia and Fiji"],"downwards":[]}
{"upwards":["Tropical cyclone scales","Alternative scales"],"downwards":[]}
{"upwards":["Landfall"],"downwards":["Tropical cyclone","Tornado or waterspout"]}
{"upwards":["Landfall","Tropical cyclone"],"downwards":[]}
{"upwards":["Landfall","Tornado or waterspout"],"downwards":[]}
{"upwards":["Martin's axiom"],"downwards":["Statement","Equivalent forms of MA(\ud835\udecb)","Consequences","Further development","References","Further reading"]}
{"upwards":["Martin's axiom","Statement"],"downwards":[]}
{"upwards":["Martin's axiom","Equivalent forms of MA(\ud835\udecb)"],"downwards":[]}
{"upwards":["Martin's axiom","Consequences"],"downwards":[]}
{"upwards":["Martin's axiom","Further development"],"downwards":[]}
{"upwards":["Martin's axiom","References"],"downwards":[]}
{"upwards":["Martin's axiom","Further reading"],"downwards":[]}
{"upwards":["Hereditarily finite set"],"downwards":["Formal definition","Discussion","Axiomatizations","Graph models"]}
{"upwards":["Hereditarily finite set","Formal definition"],"downwards":[]}
{"upwards":["Hereditarily finite set","Discussion"],"downwards":["Ackermann coding","Representation"]}
{"upwards":["Hereditarily finite set","Discussion","Ackermann coding"],"downwards":[]}
{"upwards":["Hereditarily finite set","Discussion","Representation"],"downwards":[]}
{"upwards":["Hereditarily finite set","Axiomatizations"],"downwards":["Theories of finite sets","ZF"]}
{"upwards":["Hereditarily finite set","Axiomatizations","Theories of finite sets"],"downwards":[]}
{"upwards":["Hereditarily finite set","Axiomatizations","ZF"],"downwards":[]}
{"upwards":["Hereditarily finite set","Graph models"],"downwards":[]}
{"upwards":["Suslin's problem"],"downwards":["Formulation","Implications"]}
{"upwards":["Suslin's problem","Formulation"],"downwards":[]}
{"upwards":["Suslin's problem","Implications"],"downwards":[]}
{"upwards":["Galileo's paradox"],"downwards":["Galileo on infinite sets"]}
{"upwards":["Galileo's paradox","Galileo on infinite sets"],"downwards":[]}
{"upwards":["Filter (set theory)"],"downwards":["Preliminaries, notation, and basic notions","Filters and prefilters","Set theoretic properties and constructions","Filters and nets"]}
{"upwards":["Filter (set theory)","Preliminaries, notation, and basic notions"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters"],"downwards":["Basic examples","Ultrafilters","Kernels","Finer\/coarser, subordination, and meshing"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Basic examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Ultrafilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels"],"downwards":["Classifying families by their kernels","Characterizing fixed ultra prefilters"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Classifying families by their kernels"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Kernels","Characterizing fixed ultra prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing"],"downwards":["Equivalent families of sets"]}
{"upwards":["Filter (set theory)","Filters and prefilters","Finer\/coarser, subordination, and meshing","Equivalent families of sets"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions"],"downwards":["Trace and meshing","Images and preimages under functions","Products of prefilters","Set subtraction and some examples"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Trace and meshing"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions"],"downwards":["Subordination is preserved by images and preimages"]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Images and preimages under functions","Subordination is preserved by images and preimages"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Products of prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Set theoretic properties and constructions","Set subtraction and some examples"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets"],"downwards":["Nets to prefilters","Prefilters to nets","Subordinate filters and subnets"]}
{"upwards":["Filter (set theory)","Filters and nets","Nets to prefilters"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets"],"downwards":["Partially ordered net"]}
{"upwards":["Filter (set theory)","Filters and nets","Prefilters to nets","Partially ordered net"],"downwards":[]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets"],"downwards":["Non\u2013equivalence of subnets and subordinate filters"]}
{"upwards":["Filter (set theory)","Filters and nets","Subordinate filters and subnets","Non\u2013equivalence of subnets and subordinate filters"],"downwards":[]}
{"upwards":["Larva"],"downwards":["Selected types of larvae","Insect larvae"]}
{"upwards":["Larva","Selected types of larvae"],"downwards":[]}
{"upwards":["Larva","Insect larvae"],"downwards":[]}
{"upwards":["Concept"],"downwards":["Ontology of concepts","Notable theories on the structure of concepts","Ideasthesia","Etymology"]}
{"upwards":["Concept","Ontology of concepts"],"downwards":["Concepts as mental representations","Concepts as abstract objects"]}
{"upwards":["Concept","Ontology of concepts","Concepts as mental representations"],"downwards":["The psychological view of concepts","The physicalist view of concepts"]}
{"upwards":["Concept","Ontology of concepts","Concepts as mental representations","The psychological view of concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as mental representations","The physicalist view of concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects"],"downwards":["A priori concepts","Embodied content","Realist universal concepts","Sense and reference","Concepts in calculus"]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","A priori concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Embodied content"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Realist universal concepts"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Sense and reference"],"downwards":[]}
{"upwards":["Concept","Ontology of concepts","Concepts as abstract objects","Concepts in calculus"],"downwards":[]}
{"upwards":["Concept","Notable theories on the structure of concepts"],"downwards":["Classical theory","Prototype theory","Theory-theory"]}
{"upwards":["Concept","Notable theories on the structure of concepts","Classical theory"],"downwards":["Arguments against the classical theory"]}
{"upwards":["Concept","Notable theories on the structure of concepts","Classical theory","Arguments against the classical theory"],"downwards":[]}
{"upwards":["Concept","Notable theories on the structure of concepts","Prototype theory"],"downwards":[]}
{"upwards":["Concept","Notable theories on the structure of concepts","Theory-theory"],"downwards":[]}
{"upwards":["Concept","Ideasthesia"],"downwards":[]}
{"upwards":["Concept","Etymology"],"downwards":[]}
{"upwards":["Deductive reasoning"],"downwards":["Definition","Conceptions of deduction","Rules of inference","Validity and soundness","Difference from ampliative reasoning","In various fields","History","Related concepts and theories"]}
{"upwards":["Deductive reasoning","Definition"],"downwards":[]}
{"upwards":["Deductive reasoning","Conceptions of deduction"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference"],"downwards":["Prominent rules of inference","Fallacies","Definitory and strategic rules"]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference"],"downwards":["Modus ponens","Modus tollens","Hypothetical syllogism"]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference","Modus ponens"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference","Modus tollens"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Prominent rules of inference","Hypothetical syllogism"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Fallacies"],"downwards":[]}
{"upwards":["Deductive reasoning","Rules of inference","Definitory and strategic rules"],"downwards":[]}
{"upwards":["Deductive reasoning","Validity and soundness"],"downwards":[]}
{"upwards":["Deductive reasoning","Difference from ampliative reasoning"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields"],"downwards":["Cognitive psychology","Epistemology","Probability logic"]}
{"upwards":["Deductive reasoning","In various fields","Cognitive psychology"],"downwards":["Psychological theories of deductive reasoning","Intelligence"]}
{"upwards":["Deductive reasoning","In various fields","Cognitive psychology","Psychological theories of deductive reasoning"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields","Cognitive psychology","Intelligence"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields","Epistemology"],"downwards":[]}
{"upwards":["Deductive reasoning","In various fields","Probability logic"],"downwards":[]}
{"upwards":["Deductive reasoning","History"],"downwards":[]}
{"upwards":["Deductive reasoning","Related concepts and theories"],"downwards":["Deductivism","Natural deduction","Geometrical method"]}
{"upwards":["Deductive reasoning","Related concepts and theories","Deductivism"],"downwards":[]}
{"upwards":["Deductive reasoning","Related concepts and theories","Natural deduction"],"downwards":[]}
{"upwards":["Deductive reasoning","Related concepts and theories","Geometrical method"],"downwards":[]}
{"upwards":["Scrophularia"],"downwards":["Selected species","References"]}
{"upwards":["Scrophularia","Selected species"],"downwards":[]}
{"upwards":["Scrophularia","References"],"downwards":[]}
{"upwards":["Mathematical proof"],"downwards":["History and etymology","Nature and purpose","Methods of proof","Undecidable statements","Heuristic mathematics and experimental mathematics","Related concepts","Ending a proof"]}
{"upwards":["Mathematical proof","History and etymology"],"downwards":[]}
{"upwards":["Mathematical proof","Nature and purpose"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof"],"downwards":["Direct proof","Proof by mathematical induction","Proof by contraposition","Proof by contradiction","Proof by construction","Proof by exhaustion","Probabilistic proof","Combinatorial proof","Nonconstructive proof","Statistical proofs in pure mathematics","Computer-assisted proofs"]}
{"upwards":["Mathematical proof","Methods of proof","Direct proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by mathematical induction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contraposition"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contradiction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by construction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by exhaustion"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Probabilistic proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Combinatorial proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Nonconstructive proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Statistical proofs in pure mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Computer-assisted proofs"],"downwards":[]}
{"upwards":["Mathematical proof","Undecidable statements"],"downwards":[]}
{"upwards":["Mathematical proof","Heuristic mathematics and experimental mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts"],"downwards":["Visual proof","Elementary proof","Two-column proof","Colloquial use of \"mathematical proof\"","Statistical proof using data","Inductive logic proofs and Bayesian analysis","Proofs as mental objects","Influence of mathematical proof methods outside mathematics"]}
{"upwards":["Mathematical proof","Related concepts","Visual proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Elementary proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Two-column proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Colloquial use of \"mathematical proof\""],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Statistical proof using data"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Inductive logic proofs and Bayesian analysis"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Proofs as mental objects"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Influence of mathematical proof methods outside mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Ending a proof"],"downwards":[]}
{"upwards":["Tenthredinidae"],"downwards":["Life cycle of Cladius difformis","Taxonomy","References","Bibliography","External links"]}
{"upwards":["Tenthredinidae","Life cycle of Cladius difformis"],"downwards":[]}
{"upwards":["Tenthredinidae","Taxonomy"],"downwards":["Subfamilies and genera","Phylogeny"]}
{"upwards":["Tenthredinidae","Taxonomy","Subfamilies and genera"],"downwards":[]}
{"upwards":["Tenthredinidae","Taxonomy","Phylogeny"],"downwards":[]}
{"upwards":["Tenthredinidae","References"],"downwards":[]}
{"upwards":["Tenthredinidae","Bibliography"],"downwards":[]}
{"upwards":["Tenthredinidae","External links"],"downwards":[]}
{"upwards":["June 8"],"downwards":["Events","Births","Deaths","Holidays and observances","References","External links"]}
{"upwards":["June 8","Events"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 8","Events","Pre-1600"],"downwards":[]}
{"upwards":["June 8","Events","1601\u20131900"],"downwards":[]}
{"upwards":["June 8","Events","1901\u2013present"],"downwards":[]}
{"upwards":["June 8","Births"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 8","Births","Pre-1600"],"downwards":[]}
{"upwards":["June 8","Births","1601\u20131900"],"downwards":[]}
{"upwards":["June 8","Births","1901\u2013present"],"downwards":[]}
{"upwards":["June 8","Deaths"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 8","Deaths","Pre-1600"],"downwards":[]}
{"upwards":["June 8","Deaths","1601\u20131900"],"downwards":[]}
{"upwards":["June 8","Deaths","1901\u2013present"],"downwards":[]}
{"upwards":["June 8","Holidays and observances"],"downwards":[]}
{"upwards":["June 8","References"],"downwards":[]}
{"upwards":["June 8","External links"],"downwards":[]}
{"upwards":["Number"],"downwards":["History","Main classification","Subclasses of the integers","Subclasses of the complex numbers","Extensions of the concept"]}
{"upwards":["Number","History"],"downwards":["Numerals","First use of numbers","Zero","Negative numbers","Rational numbers","Irrational numbers","Transcendental numbers and reals","Infinity and infinitesimals","Complex numbers","Prime numbers"]}
{"upwards":["Number","History","Numerals"],"downwards":[]}
{"upwards":["Number","History","First use of numbers"],"downwards":[]}
{"upwards":["Number","History","Zero"],"downwards":[]}
{"upwards":["Number","History","Negative numbers"],"downwards":[]}
{"upwards":["Number","History","Rational numbers"],"downwards":[]}
{"upwards":["Number","History","Irrational numbers"],"downwards":[]}
{"upwards":["Number","History","Transcendental numbers and reals"],"downwards":[]}
{"upwards":["Number","History","Infinity and infinitesimals"],"downwards":[]}
{"upwards":["Number","History","Complex numbers"],"downwards":[]}
{"upwards":["Number","History","Prime numbers"],"downwards":[]}
{"upwards":["Number","Main classification"],"downwards":["Natural numbers","Integers","Rational numbers","Real numbers","Complex numbers"]}
{"upwards":["Number","Main classification","Natural numbers"],"downwards":[]}
{"upwards":["Number","Main classification","Integers"],"downwards":[]}
{"upwards":["Number","Main classification","Rational numbers"],"downwards":[]}
{"upwards":["Number","Main classification","Real numbers"],"downwards":[]}
{"upwards":["Number","Main classification","Complex numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the integers"],"downwards":["Even and odd numbers","Prime numbers","Other classes of integers"]}
{"upwards":["Number","Subclasses of the integers","Even and odd numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the integers","Prime numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the integers","Other classes of integers"],"downwards":[]}
{"upwards":["Number","Subclasses of the complex numbers"],"downwards":["Algebraic, irrational and transcendental numbers","Constructible numbers","Computable numbers"]}
{"upwards":["Number","Subclasses of the complex numbers","Algebraic, irrational and transcendental numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the complex numbers","Constructible numbers"],"downwards":[]}
{"upwards":["Number","Subclasses of the complex numbers","Computable numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept"],"downwards":["p-adic numbers","Hypercomplex numbers","Transfinite numbers","Nonstandard numbers"]}
{"upwards":["Number","Extensions of the concept","p-adic numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept","Hypercomplex numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept","Transfinite numbers"],"downwards":[]}
{"upwards":["Number","Extensions of the concept","Nonstandard numbers"],"downwards":[]}
{"upwards":["Space (mathematics)"],"downwards":["History","Taxonomy of spaces","Types of spaces","Spaces and structure","List of mathematical spaces"]}
{"upwards":["Space (mathematics)","History"],"downwards":["Before the golden age of geometry","The golden age of geometry and afterwards"]}
{"upwards":["Space (mathematics)","History","Before the golden age of geometry"],"downwards":[]}
{"upwards":["Space (mathematics)","History","The golden age of geometry and afterwards"],"downwards":[]}
{"upwards":["Space (mathematics)","Taxonomy of spaces"],"downwards":["Three taxonomic ranks","Relations between species of spaces"]}
{"upwards":["Space (mathematics)","Taxonomy of spaces","Three taxonomic ranks"],"downwards":[]}
{"upwards":["Space (mathematics)","Taxonomy of spaces","Relations between species of spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces"],"downwards":["Linear and topological spaces","Affine and projective spaces","Metric and uniform spaces","Normed, Banach, inner product, and Hilbert spaces","Smooth and Riemannian manifolds","Measurable, measure, and probability spaces","Non-commutative geometry","Schemes","Topoi"]}
{"upwards":["Space (mathematics)","Types of spaces","Linear and topological spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Affine and projective spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Metric and uniform spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Normed, Banach, inner product, and Hilbert spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Smooth and Riemannian manifolds"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Measurable, measure, and probability spaces"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Non-commutative geometry"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Schemes"],"downwards":[]}
{"upwards":["Space (mathematics)","Types of spaces","Topoi"],"downwards":[]}
{"upwards":["Space (mathematics)","Spaces and structure"],"downwards":[]}
{"upwards":["Space (mathematics)","List of mathematical spaces"],"downwards":[]}
{"upwards":["Mathematical proof"],"downwards":["History and etymology","Nature and purpose","Methods of proof","Undecidable statements","Heuristic mathematics and experimental mathematics","Related concepts","Ending a proof"]}
{"upwards":["Mathematical proof","History and etymology"],"downwards":[]}
{"upwards":["Mathematical proof","Nature and purpose"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof"],"downwards":["Direct proof","Proof by mathematical induction","Proof by contraposition","Proof by contradiction","Proof by construction","Proof by exhaustion","Probabilistic proof","Combinatorial proof","Nonconstructive proof","Statistical proofs in pure mathematics","Computer-assisted proofs"]}
{"upwards":["Mathematical proof","Methods of proof","Direct proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by mathematical induction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contraposition"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by contradiction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by construction"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Proof by exhaustion"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Probabilistic proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Combinatorial proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Nonconstructive proof"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Statistical proofs in pure mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Methods of proof","Computer-assisted proofs"],"downwards":[]}
{"upwards":["Mathematical proof","Undecidable statements"],"downwards":[]}
{"upwards":["Mathematical proof","Heuristic mathematics and experimental mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts"],"downwards":["Visual proof","Elementary proof","Two-column proof","Colloquial use of \"mathematical proof\"","Statistical proof using data","Inductive logic proofs and Bayesian analysis","Proofs as mental objects","Influence of mathematical proof methods outside mathematics"]}
{"upwards":["Mathematical proof","Related concepts","Visual proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Elementary proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Two-column proof"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Colloquial use of \"mathematical proof\""],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Statistical proof using data"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Inductive logic proofs and Bayesian analysis"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Proofs as mental objects"],"downwards":[]}
{"upwards":["Mathematical proof","Related concepts","Influence of mathematical proof methods outside mathematics"],"downwards":[]}
{"upwards":["Mathematical proof","Ending a proof"],"downwards":[]}
{"upwards":["Wikipedia"],"downwards":["History","Openness","Policies and laws","Governance","Community","Language editions","Reception","Operation","Access to content","Cultural influence","Related projects"]}
{"upwards":["Wikipedia","History"],"downwards":["Nupedia","Launch and growth","Milestones"]}
{"upwards":["Wikipedia","History","Nupedia"],"downwards":[]}
{"upwards":["Wikipedia","History","Launch and growth"],"downwards":[]}
{"upwards":["Wikipedia","History","Milestones"],"downwards":[]}
{"upwards":["Wikipedia","Openness"],"downwards":["Restrictions","Review of changes","Vandalism","Edit warring"]}
{"upwards":["Wikipedia","Openness","Restrictions"],"downwards":[]}
{"upwards":["Wikipedia","Openness","Review of changes"],"downwards":[]}
{"upwards":["Wikipedia","Openness","Vandalism"],"downwards":[]}
{"upwards":["Wikipedia","Openness","Edit warring"],"downwards":[]}
{"upwards":["Wikipedia","Policies and laws"],"downwards":["Content policies and guidelines"]}
{"upwards":["Wikipedia","Policies and laws","Content policies and guidelines"],"downwards":[]}
{"upwards":["Wikipedia","Governance"],"downwards":["Administrators","Dispute resolution"]}
{"upwards":["Wikipedia","Governance","Administrators"],"downwards":[]}
{"upwards":["Wikipedia","Governance","Dispute resolution"],"downwards":["Arbitration Committee"]}
{"upwards":["Wikipedia","Governance","Dispute resolution","Arbitration Committee"],"downwards":[]}
{"upwards":["Wikipedia","Community"],"downwards":["Studies","Diversity"]}
{"upwards":["Wikipedia","Community","Studies"],"downwards":[]}
{"upwards":["Wikipedia","Community","Diversity"],"downwards":[]}
{"upwards":["Wikipedia","Language editions"],"downwards":["English Wikipedia editor numbers"]}
{"upwards":["Wikipedia","Language editions","English Wikipedia editor numbers"],"downwards":[]}
{"upwards":["Wikipedia","Reception"],"downwards":["Accuracy of content","Discouragement in education","Coverage of topics and systemic bias","Explicit content","Privacy","Sexism"]}
{"upwards":["Wikipedia","Reception","Accuracy of content"],"downwards":[]}
{"upwards":["Wikipedia","Reception","Discouragement in education"],"downwards":["Medical information"]}
{"upwards":["Wikipedia","Reception","Discouragement in education","Medical information"],"downwards":[]}
{"upwards":["Wikipedia","Reception","Coverage of topics and systemic bias"],"downwards":["Coverage of topics and bias","Systemic biases"]}
{"upwards":["Wikipedia","Reception","Coverage of topics and systemic bias","Coverage of topics and bias"],"downwards":[]}
{"upwards":["Wikipedia","Reception","Coverage of topics and systemic bias","Systemic biases"],"downwards":[]}
{"upwards":["Wikipedia","Reception","Explicit content"],"downwards":[]}
{"upwards":["Wikipedia","Reception","Privacy"],"downwards":[]}
{"upwards":["Wikipedia","Reception","Sexism"],"downwards":[]}
{"upwards":["Wikipedia","Operation"],"downwards":["Wikimedia Foundation and affiliate movements","Software operations and support","Automated editing","Hardware operations and support","Internal research and operational development","Internal news publications","The Wikipedia Library"]}
{"upwards":["Wikipedia","Operation","Wikimedia Foundation and affiliate movements"],"downwards":[]}
{"upwards":["Wikipedia","Operation","Software operations and support"],"downwards":[]}
{"upwards":["Wikipedia","Operation","Automated editing"],"downwards":[]}
{"upwards":["Wikipedia","Operation","Hardware operations and support"],"downwards":[]}
{"upwards":["Wikipedia","Operation","Internal research and operational development"],"downwards":[]}
{"upwards":["Wikipedia","Operation","Internal news publications"],"downwards":[]}
{"upwards":["Wikipedia","Operation","The Wikipedia Library"],"downwards":[]}
{"upwards":["Wikipedia","Access to content"],"downwards":["Content licensing","Methods of access","Chinese access"]}
{"upwards":["Wikipedia","Access to content","Content licensing"],"downwards":[]}
{"upwards":["Wikipedia","Access to content","Methods of access"],"downwards":["Mobile access"]}
{"upwards":["Wikipedia","Access to content","Methods of access","Mobile access"],"downwards":[]}
{"upwards":["Wikipedia","Access to content","Chinese access"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence"],"downwards":["Trusted source to combat fake news","Readership","Cultural significance","Sister projects\u00a0\u2013 Wikimedia","Publishing","Research use"]}
{"upwards":["Wikipedia","Cultural influence","Trusted source to combat fake news"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence","Readership"],"downwards":["COVID-19 pandemic"]}
{"upwards":["Wikipedia","Cultural influence","Readership","COVID-19 pandemic"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence","Cultural significance"],"downwards":["Awards","Satire"]}
{"upwards":["Wikipedia","Cultural influence","Cultural significance","Awards"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence","Cultural significance","Satire"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence","Sister projects\u00a0\u2013 Wikimedia"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence","Publishing"],"downwards":[]}
{"upwards":["Wikipedia","Cultural influence","Research use"],"downwards":[]}
{"upwards":["Wikipedia","Related projects"],"downwards":[]}
{"upwards":["English language"],"downwards":["Classification","History","Geographical distribution","Phonology","Grammar","Vocabulary","Writing system","Dialects, accents and varieties","Sample text"]}
{"upwards":["English language","Classification"],"downwards":[]}
{"upwards":["English language","History"],"downwards":["Proto-Germanic to Old English","Middle English","Early Modern English","Spread of Modern English"]}
{"upwards":["English language","History","Proto-Germanic to Old English"],"downwards":[]}
{"upwards":["English language","History","Middle English"],"downwards":[]}
{"upwards":["English language","History","Early Modern English"],"downwards":[]}
{"upwards":["English language","History","Spread of Modern English"],"downwards":[]}
{"upwards":["English language","Geographical distribution"],"downwards":["Three circles of English-speaking countries","Pluricentric English","English as a global language"]}
{"upwards":["English language","Geographical distribution","Three circles of English-speaking countries"],"downwards":[]}
{"upwards":["English language","Geographical distribution","Pluricentric English"],"downwards":[]}
{"upwards":["English language","Geographical distribution","English as a global language"],"downwards":[]}
{"upwards":["English language","Phonology"],"downwards":["Consonants","Vowels","Phonotactics","Stress, rhythm and intonation","Regional variation"]}
{"upwards":["English language","Phonology","Consonants"],"downwards":[]}
{"upwards":["English language","Phonology","Vowels"],"downwards":[]}
{"upwards":["English language","Phonology","Phonotactics"],"downwards":[]}
{"upwards":["English language","Phonology","Stress, rhythm and intonation"],"downwards":[]}
{"upwards":["English language","Phonology","Regional variation"],"downwards":[]}
{"upwards":["English language","Grammar"],"downwards":["Nouns and noun phrases","Verbs and verb phrases","Syntax"]}
{"upwards":["English language","Grammar","Nouns and noun phrases"],"downwards":["Adjectives","Determiners","Pronouns, case, and person","Prepositions"]}
{"upwards":["English language","Grammar","Nouns and noun phrases","Adjectives"],"downwards":[]}
{"upwards":["English language","Grammar","Nouns and noun phrases","Determiners"],"downwards":[]}
{"upwards":["English language","Grammar","Nouns and noun phrases","Pronouns, case, and person"],"downwards":[]}
{"upwards":["English language","Grammar","Nouns and noun phrases","Prepositions"],"downwards":[]}
{"upwards":["English language","Grammar","Verbs and verb phrases"],"downwards":["Tense, aspect and mood","Phrasal verbs","Adverbs"]}
{"upwards":["English language","Grammar","Verbs and verb phrases","Tense, aspect and mood"],"downwards":[]}
{"upwards":["English language","Grammar","Verbs and verb phrases","Phrasal verbs"],"downwards":[]}
{"upwards":["English language","Grammar","Verbs and verb phrases","Adverbs"],"downwards":[]}
{"upwards":["English language","Grammar","Syntax"],"downwards":["Basic constituent order","Clause syntax","Auxiliary verb constructions","Questions","Discourse level syntax"]}
{"upwards":["English language","Grammar","Syntax","Basic constituent order"],"downwards":[]}
{"upwards":["English language","Grammar","Syntax","Clause syntax"],"downwards":[]}
{"upwards":["English language","Grammar","Syntax","Auxiliary verb constructions"],"downwards":[]}
{"upwards":["English language","Grammar","Syntax","Questions"],"downwards":[]}
{"upwards":["English language","Grammar","Syntax","Discourse level syntax"],"downwards":[]}
{"upwards":["English language","Vocabulary"],"downwards":["Word formation processes","Word origins","English loanwords and calques in other languages"]}
{"upwards":["English language","Vocabulary","Word formation processes"],"downwards":[]}
{"upwards":["English language","Vocabulary","Word origins"],"downwards":[]}
{"upwards":["English language","Vocabulary","English loanwords and calques in other languages"],"downwards":[]}
{"upwards":["English language","Writing system"],"downwards":[]}
{"upwards":["English language","Dialects, accents and varieties"],"downwards":["Britain and Ireland","North America","Australia and New Zealand","Southeast Asia","Africa, the Caribbean, and South Asia","Non-native varieties"]}
{"upwards":["English language","Dialects, accents and varieties","Britain and Ireland"],"downwards":[]}
{"upwards":["English language","Dialects, accents and varieties","North America"],"downwards":[]}
{"upwards":["English language","Dialects, accents and varieties","Australia and New Zealand"],"downwards":[]}
{"upwards":["English language","Dialects, accents and varieties","Southeast Asia"],"downwards":[]}
{"upwards":["English language","Dialects, accents and varieties","Africa, the Caribbean, and South Asia"],"downwards":[]}
{"upwards":["English language","Dialects, accents and varieties","Non-native varieties"],"downwards":[]}
{"upwards":["English language","Sample text"],"downwards":[]}
{"upwards":["Tropical cyclone"],"downwards":["Background","Intensity","Classification and naming","Structure","Movement","Formation regions and warning centers","Preparations","Effects","Response","Climatology","Observation and forecasting","Related cyclone types"]}
{"upwards":["Tropical cyclone","Background"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity"],"downwards":["Factors that influence intensity","Formation","Rapid intensification","Dissipation","Methods for assessing intensity","Intensity metrics"]}
{"upwards":["Tropical cyclone","Intensity","Factors that influence intensity"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Formation"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Rapid intensification"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Dissipation"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Methods for assessing intensity"],"downwards":[]}
{"upwards":["Tropical cyclone","Intensity","Intensity metrics"],"downwards":[]}
{"upwards":["Tropical cyclone","Classification and naming"],"downwards":["Classification","Naming"]}
{"upwards":["Tropical cyclone","Classification and naming","Classification"],"downwards":[]}
{"upwards":["Tropical cyclone","Classification and naming","Naming"],"downwards":[]}
{"upwards":["Tropical cyclone","Structure"],"downwards":["Eye and center","Size"]}
{"upwards":["Tropical cyclone","Structure","Eye and center"],"downwards":[]}
{"upwards":["Tropical cyclone","Structure","Size"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement"],"downwards":["Environmental steering","Beta drift","Multiple storm interaction","Interaction with the mid-latitude westerlies"]}
{"upwards":["Tropical cyclone","Movement","Environmental steering"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement","Beta drift"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement","Multiple storm interaction"],"downwards":[]}
{"upwards":["Tropical cyclone","Movement","Interaction with the mid-latitude westerlies"],"downwards":[]}
{"upwards":["Tropical cyclone","Formation regions and warning centers"],"downwards":[]}
{"upwards":["Tropical cyclone","Preparations"],"downwards":[]}
{"upwards":["Tropical cyclone","Effects"],"downwards":["Natural phenomena caused or worsened by tropical cyclones","Effect on property and human life","Environmental effects"]}
{"upwards":["Tropical cyclone","Effects","Natural phenomena caused or worsened by tropical cyclones"],"downwards":[]}
{"upwards":["Tropical cyclone","Effects","Effect on property and human life"],"downwards":[]}
{"upwards":["Tropical cyclone","Effects","Environmental effects"],"downwards":[]}
{"upwards":["Tropical cyclone","Response"],"downwards":[]}
{"upwards":["Tropical cyclone","Climatology"],"downwards":["Influence of climate change"]}
{"upwards":["Tropical cyclone","Climatology","Influence of climate change"],"downwards":[]}
{"upwards":["Tropical cyclone","Observation and forecasting"],"downwards":["Observation","Forecasting","Geopotential height"]}
{"upwards":["Tropical cyclone","Observation and forecasting","Observation"],"downwards":[]}
{"upwards":["Tropical cyclone","Observation and forecasting","Forecasting"],"downwards":[]}
{"upwards":["Tropical cyclone","Observation and forecasting","Geopotential height"],"downwards":[]}
{"upwards":["Tropical cyclone","Related cyclone types"],"downwards":[]}
{"upwards":["Hyperbola"],"downwards":["Etymology and history","Definitions","In Cartesian coordinates","In polar coordinates","Parametric equations","Hyperbolic functions","Properties","Arc length","Derived curves","Elliptic coordinates","Conic section analysis of the hyperbolic appearance of circles","Applications","Hyperbolas as plane sections of quadrics"]}
{"upwards":["Hyperbola","Etymology and history"],"downwards":[]}
{"upwards":["Hyperbola","Definitions"],"downwards":["As locus of points","Hyperbola with equation y = A\/x","By the directrix property","As plane section of a cone","Pin and string construction","Steiner generation of a hyperbola","Inscribed angles for hyperbolas y = a\/(x \u2212 b) + c and the 3-point-form","As an affine image of the unit hyperbola x\u00b2 \u2212 y\u00b2 = 1","As an affine image of the hyperbola y = 1\/x","Reciprocation of a circle","Quadratic equation"]}
{"upwards":["Hyperbola","Definitions","As locus of points"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Hyperbola with equation y = A\/x"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","By the directrix property"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As plane section of a cone"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Pin and string construction"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Steiner generation of a hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Inscribed angles for hyperbolas y = a\/(x \u2212 b) + c and the 3-point-form"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the unit hyperbola x\u00b2 \u2212 y\u00b2 = 1"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x"],"downwards":["Tangent construction","Point construction","Tangent\u2013asymptotes triangle"]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x","Tangent construction"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x","Point construction"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","As an affine image of the hyperbola y = 1\/x","Tangent\u2013asymptotes triangle"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Reciprocation of a circle"],"downwards":[]}
{"upwards":["Hyperbola","Definitions","Quadratic equation"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates"],"downwards":["Equation","Asymptotes","Semi-latus rectum","Tangent","Rectangular hyperbola","Parametric representation with hyperbolic sine\/cosine","Conjugate hyperbola"]}
{"upwards":["Hyperbola","In Cartesian coordinates","Equation"],"downwards":["Eccentricity"]}
{"upwards":["Hyperbola","In Cartesian coordinates","Equation","Eccentricity"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Asymptotes"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Semi-latus rectum"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Tangent"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Rectangular hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Parametric representation with hyperbolic sine\/cosine"],"downwards":[]}
{"upwards":["Hyperbola","In Cartesian coordinates","Conjugate hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","In polar coordinates"],"downwards":[]}
{"upwards":["Hyperbola","Parametric equations"],"downwards":[]}
{"upwards":["Hyperbola","Hyperbolic functions"],"downwards":[]}
{"upwards":["Hyperbola","Properties"],"downwards":["The tangent bisects the angle between the lines to the foci","Midpoints of parallel chords","Orthogonal tangents \u2013 orthoptic","Pole-polar relation for a hyperbola","Other properties"]}
{"upwards":["Hyperbola","Properties","The tangent bisects the angle between the lines to the foci"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Midpoints of parallel chords"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Orthogonal tangents \u2013 orthoptic"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Pole-polar relation for a hyperbola"],"downwards":[]}
{"upwards":["Hyperbola","Properties","Other properties"],"downwards":[]}
{"upwards":["Hyperbola","Arc length"],"downwards":[]}
{"upwards":["Hyperbola","Derived curves"],"downwards":[]}
{"upwards":["Hyperbola","Elliptic coordinates"],"downwards":[]}
{"upwards":["Hyperbola","Conic section analysis of the hyperbolic appearance of circles"],"downwards":[]}
{"upwards":["Hyperbola","Applications"],"downwards":["Sundials","Multilateration","Path followed by a particle","Korteweg\u2013de Vries equation","Angle trisection","Efficient portfolio frontier","Biochemistry"]}
{"upwards":["Hyperbola","Applications","Sundials"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Multilateration"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Path followed by a particle"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Korteweg\u2013de Vries equation"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Angle trisection"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Efficient portfolio frontier"],"downwards":[]}
{"upwards":["Hyperbola","Applications","Biochemistry"],"downwards":[]}
{"upwards":["Hyperbola","Hyperbolas as plane sections of quadrics"],"downwards":[]}
{"upwards":["Number theory"],"downwards":["History","Main subdivisions","Other subfields","Applications","Prizes"]}
{"upwards":["Number theory","History"],"downwards":["Origins","Early modern number theory","Maturity and division into subfields"]}
{"upwards":["Number theory","History","Origins"],"downwards":["Dawn of arithmetic","Classical Greece and the early Hellenistic period","Diophantus","\u0100ryabha\u1e6da, Brahmagupta, Bh\u0101skara","Arithmetic in the Islamic golden age","Western Europe in the Middle Ages"]}
{"upwards":["Number theory","History","Origins","Dawn of arithmetic"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Classical Greece and the early Hellenistic period"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Diophantus"],"downwards":[]}
{"upwards":["Number theory","History","Origins","\u0100ryabha\u1e6da, Brahmagupta, Bh\u0101skara"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Arithmetic in the Islamic golden age"],"downwards":[]}
{"upwards":["Number theory","History","Origins","Western Europe in the Middle Ages"],"downwards":[]}
{"upwards":["Number theory","History","Early modern number theory"],"downwards":["Fermat","Euler","Lagrange, Legendre, and Gauss"]}
{"upwards":["Number theory","History","Early modern number theory","Fermat"],"downwards":[]}
{"upwards":["Number theory","History","Early modern number theory","Euler"],"downwards":[]}
{"upwards":["Number theory","History","Early modern number theory","Lagrange, Legendre, and Gauss"],"downwards":[]}
{"upwards":["Number theory","History","Maturity and division into subfields"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions"],"downwards":["Elementary number theory","Analytic number theory","Algebraic number theory","Diophantine geometry"]}
{"upwards":["Number theory","Main subdivisions","Elementary number theory"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions","Analytic number theory"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions","Algebraic number theory"],"downwards":[]}
{"upwards":["Number theory","Main subdivisions","Diophantine geometry"],"downwards":[]}
{"upwards":["Number theory","Other subfields"],"downwards":["Probabilistic number theory","Arithmetic combinatorics","Computational number theory"]}
{"upwards":["Number theory","Other subfields","Probabilistic number theory"],"downwards":[]}
{"upwards":["Number theory","Other subfields","Arithmetic combinatorics"],"downwards":[]}
{"upwards":["Number theory","Other subfields","Computational number theory"],"downwards":[]}
{"upwards":["Number theory","Applications"],"downwards":[]}
{"upwards":["Number theory","Prizes"],"downwards":[]}
{"upwards":["Square"],"downwards":["Characterizations","Properties","Coordinates and equations","Construction","Symmetry","Squares inscribed in triangles","Squaring the circle","Non-Euclidean geometry","Crossed square","Graphs"]}
{"upwards":["Square","Characterizations"],"downwards":[]}
{"upwards":["Square","Properties"],"downwards":["Perimeter and area","Other facts"]}
{"upwards":["Square","Properties","Perimeter and area"],"downwards":[]}
{"upwards":["Square","Properties","Other facts"],"downwards":[]}
{"upwards":["Square","Coordinates and equations"],"downwards":[]}
{"upwards":["Square","Construction"],"downwards":[]}
{"upwards":["Square","Symmetry"],"downwards":[]}
{"upwards":["Square","Squares inscribed in triangles"],"downwards":[]}
{"upwards":["Square","Squaring the circle"],"downwards":[]}
{"upwards":["Square","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Square","Crossed square"],"downwards":[]}
{"upwards":["Square","Graphs"],"downwards":[]}
{"upwards":["Polyhedron"],"downwards":["Definition","Characteristics","Convex polyhedra","Symmetries","Other important families of polyhedra","Generalisations of polyhedra","Alternative usages","History","In nature"]}
{"upwards":["Polyhedron","Definition"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics"],"downwards":["Number of faces","Topological classification","Duality","Vertex figures","Surface area and distances","Volume","Dehn invariant"]}
{"upwards":["Polyhedron","Characteristics","Number of faces"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Topological classification"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Duality"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Vertex figures"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Surface area and distances"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Volume"],"downwards":[]}
{"upwards":["Polyhedron","Characteristics","Dehn invariant"],"downwards":[]}
{"upwards":["Polyhedron","Convex polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries"],"downwards":["Regular polyhedra","Uniform polyhedra and their duals","Isohedra","Symmetry groups"]}
{"upwards":["Polyhedron","Symmetries","Regular polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries","Uniform polyhedra and their duals"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries","Isohedra"],"downwards":[]}
{"upwards":["Polyhedron","Symmetries","Symmetry groups"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra"],"downwards":["Polyhedra with regular faces","Pyramids","Stellations and facettings","Zonohedra","Space-filling polyhedra","Lattice polyhedra","Flexible polyhedra","Compounds","Orthogonal polyhedra","Embedded regular maps with planar faces","Canonical polyhedra"]}
{"upwards":["Polyhedron","Other important families of polyhedra","Polyhedra with regular faces"],"downwards":["Equal regular faces","Johnson solids"]}
{"upwards":["Polyhedron","Other important families of polyhedra","Polyhedra with regular faces","Equal regular faces"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Polyhedra with regular faces","Johnson solids"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Pyramids"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Stellations and facettings"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Zonohedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Space-filling polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Lattice polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Flexible polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Compounds"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Orthogonal polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Embedded regular maps with planar faces"],"downwards":[]}
{"upwards":["Polyhedron","Other important families of polyhedra","Canonical polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra"],"downwards":["Apeirohedra","Complex polyhedra","Curved polyhedra","Ideal polyhedra","Skeletons and polyhedra as graphs"]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Apeirohedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Complex polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Curved polyhedra"],"downwards":["Spherical polyhedra","Curved spacefilling polyhedra"]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Curved polyhedra","Spherical polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Curved polyhedra","Curved spacefilling polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Ideal polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Generalisations of polyhedra","Skeletons and polyhedra as graphs"],"downwards":[]}
{"upwards":["Polyhedron","Alternative usages"],"downwards":["Higher-dimensional polyhedra","Topological polyhedra","Abstract polyhedra"]}
{"upwards":["Polyhedron","Alternative usages","Higher-dimensional polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Alternative usages","Topological polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","Alternative usages","Abstract polyhedra"],"downwards":[]}
{"upwards":["Polyhedron","History"],"downwards":["Before the Greeks","Ancient Greece","Ancient China","Medieval Islam","Renaissance","17th\u201319th centuries","20th\u201321st centuries"]}
{"upwards":["Polyhedron","History","Before the Greeks"],"downwards":[]}
{"upwards":["Polyhedron","History","Ancient Greece"],"downwards":[]}
{"upwards":["Polyhedron","History","Ancient China"],"downwards":[]}
{"upwards":["Polyhedron","History","Medieval Islam"],"downwards":[]}
{"upwards":["Polyhedron","History","Renaissance"],"downwards":[]}
{"upwards":["Polyhedron","History","17th\u201319th centuries"],"downwards":[]}
{"upwards":["Polyhedron","History","20th\u201321st centuries"],"downwards":[]}
{"upwards":["Polyhedron","In nature"],"downwards":[]}
{"upwards":["Combinatorics"],"downwards":["Definition","History","Approaches and subfields of combinatorics","Related fields"]}
{"upwards":["Combinatorics","Definition"],"downwards":[]}
{"upwards":["Combinatorics","History"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics"],"downwards":["Enumerative combinatorics","Analytic combinatorics","Partition theory","Graph theory","Design theory","Finite geometry","Order theory","Matroid theory","Extremal combinatorics","Probabilistic combinatorics","Algebraic combinatorics","Combinatorics on words","Geometric combinatorics","Topological combinatorics","Arithmetic combinatorics","Infinitary combinatorics"]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Enumerative combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Analytic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Partition theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Graph theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Design theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Finite geometry"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Order theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Matroid theory"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Extremal combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Probabilistic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Algebraic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Combinatorics on words"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Geometric combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Topological combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Arithmetic combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Approaches and subfields of combinatorics","Infinitary combinatorics"],"downwards":[]}
{"upwards":["Combinatorics","Related fields"],"downwards":["Combinatorial optimization","Coding theory","Discrete and computational geometry","Combinatorics and dynamical systems","Combinatorics and physics"]}
{"upwards":["Combinatorics","Related fields","Combinatorial optimization"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Coding theory"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Discrete and computational geometry"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Combinatorics and dynamical systems"],"downwards":[]}
{"upwards":["Combinatorics","Related fields","Combinatorics and physics"],"downwards":[]}
{"upwards":["Tetrahedron"],"downwards":["Regular tetrahedron","Irregular tetrahedra","General properties","Integer tetrahedra","Related polyhedra and compounds","Applications","Tetrahedral graph"]}
{"upwards":["Tetrahedron","Regular tetrahedron"],"downwards":["Coordinates for a regular tetrahedron","Angles and distances","Isometries of the regular tetrahedron","Orthogonal projections of the regular tetrahedron","Cross section of regular tetrahedron","Spherical tiling","Helical stacking"]}
{"upwards":["Tetrahedron","Regular tetrahedron","Coordinates for a regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Angles and distances"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Isometries of the regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Orthogonal projections of the regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Cross section of regular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Spherical tiling"],"downwards":[]}
{"upwards":["Tetrahedron","Regular tetrahedron","Helical stacking"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra"],"downwards":["Trirectangular tetrahedron","Disphenoid","Orthoschemes","Space-filling tetrahedra","Fundamental domains","Isometries of irregular tetrahedra"]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Trirectangular tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Disphenoid"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Orthoschemes"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Space-filling tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Fundamental domains"],"downwards":[]}
{"upwards":["Tetrahedron","Irregular tetrahedra","Isometries of irregular tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","General properties"],"downwards":["Volume","Distance between the edges","Properties analogous to those of a triangle","Geometric relations","A law of sines for tetrahedra and the space of all shapes of tetrahedra","Law of cosines for tetrahedra","Interior point","Inradius","Circumradius","Circumcenter","Centroid","Faces"]}
{"upwards":["Tetrahedron","General properties","Volume"],"downwards":["Heron-type formula for the volume of a tetrahedron","Volume divider","Non-Euclidean volume"]}
{"upwards":["Tetrahedron","General properties","Volume","Heron-type formula for the volume of a tetrahedron"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Volume","Volume divider"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Volume","Non-Euclidean volume"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Distance between the edges"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Properties analogous to those of a triangle"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Geometric relations"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","A law of sines for tetrahedra and the space of all shapes of tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Law of cosines for tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Interior point"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Inradius"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Circumradius"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Circumcenter"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Centroid"],"downwards":[]}
{"upwards":["Tetrahedron","General properties","Faces"],"downwards":[]}
{"upwards":["Tetrahedron","Integer tetrahedra"],"downwards":[]}
{"upwards":["Tetrahedron","Related polyhedra and compounds"],"downwards":[]}
{"upwards":["Tetrahedron","Applications"],"downwards":["Numerical analysis","Structural engineering","Aviation","Chemistry","Electricity and electronics","Color space","Games","Geology","Popular culture"]}
{"upwards":["Tetrahedron","Applications","Numerical analysis"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Structural engineering"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Aviation"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Chemistry"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Electricity and electronics"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Color space"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Games"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Geology"],"downwards":[]}
{"upwards":["Tetrahedron","Applications","Popular culture"],"downwards":[]}
{"upwards":["Tetrahedron","Tetrahedral graph"],"downwards":[]}
{"upwards":["Ontology"],"downwards":["Etymology","Overview","Types of ontologies","History","Other ontological topics"]}
{"upwards":["Ontology","Etymology"],"downwards":[]}
{"upwards":["Ontology","Overview"],"downwards":["Particulars and universals","Abstract and concrete","Ontological dependence","Identity","Modality","Substances","Properties and relations","States of affairs and events","Reality of things"]}
{"upwards":["Ontology","Overview","Particulars and universals"],"downwards":[]}
{"upwards":["Ontology","Overview","Abstract and concrete"],"downwards":[]}
{"upwards":["Ontology","Overview","Ontological dependence"],"downwards":[]}
{"upwards":["Ontology","Overview","Identity"],"downwards":[]}
{"upwards":["Ontology","Overview","Modality"],"downwards":[]}
{"upwards":["Ontology","Overview","Substances"],"downwards":[]}
{"upwards":["Ontology","Overview","Properties and relations"],"downwards":[]}
{"upwards":["Ontology","Overview","States of affairs and events"],"downwards":[]}
{"upwards":["Ontology","Overview","Reality of things"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies"],"downwards":["Flat vs polycategorical vs hierarchical","Thing ontologies vs fact ontologies","Constituent ontologies vs blob theories","Information science and natural sciences"]}
{"upwards":["Ontology","Types of ontologies","Flat vs polycategorical vs hierarchical"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies","Thing ontologies vs fact ontologies"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies","Constituent ontologies vs blob theories"],"downwards":[]}
{"upwards":["Ontology","Types of ontologies","Information science and natural sciences"],"downwards":[]}
{"upwards":["Ontology","History"],"downwards":["Ancient Greek","Hindu philosophy","Medieval","Modern","20th century"]}
{"upwards":["Ontology","History","Ancient Greek"],"downwards":[]}
{"upwards":["Ontology","History","Hindu philosophy"],"downwards":[]}
{"upwards":["Ontology","History","Medieval"],"downwards":[]}
{"upwards":["Ontology","History","Modern"],"downwards":[]}
{"upwards":["Ontology","History","20th century"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics"],"downwards":["Ontological formations","Ontology of fictional characters","Ontological and epistemological certainty","Body and environment, questioning the meaning of being","Ontology and language","Ontology and human geography","Anthropology","Reality and actuality","Microcosmic ontology","Ontological argument","Hintikka's locution for existence"]}
{"upwards":["Ontology","Other ontological topics","Ontological formations"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontology of fictional characters"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontological and epistemological certainty"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Body and environment, questioning the meaning of being"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontology and language"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontology and human geography"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Anthropology"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Reality and actuality"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Microcosmic ontology"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Ontological argument"],"downwards":[]}
{"upwards":["Ontology","Other ontological topics","Hintikka's locution for existence"],"downwards":[]}
{"upwards":["Point (geometry)"],"downwards":["Points in Euclidean geometry","Dimension of a point","Geometry without points","Point masses and the Dirac delta function"]}
{"upwards":["Point (geometry)","Points in Euclidean geometry"],"downwards":[]}
{"upwards":["Point (geometry)","Dimension of a point"],"downwards":["Vector space dimension","Topological dimension","Hausdorff dimension"]}
{"upwards":["Point (geometry)","Dimension of a point","Vector space dimension"],"downwards":[]}
{"upwards":["Point (geometry)","Dimension of a point","Topological dimension"],"downwards":[]}
{"upwards":["Point (geometry)","Dimension of a point","Hausdorff dimension"],"downwards":[]}
{"upwards":["Point (geometry)","Geometry without points"],"downwards":[]}
{"upwards":["Point (geometry)","Point masses and the Dirac delta function"],"downwards":[]}
{"upwards":["Circle"],"downwards":["Euclid's definition","Topological definition","Terminology","History","Analytic results","Properties","Compass and straightedge constructions","Circle of Apollonius","Inscription in or circumscription about other figures","Limiting case of other figures","In other p-norms","Locus of constant sum","Squaring the circle","Significance in art and symbolism","Unicode"]}
{"upwards":["Circle","Euclid's definition"],"downwards":[]}
{"upwards":["Circle","Topological definition"],"downwards":[]}
{"upwards":["Circle","Terminology"],"downwards":[]}
{"upwards":["Circle","History"],"downwards":[]}
{"upwards":["Circle","Analytic results"],"downwards":["Circumference","Area enclosed","Equations","Tangent lines"]}
{"upwards":["Circle","Analytic results","Circumference"],"downwards":[]}
{"upwards":["Circle","Analytic results","Area enclosed"],"downwards":[]}
{"upwards":["Circle","Analytic results","Equations"],"downwards":["Cartesian coordinates","Polar coordinates","Complex plane"]}
{"upwards":["Circle","Analytic results","Equations","Cartesian coordinates"],"downwards":[]}
{"upwards":["Circle","Analytic results","Equations","Polar coordinates"],"downwards":[]}
{"upwards":["Circle","Analytic results","Equations","Complex plane"],"downwards":[]}
{"upwards":["Circle","Analytic results","Tangent lines"],"downwards":[]}
{"upwards":["Circle","Properties"],"downwards":["Chord","Tangent","Theorems","Inscribed angles","Sagitta"]}
{"upwards":["Circle","Properties","Chord"],"downwards":[]}
{"upwards":["Circle","Properties","Tangent"],"downwards":[]}
{"upwards":["Circle","Properties","Theorems"],"downwards":[]}
{"upwards":["Circle","Properties","Inscribed angles"],"downwards":[]}
{"upwards":["Circle","Properties","Sagitta"],"downwards":[]}
{"upwards":["Circle","Compass and straightedge constructions"],"downwards":["Construction with given diameter","Construction through three noncollinear points"]}
{"upwards":["Circle","Compass and straightedge constructions","Construction with given diameter"],"downwards":[]}
{"upwards":["Circle","Compass and straightedge constructions","Construction through three noncollinear points"],"downwards":[]}
{"upwards":["Circle","Circle of Apollonius"],"downwards":["Cross-ratios","Generalised circles"]}
{"upwards":["Circle","Circle of Apollonius","Cross-ratios"],"downwards":[]}
{"upwards":["Circle","Circle of Apollonius","Generalised circles"],"downwards":[]}
{"upwards":["Circle","Inscription in or circumscription about other figures"],"downwards":[]}
{"upwards":["Circle","Limiting case of other figures"],"downwards":[]}
{"upwards":["Circle","In other p-norms"],"downwards":[]}
{"upwards":["Circle","Locus of constant sum"],"downwards":[]}
{"upwards":["Circle","Squaring the circle"],"downwards":[]}
{"upwards":["Circle","Significance in art and symbolism"],"downwards":[]}
{"upwards":["Circle","Unicode"],"downwards":[]}
{"upwards":["Triangle"],"downwards":["Types of triangle","Basic facts","Existence of a triangle","Points, lines, and circles associated with a triangle","Computing the sides and angles","Area","Further formulas for general Euclidean triangles","Figures inscribed in a triangle","Figures circumscribed about a triangle","Specifying the location of a point in a triangle","Non-planar triangles","Triangles in construction"]}
{"upwards":["Triangle","Types of triangle"],"downwards":["By lengths of sides","By internal angles"]}
{"upwards":["Triangle","Types of triangle","By lengths of sides"],"downwards":[]}
{"upwards":["Triangle","Types of triangle","By internal angles"],"downwards":[]}
{"upwards":["Triangle","Basic facts"],"downwards":["Similarity and congruence","Right triangles"]}
{"upwards":["Triangle","Basic facts","Similarity and congruence"],"downwards":[]}
{"upwards":["Triangle","Basic facts","Right triangles"],"downwards":[]}
{"upwards":["Triangle","Existence of a triangle"],"downwards":["Condition on the sides","Conditions on the angles"]}
{"upwards":["Triangle","Existence of a triangle","Condition on the sides"],"downwards":[]}
{"upwards":["Triangle","Existence of a triangle","Conditions on the angles"],"downwards":["Trigonometric conditions"]}
{"upwards":["Triangle","Existence of a triangle","Conditions on the angles","Trigonometric conditions"],"downwards":[]}
{"upwards":["Triangle","Points, lines, and circles associated with a triangle"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles"],"downwards":["Trigonometric ratios in right triangles","Sine, cosine and tangent rules","Solution of triangles"]}
{"upwards":["Triangle","Computing the sides and angles","Trigonometric ratios in right triangles"],"downwards":["Sine, cosine and tangent","Inverse functions"]}
{"upwards":["Triangle","Computing the sides and angles","Trigonometric ratios in right triangles","Sine, cosine and tangent"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles","Trigonometric ratios in right triangles","Inverse functions"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles","Sine, cosine and tangent rules"],"downwards":[]}
{"upwards":["Triangle","Computing the sides and angles","Solution of triangles"],"downwards":[]}
{"upwards":["Triangle","Area"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles"],"downwards":["Medians, angle bisectors, perpendicular side bisectors, and altitudes","Circumradius and inradius","Adjacent triangles","Centroid","Circumcenter, incenter, and orthocenter","Angles","Morley's trisector theorem"]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Medians, angle bisectors, perpendicular side bisectors, and altitudes"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Circumradius and inradius"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Adjacent triangles"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Centroid"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Circumcenter, incenter, and orthocenter"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Angles"],"downwards":[]}
{"upwards":["Triangle","Further formulas for general Euclidean triangles","Morley's trisector theorem"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle"],"downwards":["Conics","Convex polygon","Hexagon","Squares","Triangles"]}
{"upwards":["Triangle","Figures inscribed in a triangle","Conics"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Convex polygon"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Hexagon"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Squares"],"downwards":[]}
{"upwards":["Triangle","Figures inscribed in a triangle","Triangles"],"downwards":[]}
{"upwards":["Triangle","Figures circumscribed about a triangle"],"downwards":[]}
{"upwards":["Triangle","Specifying the location of a point in a triangle"],"downwards":[]}
{"upwards":["Triangle","Non-planar triangles"],"downwards":[]}
{"upwards":["Triangle","Triangles in construction"],"downwards":[]}
{"upwards":["Line (geometry)"],"downwards":["Properties","Definition","Other representations","Generalizations of the Euclidean line","Extensions"]}
{"upwards":["Line (geometry)","Properties"],"downwards":["In higher dimensions","Types","In axiomatic systems"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions"],"downwards":["Collinear points"]}
{"upwards":["Line (geometry)","Properties","In higher dimensions","Collinear points"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","Types"],"downwards":[]}
{"upwards":["Line (geometry)","Properties","In axiomatic systems"],"downwards":[]}
{"upwards":["Line (geometry)","Definition"],"downwards":["Linear equation","Parametric equation","Hesse normal form"]}
{"upwards":["Line (geometry)","Definition","Linear equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Parametric equation"],"downwards":[]}
{"upwards":["Line (geometry)","Definition","Hesse normal form"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations"],"downwards":["Vectors","Polar coordinates"]}
{"upwards":["Line (geometry)","Other representations","Vectors"],"downwards":[]}
{"upwards":["Line (geometry)","Other representations","Polar coordinates"],"downwards":[]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line"],"downwards":["Projective geometry"]}
{"upwards":["Line (geometry)","Generalizations of the Euclidean line","Projective geometry"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions"],"downwards":["Ray","Line segment","Number line"]}
{"upwards":["Line (geometry)","Extensions","Ray"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Line segment"],"downwards":[]}
{"upwards":["Line (geometry)","Extensions","Number line"],"downwards":[]}
{"upwards":["Parabola"],"downwards":["History","Definition as a locus of points","In a Cartesian coordinate system","As a graph of a function","Similarity to the unit parabola","As a special conic section","In polar coordinates","Conic section and quadratic form","Proof of the reflective property","Pin and string construction","Properties related to Pascal's theorem","Steiner generation","Inscribed angles and the 3-point form","Pole\u2013polar relation","Tangent properties","Facts related to chords and arcs","A geometrical construction to find a sector area","Focal length and radius of curvature at the vertex","As the affine image of the unit parabola","As quadratic B\u00e9zier curve","Numerical integration","As plane section of quadric","As trisectrix","Generalizations","In the physical world"]}
{"upwards":["Parabola","History"],"downwards":[]}
{"upwards":["Parabola","Definition as a locus of points"],"downwards":[]}
{"upwards":["Parabola","In a Cartesian coordinate system"],"downwards":["Axis of symmetry parallel to the y axis","General position"]}
{"upwards":["Parabola","In a Cartesian coordinate system","Axis of symmetry parallel to the y axis"],"downwards":[]}
{"upwards":["Parabola","In a Cartesian coordinate system","General position"],"downwards":[]}
{"upwards":["Parabola","As a graph of a function"],"downwards":[]}
{"upwards":["Parabola","Similarity to the unit parabola"],"downwards":[]}
{"upwards":["Parabola","As a special conic section"],"downwards":[]}
{"upwards":["Parabola","In polar coordinates"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form"],"downwards":["Diagram, description, and definitions","Derivation of quadratic equation","Focal length","Position of the focus","Alternative proof with Dandelin spheres"]}
{"upwards":["Parabola","Conic section and quadratic form","Diagram, description, and definitions"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Derivation of quadratic equation"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Focal length"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Position of the focus"],"downwards":[]}
{"upwards":["Parabola","Conic section and quadratic form","Alternative proof with Dandelin spheres"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property"],"downwards":["Construction and definitions","Deductions","Other consequences","Alternative proofs"]}
{"upwards":["Parabola","Proof of the reflective property","Construction and definitions"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Deductions"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences"],"downwards":["Tangent bisection property","Intersection of a tangent and perpendicular from focus","Reflection of light striking the convex side"]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences","Tangent bisection property"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences","Intersection of a tangent and perpendicular from focus"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Other consequences","Reflection of light striking the convex side"],"downwards":[]}
{"upwards":["Parabola","Proof of the reflective property","Alternative proofs"],"downwards":[]}
{"upwards":["Parabola","Pin and string construction"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem"],"downwards":["4-points property","3-points\u20131-tangent property","2-points\u20132-tangents property","Axis direction"]}
{"upwards":["Parabola","Properties related to Pascal's theorem","4-points property"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem","3-points\u20131-tangent property"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem","2-points\u20132-tangents property"],"downwards":[]}
{"upwards":["Parabola","Properties related to Pascal's theorem","Axis direction"],"downwards":[]}
{"upwards":["Parabola","Steiner generation"],"downwards":["Parabola","Dual parabola"]}
{"upwards":["Parabola","Steiner generation","Parabola"],"downwards":[]}
{"upwards":["Parabola","Steiner generation","Dual parabola"],"downwards":[]}
{"upwards":["Parabola","Inscribed angles and the 3-point form"],"downwards":[]}
{"upwards":["Parabola","Pole\u2013polar relation"],"downwards":[]}
{"upwards":["Parabola","Tangent properties"],"downwards":["Two tangent properties related to the latus rectum","Orthoptic property","Lambert's theorem"]}
{"upwards":["Parabola","Tangent properties","Two tangent properties related to the latus rectum"],"downwards":[]}
{"upwards":["Parabola","Tangent properties","Orthoptic property"],"downwards":[]}
{"upwards":["Parabola","Tangent properties","Lambert's theorem"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs"],"downwards":["Focal length calculated from parameters of a chord","Area enclosed between a parabola and a chord","Corollary concerning midpoints and endpoints of chords","Arc length"]}
{"upwards":["Parabola","Facts related to chords and arcs","Focal length calculated from parameters of a chord"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs","Area enclosed between a parabola and a chord"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs","Corollary concerning midpoints and endpoints of chords"],"downwards":[]}
{"upwards":["Parabola","Facts related to chords and arcs","Arc length"],"downwards":[]}
{"upwards":["Parabola","A geometrical construction to find a sector area"],"downwards":[]}
{"upwards":["Parabola","Focal length and radius of curvature at the vertex"],"downwards":[]}
{"upwards":["Parabola","As the affine image of the unit parabola"],"downwards":[]}
{"upwards":["Parabola","As quadratic B\u00e9zier curve"],"downwards":[]}
{"upwards":["Parabola","Numerical integration"],"downwards":[]}
{"upwards":["Parabola","As plane section of quadric"],"downwards":[]}
{"upwards":["Parabola","As trisectrix"],"downwards":[]}
{"upwards":["Parabola","Generalizations"],"downwards":[]}
{"upwards":["Parabola","In the physical world"],"downwards":["Gallery"]}
{"upwards":["Parabola","In the physical world","Gallery"],"downwards":[]}
{"upwards":["Polygon"],"downwards":["Etymology","Classification","Properties and formulas","Generalizations","Naming","History","In nature","Computer graphics"]}
{"upwards":["Polygon","Etymology"],"downwards":[]}
{"upwards":["Polygon","Classification"],"downwards":["Number of sides","Convexity and intersection","Equality and symmetry","Miscellaneous"]}
{"upwards":["Polygon","Classification","Number of sides"],"downwards":[]}
{"upwards":["Polygon","Classification","Convexity and intersection"],"downwards":[]}
{"upwards":["Polygon","Classification","Equality and symmetry"],"downwards":[]}
{"upwards":["Polygon","Classification","Miscellaneous"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas"],"downwards":["Angles","Area","Centroid"]}
{"upwards":["Polygon","Properties and formulas","Angles"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Area"],"downwards":["Simple polygons","Regular polygons","Self-intersecting"]}
{"upwards":["Polygon","Properties and formulas","Area","Simple polygons"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Area","Regular polygons"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Area","Self-intersecting"],"downwards":[]}
{"upwards":["Polygon","Properties and formulas","Centroid"],"downwards":[]}
{"upwards":["Polygon","Generalizations"],"downwards":[]}
{"upwards":["Polygon","Naming"],"downwards":[]}
{"upwards":["Polygon","History"],"downwards":[]}
{"upwards":["Polygon","In nature"],"downwards":[]}
{"upwards":["Polygon","Computer graphics"],"downwards":[]}
{"upwards":["Hexagon"],"downwards":["Regular hexagon","Parameters","Point in plane","Symmetry","Dissection","Related polygons and tilings","Hexagonal structures","Tesselations by hexagons","Hexagon inscribed in a conic section","Hexagon tangential to a conic section","Equilateral triangles on the sides of an arbitrary hexagon","Skew hexagon","Convex equilateral hexagon","Gallery of natural and artificial hexagons"]}
{"upwards":["Hexagon","Regular hexagon"],"downwards":[]}
{"upwards":["Hexagon","Parameters"],"downwards":[]}
{"upwards":["Hexagon","Point in plane"],"downwards":[]}
{"upwards":["Hexagon","Symmetry"],"downwards":["A2 and G2 groups"]}
{"upwards":["Hexagon","Symmetry","A2 and G2 groups"],"downwards":[]}
{"upwards":["Hexagon","Dissection"],"downwards":[]}
{"upwards":["Hexagon","Related polygons and tilings"],"downwards":["Self-crossing hexagons"]}
{"upwards":["Hexagon","Related polygons and tilings","Self-crossing hexagons"],"downwards":[]}
{"upwards":["Hexagon","Hexagonal structures"],"downwards":[]}
{"upwards":["Hexagon","Tesselations by hexagons"],"downwards":[]}
{"upwards":["Hexagon","Hexagon inscribed in a conic section"],"downwards":["Cyclic hexagon"]}
{"upwards":["Hexagon","Hexagon inscribed in a conic section","Cyclic hexagon"],"downwards":[]}
{"upwards":["Hexagon","Hexagon tangential to a conic section"],"downwards":[]}
{"upwards":["Hexagon","Equilateral triangles on the sides of an arbitrary hexagon"],"downwards":[]}
{"upwards":["Hexagon","Skew hexagon"],"downwards":["Petrie polygons"]}
{"upwards":["Hexagon","Skew hexagon","Petrie polygons"],"downwards":[]}
{"upwards":["Hexagon","Convex equilateral hexagon"],"downwards":["Polyhedra with hexagons"]}
{"upwards":["Hexagon","Convex equilateral hexagon","Polyhedra with hexagons"],"downwards":[]}
{"upwards":["Hexagon","Gallery of natural and artificial hexagons"],"downwards":[]}
{"upwards":["Pentagon"],"downwards":["Regular pentagons","Equilateral pentagons","Cyclic pentagons","General convex pentagons","Pentagons in tiling","Pentagons in polyhedra","Pentagons in nature","Other examples"]}
{"upwards":["Pentagon","Regular pentagons"],"downwards":["Derivation of the area formula","Inradius","Chords from the circumscribed circle to the vertices","Point in plane","Geometrical constructions","Physical construction methods","Symmetry","Regular pentagram"]}
{"upwards":["Pentagon","Regular pentagons","Derivation of the area formula"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Inradius"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Chords from the circumscribed circle to the vertices"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Point in plane"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions"],"downwards":["Richmond's method","Carlyle circles","Euclid's method"]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions","Richmond's method"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions","Carlyle circles"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Geometrical constructions","Euclid's method"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Physical construction methods"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Symmetry"],"downwards":[]}
{"upwards":["Pentagon","Regular pentagons","Regular pentagram"],"downwards":[]}
{"upwards":["Pentagon","Equilateral pentagons"],"downwards":[]}
{"upwards":["Pentagon","Cyclic pentagons"],"downwards":[]}
{"upwards":["Pentagon","General convex pentagons"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in tiling"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in polyhedra"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in nature"],"downwards":["Plants","Animals","Minerals"]}
{"upwards":["Pentagon","Pentagons in nature","Plants"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in nature","Animals"],"downwards":[]}
{"upwards":["Pentagon","Pentagons in nature","Minerals"],"downwards":[]}
{"upwards":["Pentagon","Other examples"],"downwards":[]}
{"upwards":["Ellipse"],"downwards":["Definition as locus of points","In Cartesian coordinates","Parametric representation","Polar forms","Eccentricity and the directrix property","Focus-to-focus reflection property","Conjugate diameters","Orthogonal tangents","Drawing ellipses","Inscribed angles and three-point form","Pole-polar relation","Metric properties","In triangle geometry","As plane sections of quadrics","Applications"]}
{"upwards":["Ellipse","Definition as locus of points"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates"],"downwards":["Standard equation","Parameters","Tangent","Shifted ellipse","General ellipse"]}
{"upwards":["Ellipse","In Cartesian coordinates","Standard equation"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters"],"downwards":["Principal axes","Linear eccentricity","Eccentricity","Semi-latus rectum"]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Principal axes"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Linear eccentricity"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Eccentricity"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Parameters","Semi-latus rectum"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Tangent"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","Shifted ellipse"],"downwards":[]}
{"upwards":["Ellipse","In Cartesian coordinates","General ellipse"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation"],"downwards":["Standard parametric representation","Rational representation","Tangent slope as parameter","General ellipse"]}
{"upwards":["Ellipse","Parametric representation","Standard parametric representation"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation","Rational representation"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation","Tangent slope as parameter"],"downwards":[]}
{"upwards":["Ellipse","Parametric representation","General ellipse"],"downwards":[]}
{"upwards":["Ellipse","Polar forms"],"downwards":["Polar form relative to center","Polar form relative to focus"]}
{"upwards":["Ellipse","Polar forms","Polar form relative to center"],"downwards":[]}
{"upwards":["Ellipse","Polar forms","Polar form relative to focus"],"downwards":[]}
{"upwards":["Ellipse","Eccentricity and the directrix property"],"downwards":[]}
{"upwards":["Ellipse","Focus-to-focus reflection property"],"downwards":[]}
{"upwards":["Ellipse","Conjugate diameters"],"downwards":["Definition of conjugate diameters","Theorem of Apollonios on conjugate diameters"]}
{"upwards":["Ellipse","Conjugate diameters","Definition of conjugate diameters"],"downwards":[]}
{"upwards":["Ellipse","Conjugate diameters","Theorem of Apollonios on conjugate diameters"],"downwards":[]}
{"upwards":["Ellipse","Orthogonal tangents"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses"],"downwards":["de La Hire's point construction","Pins-and-string method","Paper strip methods","Approximation by osculating circles","Steiner generation","As hypotrochoid"]}
{"upwards":["Ellipse","Drawing ellipses","de La Hire's point construction"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Pins-and-string method"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Paper strip methods"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Approximation by osculating circles"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","Steiner generation"],"downwards":[]}
{"upwards":["Ellipse","Drawing ellipses","As hypotrochoid"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form"],"downwards":["Circles","Ellipses"]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Circles"],"downwards":["Inscribed angle theorem for circles","Three-point form of circle equation"]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Circles","Inscribed angle theorem for circles"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Circles","Three-point form of circle equation"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Ellipses"],"downwards":["Inscribed angle theorem for ellipses","Three-point form of ellipse equation"]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Ellipses","Inscribed angle theorem for ellipses"],"downwards":[]}
{"upwards":["Ellipse","Inscribed angles and three-point form","Ellipses","Three-point form of ellipse equation"],"downwards":[]}
{"upwards":["Ellipse","Pole-polar relation"],"downwards":[]}
{"upwards":["Ellipse","Metric properties"],"downwards":["Area","Circumference","Arc length","Curvature"]}
{"upwards":["Ellipse","Metric properties","Area"],"downwards":[]}
{"upwards":["Ellipse","Metric properties","Circumference"],"downwards":[]}
{"upwards":["Ellipse","Metric properties","Arc length"],"downwards":[]}
{"upwards":["Ellipse","Metric properties","Curvature"],"downwards":[]}
{"upwards":["Ellipse","In triangle geometry"],"downwards":[]}
{"upwards":["Ellipse","As plane sections of quadrics"],"downwards":[]}
{"upwards":["Ellipse","Applications"],"downwards":["Physics","Statistics and finance","Computer graphics","Optimization theory"]}
{"upwards":["Ellipse","Applications","Physics"],"downwards":["Elliptical reflectors and acoustics","Planetary orbits","Harmonic oscillators","Phase visualization","Elliptical gears","Optics"]}
{"upwards":["Ellipse","Applications","Physics","Elliptical reflectors and acoustics"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Planetary orbits"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Harmonic oscillators"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Phase visualization"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Elliptical gears"],"downwards":[]}
{"upwards":["Ellipse","Applications","Physics","Optics"],"downwards":[]}
{"upwards":["Ellipse","Applications","Statistics and finance"],"downwards":[]}
{"upwards":["Ellipse","Applications","Computer graphics"],"downwards":[]}
{"upwards":["Ellipse","Applications","Optimization theory"],"downwards":[]}
{"upwards":["Dodecahedron"],"downwards":["Regular dodecahedron","Other pentagonal dodecahedra","Rhombic dodecahedron","Other dodecahedra","Practical usage"]}
{"upwards":["Dodecahedron","Regular dodecahedron"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra"],"downwards":["Pyritohedron","Tetartoid","Dual of triangular gyrobianticupola"]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron"],"downwards":["Crystal pyrite","Cartesian coordinates","Geometric freedom"]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron","Crystal pyrite"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron","Cartesian coordinates"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Pyritohedron","Geometric freedom"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Tetartoid"],"downwards":["Cartesian coordinates","Geometric freedom"]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Tetartoid","Cartesian coordinates"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Tetartoid","Geometric freedom"],"downwards":[]}
{"upwards":["Dodecahedron","Other pentagonal dodecahedra","Dual of triangular gyrobianticupola"],"downwards":[]}
{"upwards":["Dodecahedron","Rhombic dodecahedron"],"downwards":[]}
{"upwards":["Dodecahedron","Other dodecahedra"],"downwards":[]}
{"upwards":["Dodecahedron","Practical usage"],"downwards":[]}
{"upwards":["Cube"],"downwards":["Orthogonal projections","Spherical tiling","Cartesian coordinates","Equation in three dimensional space","Formulas","Doubling the cube","Uniform colorings and symmetry","Geometric relations","In theology","Other dimensions","Related polyhedra","Cubical graph"]}
{"upwards":["Cube","Orthogonal projections"],"downwards":[]}
{"upwards":["Cube","Spherical tiling"],"downwards":[]}
{"upwards":["Cube","Cartesian coordinates"],"downwards":[]}
{"upwards":["Cube","Equation in three dimensional space"],"downwards":[]}
{"upwards":["Cube","Formulas"],"downwards":["Point in space"]}
{"upwards":["Cube","Formulas","Point in space"],"downwards":[]}
{"upwards":["Cube","Doubling the cube"],"downwards":[]}
{"upwards":["Cube","Uniform colorings and symmetry"],"downwards":[]}
{"upwards":["Cube","Geometric relations"],"downwards":[]}
{"upwards":["Cube","In theology"],"downwards":[]}
{"upwards":["Cube","Other dimensions"],"downwards":[]}
{"upwards":["Cube","Related polyhedra"],"downwards":["In uniform honeycombs and polychora"]}
{"upwards":["Cube","Related polyhedra","In uniform honeycombs and polychora"],"downwards":[]}
{"upwards":["Cube","Cubical graph"],"downwards":[]}
{"upwards":["Octahedron"],"downwards":["Regular octahedron","Irregular octahedra","Octahedra in the physical world","Related polyhedra"]}
{"upwards":["Octahedron","Regular octahedron"],"downwards":["Dimensions","Orthogonal projections","Spherical tiling","Cartesian coordinates","Area and volume","Geometric relations","Characteristic orthoscheme","Topology","Nets","Faceting","Uniform colorings and symmetry"]}
{"upwards":["Octahedron","Regular octahedron","Dimensions"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Orthogonal projections"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Spherical tiling"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Cartesian coordinates"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Area and volume"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations"],"downwards":["Dual","Stellation","Snub octahedron","Tessellations"]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Dual"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Stellation"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Snub octahedron"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Geometric relations","Tessellations"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Characteristic orthoscheme"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Topology"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Nets"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Faceting"],"downwards":[]}
{"upwards":["Octahedron","Regular octahedron","Uniform colorings and symmetry"],"downwards":[]}
{"upwards":["Octahedron","Irregular octahedra"],"downwards":[]}
{"upwards":["Octahedron","Octahedra in the physical world"],"downwards":["Octahedra in nature","Octahedra in art and culture","Tetrahedral octet truss"]}
{"upwards":["Octahedron","Octahedra in the physical world","Octahedra in nature"],"downwards":[]}
{"upwards":["Octahedron","Octahedra in the physical world","Octahedra in art and culture"],"downwards":[]}
{"upwards":["Octahedron","Octahedra in the physical world","Tetrahedral octet truss"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra"],"downwards":["Tetratetrahedron","Trigonal antiprism","Square bipyramid","Other related polyhedra"]}
{"upwards":["Octahedron","Related polyhedra","Tetratetrahedron"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra","Trigonal antiprism"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra","Square bipyramid"],"downwards":[]}
{"upwards":["Octahedron","Related polyhedra","Other related polyhedra"],"downwards":[]}
{"upwards":["Icosahedron"],"downwards":["Regular icosahedra","Stellated icosahedra","Pyritohedral symmetry","Other icosahedra"]}
{"upwards":["Icosahedron","Regular icosahedra"],"downwards":["Convex regular icosahedron","Great icosahedron"]}
{"upwards":["Icosahedron","Regular icosahedra","Convex regular icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Regular icosahedra","Great icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Stellated icosahedra"],"downwards":[]}
{"upwards":["Icosahedron","Pyritohedral symmetry"],"downwards":["Cartesian coordinates","Jessen's icosahedron","Cuboctahedron"]}
{"upwards":["Icosahedron","Pyritohedral symmetry","Cartesian coordinates"],"downwards":[]}
{"upwards":["Icosahedron","Pyritohedral symmetry","Jessen's icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Pyritohedral symmetry","Cuboctahedron"],"downwards":[]}
{"upwards":["Icosahedron","Other icosahedra"],"downwards":["Rhombic icosahedron","Pyramid and prism symmetries","Johnson solids"]}
{"upwards":["Icosahedron","Other icosahedra","Rhombic icosahedron"],"downwards":[]}
{"upwards":["Icosahedron","Other icosahedra","Pyramid and prism symmetries"],"downwards":[]}
{"upwards":["Icosahedron","Other icosahedra","Johnson solids"],"downwards":[]}
{"upwards":["Paraboloid"],"downwards":["Properties and applications","Cylinder between pencils of elliptic and hyperbolic paraboloids","Curvature","Geometric representation of multiplication table","Dimensions of a paraboloidal dish"]}
{"upwards":["Paraboloid","Properties and applications"],"downwards":["Elliptic paraboloid","Hyperbolic paraboloid"]}
{"upwards":["Paraboloid","Properties and applications","Elliptic paraboloid"],"downwards":["Parabolic reflector"]}
{"upwards":["Paraboloid","Properties and applications","Elliptic paraboloid","Parabolic reflector"],"downwards":[]}
{"upwards":["Paraboloid","Properties and applications","Hyperbolic paraboloid"],"downwards":["Examples in architecture"]}
{"upwards":["Paraboloid","Properties and applications","Hyperbolic paraboloid","Examples in architecture"],"downwards":[]}
{"upwards":["Paraboloid","Cylinder between pencils of elliptic and hyperbolic paraboloids"],"downwards":[]}
{"upwards":["Paraboloid","Curvature"],"downwards":[]}
{"upwards":["Paraboloid","Geometric representation of multiplication table"],"downwards":[]}
{"upwards":["Paraboloid","Dimensions of a paraboloidal dish"],"downwards":[]}
{"upwards":["Sphere"],"downwards":["Basic terminology","Equations","Properties","Treatment by area of mathematics","Curves on a sphere","Generalizations","History","Gallery","Regions"]}
{"upwards":["Sphere","Basic terminology"],"downwards":[]}
{"upwards":["Sphere","Equations"],"downwards":["Parametric"]}
{"upwards":["Sphere","Equations","Parametric"],"downwards":[]}
{"upwards":["Sphere","Properties"],"downwards":["Enclosed volume","Surface area","Other geometric properties","Properties of the sphere"]}
{"upwards":["Sphere","Properties","Enclosed volume"],"downwards":[]}
{"upwards":["Sphere","Properties","Surface area"],"downwards":[]}
{"upwards":["Sphere","Properties","Other geometric properties"],"downwards":["Pencil of spheres"]}
{"upwards":["Sphere","Properties","Other geometric properties","Pencil of spheres"],"downwards":[]}
{"upwards":["Sphere","Properties","Properties of the sphere"],"downwards":[]}
{"upwards":["Sphere","Treatment by area of mathematics"],"downwards":["Spherical geometry","Differential geometry","Topology"]}
{"upwards":["Sphere","Treatment by area of mathematics","Spherical geometry"],"downwards":[]}
{"upwards":["Sphere","Treatment by area of mathematics","Differential geometry"],"downwards":[]}
{"upwards":["Sphere","Treatment by area of mathematics","Topology"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere"],"downwards":["Circles","Loxodrome","Clelia curves","Spherical conics","Intersection of a sphere with a more general surface"]}
{"upwards":["Sphere","Curves on a sphere","Circles"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Loxodrome"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Clelia curves"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Spherical conics"],"downwards":[]}
{"upwards":["Sphere","Curves on a sphere","Intersection of a sphere with a more general surface"],"downwards":[]}
{"upwards":["Sphere","Generalizations"],"downwards":["Ellipsoids","Dimensionality","Metric spaces"]}
{"upwards":["Sphere","Generalizations","Ellipsoids"],"downwards":[]}
{"upwards":["Sphere","Generalizations","Dimensionality"],"downwards":[]}
{"upwards":["Sphere","Generalizations","Metric spaces"],"downwards":[]}
{"upwards":["Sphere","History"],"downwards":[]}
{"upwards":["Sphere","Gallery"],"downwards":[]}
{"upwards":["Sphere","Regions"],"downwards":[]}
{"upwards":["Geometry"],"downwards":["History","Main concepts","Contemporary geometry","Applications"]}
{"upwards":["Geometry","History"],"downwards":[]}
{"upwards":["Geometry","Main concepts"],"downwards":["Axioms","Objects","Length, area, and volume","Congruence and similarity","Compass and straightedge constructions","Dimension","Symmetry"]}
{"upwards":["Geometry","Main concepts","Axioms"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects"],"downwards":["Points","Lines","Planes","Angles","Curves","Surfaces","Manifolds"]}
{"upwards":["Geometry","Main concepts","Objects","Points"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Lines"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Planes"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Angles"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Curves"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Surfaces"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Objects","Manifolds"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Length, area, and volume"],"downwards":["Metrics and measures"]}
{"upwards":["Geometry","Main concepts","Length, area, and volume","Metrics and measures"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Congruence and similarity"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Compass and straightedge constructions"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Dimension"],"downwards":[]}
{"upwards":["Geometry","Main concepts","Symmetry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry"],"downwards":["Euclidean geometry","Differential geometry","Topology","Algebraic geometry","Complex geometry","Discrete geometry","Computational geometry","Geometric group theory","Convex geometry"]}
{"upwards":["Geometry","Contemporary geometry","Euclidean geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Differential geometry"],"downwards":["Non-Euclidean geometry"]}
{"upwards":["Geometry","Contemporary geometry","Differential geometry","Non-Euclidean geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Topology"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Algebraic geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Complex geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Discrete geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Computational geometry"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Geometric group theory"],"downwards":[]}
{"upwards":["Geometry","Contemporary geometry","Convex geometry"],"downwards":[]}
{"upwards":["Geometry","Applications"],"downwards":["Art","Architecture","Physics","Other fields of mathematics"]}
{"upwards":["Geometry","Applications","Art"],"downwards":[]}
{"upwards":["Geometry","Applications","Architecture"],"downwards":[]}
{"upwards":["Geometry","Applications","Physics"],"downwards":[]}
{"upwards":["Geometry","Applications","Other fields of mathematics"],"downwards":[]}
{"upwards":["Ellipsoid"],"downwards":["Standard equation","Parameterization","Volume","Surface area","Plane sections","Pins-and-string construction","In general position","Applications","In higher dimensions"]}
{"upwards":["Ellipsoid","Standard equation"],"downwards":[]}
{"upwards":["Ellipsoid","Parameterization"],"downwards":[]}
{"upwards":["Ellipsoid","Volume"],"downwards":[]}
{"upwards":["Ellipsoid","Surface area"],"downwards":["Approximate formula"]}
{"upwards":["Ellipsoid","Surface area","Approximate formula"],"downwards":[]}
{"upwards":["Ellipsoid","Plane sections"],"downwards":["Determining the ellipse of a plane section"]}
{"upwards":["Ellipsoid","Plane sections","Determining the ellipse of a plane section"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction"],"downwards":["Steps of the construction","Semi-axes","Converse","Confocal ellipsoids","Limit case, ellipsoid of revolution","Properties of the focal hyperbola","Property of the focal ellipse"]}
{"upwards":["Ellipsoid","Pins-and-string construction","Steps of the construction"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Semi-axes"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Converse"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Confocal ellipsoids"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Limit case, ellipsoid of revolution"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Properties of the focal hyperbola"],"downwards":[]}
{"upwards":["Ellipsoid","Pins-and-string construction","Property of the focal ellipse"],"downwards":[]}
{"upwards":["Ellipsoid","In general position"],"downwards":["As a quadric","Parametric representation"]}
{"upwards":["Ellipsoid","In general position","As a quadric"],"downwards":[]}
{"upwards":["Ellipsoid","In general position","Parametric representation"],"downwards":[]}
{"upwards":["Ellipsoid","Applications"],"downwards":["Dynamical properties","Fluid dynamics","In probability and statistics"]}
{"upwards":["Ellipsoid","Applications","Dynamical properties"],"downwards":[]}
{"upwards":["Ellipsoid","Applications","Fluid dynamics"],"downwards":[]}
{"upwards":["Ellipsoid","Applications","In probability and statistics"],"downwards":[]}
{"upwards":["Ellipsoid","In higher dimensions"],"downwards":[]}
{"upwards":["Lattice (order)"],"downwards":["Definition","Bounded lattice","Connection to other algebraic structures","Examples","Examples of non-lattices","Morphisms of lattices","Sublattices","Properties of lattices","Free lattices","Important lattice-theoretic notions"]}
{"upwards":["Lattice (order)","Definition"],"downwards":["As partially ordered set","As algebraic structure","Connection between the two definitions"]}
{"upwards":["Lattice (order)","Definition","As partially ordered set"],"downwards":[]}
{"upwards":["Lattice (order)","Definition","As algebraic structure"],"downwards":[]}
{"upwards":["Lattice (order)","Definition","Connection between the two definitions"],"downwards":[]}
{"upwards":["Lattice (order)","Bounded lattice"],"downwards":[]}
{"upwards":["Lattice (order)","Connection to other algebraic structures"],"downwards":[]}
{"upwards":["Lattice (order)","Examples"],"downwards":[]}
{"upwards":["Lattice (order)","Examples of non-lattices"],"downwards":[]}
{"upwards":["Lattice (order)","Morphisms of lattices"],"downwards":[]}
{"upwards":["Lattice (order)","Sublattices"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices"],"downwards":["Completeness","Conditional completeness","Distributivity","Modularity","Semimodularity","Continuity and algebraicity","Complements and pseudo-complements","Jordan\u2013Dedekind chain condition","Graded\/ranked"]}
{"upwards":["Lattice (order)","Properties of lattices","Completeness"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Conditional completeness"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Distributivity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Modularity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Semimodularity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Continuity and algebraicity"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Complements and pseudo-complements"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Jordan\u2013Dedekind chain condition"],"downwards":[]}
{"upwards":["Lattice (order)","Properties of lattices","Graded\/ranked"],"downwards":[]}
{"upwards":["Lattice (order)","Free lattices"],"downwards":[]}
{"upwards":["Lattice (order)","Important lattice-theoretic notions"],"downwards":[]}
{"upwards":["Lattice (group)"],"downwards":["Symmetry considerations and examples","Dividing space according to a lattice","Lattice points in convex sets","Computational lattice problems","Lattices in two dimensions: detailed discussion","Lattices in three dimensions","Lattices in complex space","In Lie groups","Lattices in general vector spaces","Related notions"]}
{"upwards":["Lattice (group)","Symmetry considerations and examples"],"downwards":[]}
{"upwards":["Lattice (group)","Dividing space according to a lattice"],"downwards":[]}
{"upwards":["Lattice (group)","Lattice points in convex sets"],"downwards":[]}
{"upwards":["Lattice (group)","Computational lattice problems"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in two dimensions: detailed discussion"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in three dimensions"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in complex space"],"downwards":[]}
{"upwards":["Lattice (group)","In Lie groups"],"downwards":[]}
{"upwards":["Lattice (group)","Lattices in general vector spaces"],"downwards":[]}
{"upwards":["Lattice (group)","Related notions"],"downwards":[]}
{"upwards":["Mathematical structure"],"downwards":["History","Example: the real numbers"]}
{"upwards":["Mathematical structure","History"],"downwards":[]}
{"upwards":["Mathematical structure","Example: the real numbers"],"downwards":[]}
{"upwards":["Oxford University Press"],"downwards":["History","Museum","Scholarly journals","Series and titles","Clarendon Scholarships","Controversies"]}
{"upwards":["Oxford University Press","History"],"downwards":[]}
{"upwards":["Oxford University Press","Museum"],"downwards":["Clarendon Press"]}
{"upwards":["Oxford University Press","Museum","Clarendon Press"],"downwards":[]}
{"upwards":["Oxford University Press","Scholarly journals"],"downwards":[]}
{"upwards":["Oxford University Press","Series and titles"],"downwards":[]}
{"upwards":["Oxford University Press","Clarendon Scholarships"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies"],"downwards":["Tehran Book Fair Controversy","Malcolm vs Oxford University 1986-1992","Closure of Poetry List","Tax Exemption Controversy","East African Bribery Scandal"]}
{"upwards":["Oxford University Press","Controversies","Tehran Book Fair Controversy"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","Malcolm vs Oxford University 1986-1992"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","Closure of Poetry List"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","Tax Exemption Controversy"],"downwards":[]}
{"upwards":["Oxford University Press","Controversies","East African Bribery Scandal"],"downwards":[]}
{"upwards":["Vector space"],"downwards":["Definition and basic properties","Related concepts and properties","History","Examples","Linear maps and matrices","Basic constructions","Vector spaces with additional structure","Related structures","Related concepts"]}
{"upwards":["Vector space","Definition and basic properties"],"downwards":[]}
{"upwards":["Vector space","Related concepts and properties"],"downwards":[]}
{"upwards":["Vector space","History"],"downwards":[]}
{"upwards":["Vector space","Examples"],"downwards":["Arrows in the plane","Second example: ordered pairs of numbers","Coordinate space","Complex numbers and other field extensions","Function spaces","Linear equations"]}
{"upwards":["Vector space","Examples","Arrows in the plane"],"downwards":[]}
{"upwards":["Vector space","Examples","Second example: ordered pairs of numbers"],"downwards":[]}
{"upwards":["Vector space","Examples","Coordinate space"],"downwards":[]}
{"upwards":["Vector space","Examples","Complex numbers and other field extensions"],"downwards":[]}
{"upwards":["Vector space","Examples","Function spaces"],"downwards":[]}
{"upwards":["Vector space","Examples","Linear equations"],"downwards":[]}
{"upwards":["Vector space","Linear maps and matrices"],"downwards":["Matrices","Eigenvalues and eigenvectors"]}
{"upwards":["Vector space","Linear maps and matrices","Matrices"],"downwards":[]}
{"upwards":["Vector space","Linear maps and matrices","Eigenvalues and eigenvectors"],"downwards":[]}
{"upwards":["Vector space","Basic constructions"],"downwards":["Subspaces and quotient spaces","Direct product and direct sum","Tensor product"]}
{"upwards":["Vector space","Basic constructions","Subspaces and quotient spaces"],"downwards":[]}
{"upwards":["Vector space","Basic constructions","Direct product and direct sum"],"downwards":[]}
{"upwards":["Vector space","Basic constructions","Tensor product"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure"],"downwards":["Normed vector spaces and inner product spaces","Topological vector spaces","Algebras over fields"]}
{"upwards":["Vector space","Vector spaces with additional structure","Normed vector spaces and inner product spaces"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure","Topological vector spaces"],"downwards":["Banach spaces","Hilbert spaces"]}
{"upwards":["Vector space","Vector spaces with additional structure","Topological vector spaces","Banach spaces"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure","Topological vector spaces","Hilbert spaces"],"downwards":[]}
{"upwards":["Vector space","Vector spaces with additional structure","Algebras over fields"],"downwards":[]}
{"upwards":["Vector space","Related structures"],"downwards":["Vector bundles","Modules","Affine and projective spaces"]}
{"upwards":["Vector space","Related structures","Vector bundles"],"downwards":[]}
{"upwards":["Vector space","Related structures","Modules"],"downwards":[]}
{"upwards":["Vector space","Related structures","Affine and projective spaces"],"downwards":[]}
{"upwards":["Vector space","Related concepts"],"downwards":[]}
{"upwards":["Module (mathematics)"],"downwards":["Introduction and definition","Examples","Submodules and homomorphisms","Types of modules","Further notions"]}
{"upwards":["Module (mathematics)","Introduction and definition"],"downwards":["Motivation","Formal definition"]}
{"upwards":["Module (mathematics)","Introduction and definition","Motivation"],"downwards":[]}
{"upwards":["Module (mathematics)","Introduction and definition","Formal definition"],"downwards":[]}
{"upwards":["Module (mathematics)","Examples"],"downwards":[]}
{"upwards":["Module (mathematics)","Submodules and homomorphisms"],"downwards":[]}
{"upwards":["Module (mathematics)","Types of modules"],"downwards":[]}
{"upwards":["Module (mathematics)","Further notions"],"downwards":["Relation to representation theory","Generalizations"]}
{"upwards":["Module (mathematics)","Further notions","Relation to representation theory"],"downwards":[]}
{"upwards":["Module (mathematics)","Further notions","Generalizations"],"downwards":[]}
{"upwards":["Tensor"],"downwards":["Definition","Examples","Properties","Notation","Operations","Applications","Generalizations","History"]}
{"upwards":["Tensor","Definition"],"downwards":["As multidimensional arrays","As multilinear maps","Using tensor products","Tensors in infinite dimensions","Tensor fields"]}
{"upwards":["Tensor","Definition","As multidimensional arrays"],"downwards":[]}
{"upwards":["Tensor","Definition","As multilinear maps"],"downwards":[]}
{"upwards":["Tensor","Definition","Using tensor products"],"downwards":[]}
{"upwards":["Tensor","Definition","Tensors in infinite dimensions"],"downwards":[]}
{"upwards":["Tensor","Definition","Tensor fields"],"downwards":[]}
{"upwards":["Tensor","Examples"],"downwards":[]}
{"upwards":["Tensor","Properties"],"downwards":[]}
{"upwards":["Tensor","Notation"],"downwards":["Ricci calculus","Einstein summation convention","Penrose graphical notation","Abstract index notation","Component-free notation"]}
{"upwards":["Tensor","Notation","Ricci calculus"],"downwards":[]}
{"upwards":["Tensor","Notation","Einstein summation convention"],"downwards":[]}
{"upwards":["Tensor","Notation","Penrose graphical notation"],"downwards":[]}
{"upwards":["Tensor","Notation","Abstract index notation"],"downwards":[]}
{"upwards":["Tensor","Notation","Component-free notation"],"downwards":[]}
{"upwards":["Tensor","Operations"],"downwards":["Tensor product","Contraction","Raising or lowering an index"]}
{"upwards":["Tensor","Operations","Tensor product"],"downwards":[]}
{"upwards":["Tensor","Operations","Contraction"],"downwards":[]}
{"upwards":["Tensor","Operations","Raising or lowering an index"],"downwards":[]}
{"upwards":["Tensor","Applications"],"downwards":["Continuum mechanics","Other examples from physics","Computer vision and optics","Machine learning"]}
{"upwards":["Tensor","Applications","Continuum mechanics"],"downwards":[]}
{"upwards":["Tensor","Applications","Other examples from physics"],"downwards":[]}
{"upwards":["Tensor","Applications","Computer vision and optics"],"downwards":[]}
{"upwards":["Tensor","Applications","Machine learning"],"downwards":[]}
{"upwards":["Tensor","Generalizations"],"downwards":["Tensor products of vector spaces","Tensors in infinite dimensions","Tensor densities","Geometric objects","Spinors"]}
{"upwards":["Tensor","Generalizations","Tensor products of vector spaces"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Tensors in infinite dimensions"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Tensor densities"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Geometric objects"],"downwards":[]}
{"upwards":["Tensor","Generalizations","Spinors"],"downwards":[]}
{"upwards":["Tensor","History"],"downwards":[]}
{"upwards":["Philip J. Davis"],"downwards":["Publications (Books Only)","Notes","References","External links"]}
{"upwards":["Philip J. Davis","Publications (Books Only)"],"downwards":[]}
{"upwards":["Philip J. Davis","Notes"],"downwards":[]}
{"upwards":["Philip J. Davis","References"],"downwards":[]}
{"upwards":["Philip J. Davis","External links"],"downwards":[]}
{"upwards":["Matrix (mathematics)"],"downwards":["Definition","Notation","Basic operations","Linear equations","Linear transformations","Square matrix","Computational aspects","Decomposition","Abstract algebraic aspects and generalizations","Applications","History"]}
{"upwards":["Matrix (mathematics)","Definition"],"downwards":["Size"]}
{"upwards":["Matrix (mathematics)","Definition","Size"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Notation"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations"],"downwards":["Addition, scalar multiplication, and transposition","Matrix multiplication","Row operations","Submatrix"]}
{"upwards":["Matrix (mathematics)","Basic operations","Addition, scalar multiplication, and transposition"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations","Matrix multiplication"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations","Row operations"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Basic operations","Submatrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Linear equations"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Linear transformations"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix"],"downwards":["Main types","Main operations"]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types"],"downwards":["Diagonal and triangular matrix","Identity matrix","Symmetric or skew-symmetric matrix","Invertible matrix and its inverse","Definite matrix","Orthogonal matrix"]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Diagonal and triangular matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Identity matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Symmetric or skew-symmetric matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Invertible matrix and its inverse"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Definite matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main types","Orthogonal matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations"],"downwards":["Trace","Determinant","Eigenvalues and eigenvectors"]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations","Trace"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations","Determinant"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Square matrix","Main operations","Eigenvalues and eigenvectors"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Computational aspects"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Decomposition"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations"],"downwards":["Matrices with more general entries","Relationship to linear maps","Matrix groups","Infinite matrices","Empty matrix"]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Matrices with more general entries"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Relationship to linear maps"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Matrix groups"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Infinite matrices"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Abstract algebraic aspects and generalizations","Empty matrix"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications"],"downwards":["Graph theory","Analysis and geometry","Probability theory and statistics","Symmetries and transformations in physics","Linear combinations of quantum states","Normal modes","Geometrical optics","Electronics"]}
{"upwards":["Matrix (mathematics)","Applications","Graph theory"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Analysis and geometry"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Probability theory and statistics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Symmetries and transformations in physics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Linear combinations of quantum states"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Normal modes"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Geometrical optics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","Applications","Electronics"],"downwards":[]}
{"upwards":["Matrix (mathematics)","History"],"downwards":["Other historical usages of the word \"matrix\" in mathematics"]}
{"upwards":["Matrix (mathematics)","History","Other historical usages of the word \"matrix\" in mathematics"],"downwards":[]}
{"upwards":["Stanford Encyclopedia of Philosophy"],"downwards":["Approach and history"]}
{"upwards":["Stanford Encyclopedia of Philosophy","Approach and history"],"downwards":[]}
{"upwards":["Analytic geometry"],"downwards":["History","Coordinates","Equations and curves","Distance and angle","Transformations","Finding intersections of geometric objects","Geometric axis","Spherical and nonlinear planes and their tangents"]}
{"upwards":["Analytic geometry","History"],"downwards":["Ancient Greece","Persia","Western Europe"]}
{"upwards":["Analytic geometry","History","Ancient Greece"],"downwards":[]}
{"upwards":["Analytic geometry","History","Persia"],"downwards":[]}
{"upwards":["Analytic geometry","History","Western Europe"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates"],"downwards":["Cartesian coordinates (in a plane or space)","Polar coordinates (in a plane)","Cylindrical coordinates (in a space)","Spherical coordinates (in a space)"]}
{"upwards":["Analytic geometry","Coordinates","Cartesian coordinates (in a plane or space)"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates","Polar coordinates (in a plane)"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates","Cylindrical coordinates (in a space)"],"downwards":[]}
{"upwards":["Analytic geometry","Coordinates","Spherical coordinates (in a space)"],"downwards":[]}
{"upwards":["Analytic geometry","Equations and curves"],"downwards":["Lines and planes","Conic sections","Quadric surfaces"]}
{"upwards":["Analytic geometry","Equations and curves","Lines and planes"],"downwards":[]}
{"upwards":["Analytic geometry","Equations and curves","Conic sections"],"downwards":[]}
{"upwards":["Analytic geometry","Equations and curves","Quadric surfaces"],"downwards":[]}
{"upwards":["Analytic geometry","Distance and angle"],"downwards":[]}
{"upwards":["Analytic geometry","Transformations"],"downwards":[]}
{"upwards":["Analytic geometry","Finding intersections of geometric objects"],"downwards":["Finding intercepts"]}
{"upwards":["Analytic geometry","Finding intersections of geometric objects","Finding intercepts"],"downwards":[]}
{"upwards":["Analytic geometry","Geometric axis"],"downwards":[]}
{"upwards":["Analytic geometry","Spherical and nonlinear planes and their tangents"],"downwards":["Tangent lines and planes"]}
{"upwards":["Analytic geometry","Spherical and nonlinear planes and their tangents","Tangent lines and planes"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)"],"downwards":["Vectors in Euclidean geometry","Vector spaces","Vectors in algebra","Data represented by vectors"]}
{"upwards":["Vector (mathematics and physics)","Vectors in Euclidean geometry"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vector spaces"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vectors in algebra"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Data represented by vectors"],"downwards":[]}
{"upwards":["Recursive definition"],"downwards":["Form of recursive definitions","Examples of recursive definitions"]}
{"upwards":["Recursive definition","Form of recursive definitions"],"downwards":["Principle of recursive definition"]}
{"upwards":["Recursive definition","Form of recursive definitions","Principle of recursive definition"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions"],"downwards":["Elementary functions","Prime numbers","Non-negative even numbers","Well formed formulas"]}
{"upwards":["Recursive definition","Examples of recursive definitions","Elementary functions"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions","Prime numbers"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions","Non-negative even numbers"],"downwards":[]}
{"upwards":["Recursive definition","Examples of recursive definitions","Well formed formulas"],"downwards":[]}
{"upwards":["Class (set theory)"],"downwards":["Examples","Paradoxes","Classes in formal set theories","Notes","References","External links"]}
{"upwards":["Class (set theory)","Examples"],"downwards":[]}
{"upwards":["Class (set theory)","Paradoxes"],"downwards":[]}
{"upwards":["Class (set theory)","Classes in formal set theories"],"downwards":[]}
{"upwards":["Class (set theory)","Notes"],"downwards":[]}
{"upwards":["Class (set theory)","References"],"downwards":[]}
{"upwards":["Class (set theory)","External links"],"downwards":[]}
{"upwards":["Binary relation"],"downwards":["Definition","Operations","Examples","Special types of binary relations","Sets versus classes","Homogeneous relation","Heterogeneous relation","Calculus of relations","Induced concept lattice","Particular relations","Preorder R\\R","Fringe of a relation","Mathematical heaps"]}
{"upwards":["Binary relation","Definition"],"downwards":[]}
{"upwards":["Binary relation","Operations"],"downwards":["Union","Intersection","Composition","Converse","Complement","Restriction","Matrix representation"]}
{"upwards":["Binary relation","Operations","Union"],"downwards":[]}
{"upwards":["Binary relation","Operations","Intersection"],"downwards":[]}
{"upwards":["Binary relation","Operations","Composition"],"downwards":[]}
{"upwards":["Binary relation","Operations","Converse"],"downwards":[]}
{"upwards":["Binary relation","Operations","Complement"],"downwards":[]}
{"upwards":["Binary relation","Operations","Restriction"],"downwards":[]}
{"upwards":["Binary relation","Operations","Matrix representation"],"downwards":[]}
{"upwards":["Binary relation","Examples"],"downwards":[]}
{"upwards":["Binary relation","Special types of binary relations"],"downwards":[]}
{"upwards":["Binary relation","Sets versus classes"],"downwards":[]}
{"upwards":["Binary relation","Homogeneous relation"],"downwards":[]}
{"upwards":["Binary relation","Heterogeneous relation"],"downwards":[]}
{"upwards":["Binary relation","Calculus of relations"],"downwards":[]}
{"upwards":["Binary relation","Induced concept lattice"],"downwards":[]}
{"upwards":["Binary relation","Particular relations"],"downwards":["Difunctional","Ferrers type","Contact"]}
{"upwards":["Binary relation","Particular relations","Difunctional"],"downwards":[]}
{"upwards":["Binary relation","Particular relations","Ferrers type"],"downwards":[]}
{"upwards":["Binary relation","Particular relations","Contact"],"downwards":[]}
{"upwards":["Binary relation","Preorder R\\R"],"downwards":[]}
{"upwards":["Binary relation","Fringe of a relation"],"downwards":[]}
{"upwards":["Binary relation","Mathematical heaps"],"downwards":[]}
{"upwards":["Nicolas Bourbaki"],"downwards":["Background","The Bourbaki collective","Working method","Works","Membership","Influence and criticism"]}
{"upwards":["Nicolas Bourbaki","Background"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective"],"downwards":["Founding","World War II","Postwar until the present"]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective","Founding"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective","World War II"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","The Bourbaki collective","Postwar until the present"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Working method"],"downwards":["Humor"]}
{"upwards":["Nicolas Bourbaki","Working method","Humor"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works"],"downwards":["\u00c9l\u00e9ments de math\u00e9matique","S\u00e9minaire Bourbaki","Articles","La Tribu"]}
{"upwards":["Nicolas Bourbaki","Works","\u00c9l\u00e9ments de math\u00e9matique"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works","S\u00e9minaire Bourbaki"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works","Articles"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Works","La Tribu"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Membership"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Influence and criticism"],"downwards":["Influence","Praise","Criticism"]}
{"upwards":["Nicolas Bourbaki","Influence and criticism","Influence"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Influence and criticism","Praise"],"downwards":[]}
{"upwards":["Nicolas Bourbaki","Influence and criticism","Criticism"],"downwards":[]}
{"upwards":["Natural transformation"],"downwards":["Definition","Examples","Unnatural isomorphism","Operations with natural transformations","Functor categories","Yoneda lemma","Historical notes"]}
{"upwards":["Natural transformation","Definition"],"downwards":[]}
{"upwards":["Natural transformation","Examples"],"downwards":["Opposite group","Abelianization","Hurewicz homomorphism","Determinant","Double dual of a vector space","Finite calculus","Tensor-hom adjunction"]}
{"upwards":["Natural transformation","Examples","Opposite group"],"downwards":[]}
{"upwards":["Natural transformation","Examples","Abelianization"],"downwards":[]}
{"upwards":["Natural transformation","Examples","Hurewicz homomorphism"],"downwards":[]}
{"upwards":["Natural transformation","Examples","Determinant"],"downwards":[]}
{"upwards":["Natural transformation","Examples","Double dual of a vector space"],"downwards":[]}
{"upwards":["Natural transformation","Examples","Finite calculus"],"downwards":[]}
{"upwards":["Natural transformation","Examples","Tensor-hom adjunction"],"downwards":[]}
{"upwards":["Natural transformation","Unnatural isomorphism"],"downwards":["Example: fundamental group of torus","Example: dual of a finite-dimensional vector space"]}
{"upwards":["Natural transformation","Unnatural isomorphism","Example: fundamental group of torus"],"downwards":[]}
{"upwards":["Natural transformation","Unnatural isomorphism","Example: dual of a finite-dimensional vector space"],"downwards":[]}
{"upwards":["Natural transformation","Operations with natural transformations"],"downwards":["Vertical composition","Horizontal composition","Whiskering","Interchange law"]}
{"upwards":["Natural transformation","Operations with natural transformations","Vertical composition"],"downwards":[]}
{"upwards":["Natural transformation","Operations with natural transformations","Horizontal composition"],"downwards":[]}
{"upwards":["Natural transformation","Operations with natural transformations","Whiskering"],"downwards":[]}
{"upwards":["Natural transformation","Operations with natural transformations","Interchange law"],"downwards":[]}
{"upwards":["Natural transformation","Functor categories"],"downwards":["More examples"]}
{"upwards":["Natural transformation","Functor categories","More examples"],"downwards":[]}
{"upwards":["Natural transformation","Yoneda lemma"],"downwards":[]}
{"upwards":["Natural transformation","Historical notes"],"downwards":[]}
{"upwards":["Universal property"],"downwards":["Motivation","Formal definition","Connection with comma categories","Examples","Properties","History"]}
{"upwards":["Universal property","Motivation"],"downwards":[]}
{"upwards":["Universal property","Formal definition"],"downwards":[]}
{"upwards":["Universal property","Connection with comma categories"],"downwards":[]}
{"upwards":["Universal property","Examples"],"downwards":["Tensor algebras","Products","Limits and colimits"]}
{"upwards":["Universal property","Examples","Tensor algebras"],"downwards":[]}
{"upwards":["Universal property","Examples","Products"],"downwards":[]}
{"upwards":["Universal property","Examples","Limits and colimits"],"downwards":[]}
{"upwards":["Universal property","Properties"],"downwards":["Existence and uniqueness","Equivalent formulations","Relation to adjoint functors"]}
{"upwards":["Universal property","Properties","Existence and uniqueness"],"downwards":[]}
{"upwards":["Universal property","Properties","Equivalent formulations"],"downwards":[]}
{"upwards":["Universal property","Properties","Relation to adjoint functors"],"downwards":[]}
{"upwards":["Universal property","History"],"downwards":[]}
{"upwards":["Word and Object"],"downwards":["Synopsis","Behaviorism","Translation and meaning","Reference","Semantic ascent"]}
{"upwards":["Word and Object","Synopsis"],"downwards":[]}
{"upwards":["Word and Object","Behaviorism"],"downwards":[]}
{"upwards":["Word and Object","Translation and meaning"],"downwards":["Indeterminacy of translation","Analytical hypotheses"]}
{"upwards":["Word and Object","Translation and meaning","Indeterminacy of translation"],"downwards":[]}
{"upwards":["Word and Object","Translation and meaning","Analytical hypotheses"],"downwards":[]}
{"upwards":["Word and Object","Reference"],"downwards":["Acquiring reference","Vagaries of reference and referential transparency","Canonical notation"]}
{"upwards":["Word and Object","Reference","Acquiring reference"],"downwards":[]}
{"upwards":["Word and Object","Reference","Vagaries of reference and referential transparency"],"downwards":[]}
{"upwards":["Word and Object","Reference","Canonical notation"],"downwards":[]}
{"upwards":["Word and Object","Semantic ascent"],"downwards":[]}
{"upwards":["Commutative diagram"],"downwards":["Description","Examples","Diagram chasing","In higher category theory","Diagrams as functors"]}
{"upwards":["Commutative diagram","Description"],"downwards":["Arrow symbols","Verifying commutativity"]}
{"upwards":["Commutative diagram","Description","Arrow symbols"],"downwards":[]}
{"upwards":["Commutative diagram","Description","Verifying commutativity"],"downwards":[]}
{"upwards":["Commutative diagram","Examples"],"downwards":["Example 1","Example 2"]}
{"upwards":["Commutative diagram","Examples","Example 1"],"downwards":[]}
{"upwards":["Commutative diagram","Examples","Example 2"],"downwards":[]}
{"upwards":["Commutative diagram","Diagram chasing"],"downwards":[]}
{"upwards":["Commutative diagram","In higher category theory"],"downwards":[]}
{"upwards":["Commutative diagram","Diagrams as functors"],"downwards":[]}
{"upwards":["Set (mathematics)"],"downwards":["History","How sets are defined and set notation","Membership","The empty set","Singleton sets","Subsets","Euler and Venn diagrams","Special sets of numbers in mathematics","Functions","Cardinality","Power sets","Partitions","Basic operations","Applications","Principle of inclusion and exclusion"]}
{"upwards":["Set (mathematics)","History"],"downwards":["Naive set theory","Axiomatic set theory"]}
{"upwards":["Set (mathematics)","History","Naive set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","History","Axiomatic set theory"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation"],"downwards":["Roster notation","Semantic definition","Set-builder notation","Classifying methods of definition"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation"],"downwards":["Infinite sets in roster notation"]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Roster notation","Infinite sets in roster notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Semantic definition"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Set-builder notation"],"downwards":[]}
{"upwards":["Set (mathematics)","How sets are defined and set notation","Classifying methods of definition"],"downwards":[]}
{"upwards":["Set (mathematics)","Membership"],"downwards":[]}
{"upwards":["Set (mathematics)","The empty set"],"downwards":[]}
{"upwards":["Set (mathematics)","Singleton sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Subsets"],"downwards":[]}
{"upwards":["Set (mathematics)","Euler and Venn diagrams"],"downwards":[]}
{"upwards":["Set (mathematics)","Special sets of numbers in mathematics"],"downwards":[]}
{"upwards":["Set (mathematics)","Functions"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality"],"downwards":["Infinite sets and infinite cardinality","The continuum hypothesis"]}
{"upwards":["Set (mathematics)","Cardinality","Infinite sets and infinite cardinality"],"downwards":[]}
{"upwards":["Set (mathematics)","Cardinality","The continuum hypothesis"],"downwards":[]}
{"upwards":["Set (mathematics)","Power sets"],"downwards":[]}
{"upwards":["Set (mathematics)","Partitions"],"downwards":[]}
{"upwards":["Set (mathematics)","Basic operations"],"downwards":[]}
{"upwards":["Set (mathematics)","Applications"],"downwards":[]}
{"upwards":["Set (mathematics)","Principle of inclusion and exclusion"],"downwards":[]}
{"upwards":["Fundamenta Mathematicae"],"downwards":["History","Abstracting and indexing","Notes","References","External links"]}
{"upwards":["Fundamenta Mathematicae","History"],"downwards":[]}
{"upwards":["Fundamenta Mathematicae","Abstracting and indexing"],"downwards":[]}
{"upwards":["Fundamenta Mathematicae","Notes"],"downwards":[]}
{"upwards":["Fundamenta Mathematicae","References"],"downwards":[]}
{"upwards":["Fundamenta Mathematicae","External links"],"downwards":[]}
{"upwards":["Wayback Machine"],"downwards":["History","Technical information","Uses","Legal status","Archived content legal issues","Censorship and other threats"]}
{"upwards":["Wayback Machine","History"],"downwards":[]}
{"upwards":["Wayback Machine","Technical information"],"downwards":["Storage capacity and growth","Wayback Machine APIs","Website exclusion policy"]}
{"upwards":["Wayback Machine","Technical information","Storage capacity and growth"],"downwards":[]}
{"upwards":["Wayback Machine","Technical information","Wayback Machine APIs"],"downwards":[]}
{"upwards":["Wayback Machine","Technical information","Website exclusion policy"],"downwards":["Oakland Archive Policy"]}
{"upwards":["Wayback Machine","Technical information","Website exclusion policy","Oakland Archive Policy"],"downwards":[]}
{"upwards":["Wayback Machine","Uses"],"downwards":["Limitations","In legal evidence"]}
{"upwards":["Wayback Machine","Uses","Limitations"],"downwards":[]}
{"upwards":["Wayback Machine","Uses","In legal evidence"],"downwards":["Civil litigation","Patent law","Limitations of utility"]}
{"upwards":["Wayback Machine","Uses","In legal evidence","Civil litigation"],"downwards":[]}
{"upwards":["Wayback Machine","Uses","In legal evidence","Patent law"],"downwards":[]}
{"upwards":["Wayback Machine","Uses","In legal evidence","Limitations of utility"],"downwards":[]}
{"upwards":["Wayback Machine","Legal status"],"downwards":[]}
{"upwards":["Wayback Machine","Archived content legal issues"],"downwards":["Scientology","Healthcare Advocates, Inc.","Suzanne Shell","Daniel Davydiuk","FlexiSpy"]}
{"upwards":["Wayback Machine","Archived content legal issues","Scientology"],"downwards":[]}
{"upwards":["Wayback Machine","Archived content legal issues","Healthcare Advocates, Inc."],"downwards":[]}
{"upwards":["Wayback Machine","Archived content legal issues","Suzanne Shell"],"downwards":[]}
{"upwards":["Wayback Machine","Archived content legal issues","Daniel Davydiuk"],"downwards":[]}
{"upwards":["Wayback Machine","Archived content legal issues","FlexiSpy"],"downwards":[]}
{"upwards":["Wayback Machine","Censorship and other threats"],"downwards":[]}
{"upwards":["Metamath"],"downwards":["Metamath language","Metamath proof checker","Metamath databases","Natural deduction","Other works connected to Metamath"]}
{"upwards":["Metamath","Metamath language"],"downwards":["Language basics","Proofs","Substitution"]}
{"upwards":["Metamath","Metamath language","Language basics"],"downwards":[]}
{"upwards":["Metamath","Metamath language","Proofs"],"downwards":[]}
{"upwards":["Metamath","Metamath language","Substitution"],"downwards":[]}
{"upwards":["Metamath","Metamath proof checker"],"downwards":[]}
{"upwards":["Metamath","Metamath databases"],"downwards":["Metamath Proof Explorer","Intuitionistic Logic Explorer","New Foundations Explorer","Higher-Order Logic Explorer","Databases without explorers","Older explorers"]}
{"upwards":["Metamath","Metamath databases","Metamath Proof Explorer"],"downwards":[]}
{"upwards":["Metamath","Metamath databases","Intuitionistic Logic Explorer"],"downwards":[]}
{"upwards":["Metamath","Metamath databases","New Foundations Explorer"],"downwards":[]}
{"upwards":["Metamath","Metamath databases","Higher-Order Logic Explorer"],"downwards":[]}
{"upwards":["Metamath","Metamath databases","Databases without explorers"],"downwards":[]}
{"upwards":["Metamath","Metamath databases","Older explorers"],"downwards":[]}
{"upwards":["Metamath","Natural deduction"],"downwards":[]}
{"upwards":["Metamath","Other works connected to Metamath"],"downwards":["Proof checkers","Editors"]}
{"upwards":["Metamath","Other works connected to Metamath","Proof checkers"],"downwards":[]}
{"upwards":["Metamath","Other works connected to Metamath","Editors"],"downwards":[]}
{"upwards":["Naive Set Theory (book)"],"downwards":["Absence of the Axiom of Foundation","Errata"]}
{"upwards":["Naive Set Theory (book)","Absence of the Axiom of Foundation"],"downwards":[]}
{"upwards":["Naive Set Theory (book)","Errata"],"downwards":[]}
{"upwards":["Boolean algebra"],"downwards":["History","Values","Operations","Laws","Diagrammatic representations","Boolean algebras","Axiomatizing Boolean algebra","Propositional logic","Applications"]}
{"upwards":["Boolean algebra","History"],"downwards":[]}
{"upwards":["Boolean algebra","Values"],"downwards":[]}
{"upwards":["Boolean algebra","Operations"],"downwards":["Basic operations","Secondary operations"]}
{"upwards":["Boolean algebra","Operations","Basic operations"],"downwards":[]}
{"upwards":["Boolean algebra","Operations","Secondary operations"],"downwards":[]}
{"upwards":["Boolean algebra","Laws"],"downwards":["Monotone laws","Nonmonotone laws","Completeness","Duality principle"]}
{"upwards":["Boolean algebra","Laws","Monotone laws"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Nonmonotone laws"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Completeness"],"downwards":[]}
{"upwards":["Boolean algebra","Laws","Duality principle"],"downwards":[]}
{"upwards":["Boolean algebra","Diagrammatic representations"],"downwards":["Venn diagrams","Digital logic gates"]}
{"upwards":["Boolean algebra","Diagrammatic representations","Venn diagrams"],"downwards":[]}
{"upwards":["Boolean algebra","Diagrammatic representations","Digital logic gates"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras"],"downwards":["Concrete Boolean algebras","Subsets as bit vectors","The prototypical Boolean algebra","Boolean algebras: the definition","Representable Boolean algebras"]}
{"upwards":["Boolean algebra","Boolean algebras","Concrete Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Subsets as bit vectors"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","The prototypical Boolean algebra"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Boolean algebras: the definition"],"downwards":[]}
{"upwards":["Boolean algebra","Boolean algebras","Representable Boolean algebras"],"downwards":[]}
{"upwards":["Boolean algebra","Axiomatizing Boolean algebra"],"downwards":[]}
{"upwards":["Boolean algebra","Propositional logic"],"downwards":["Applications","Deductive systems for propositional logic"]}
{"upwards":["Boolean algebra","Propositional logic","Applications"],"downwards":[]}
{"upwards":["Boolean algebra","Propositional logic","Deductive systems for propositional logic"],"downwards":["Sequent calculus"]}
{"upwards":["Boolean algebra","Propositional logic","Deductive systems for propositional logic","Sequent calculus"],"downwards":[]}
{"upwards":["Boolean algebra","Applications"],"downwards":["Computers","Two-valued logic","Boolean operations"]}
{"upwards":["Boolean algebra","Applications","Computers"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Two-valued logic"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations"],"downwards":["Natural language","Digital logic","Naive set theory","Video cards","Modeling and CAD","Boolean searches"]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Natural language"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Digital logic"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Naive set theory"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Video cards"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Modeling and CAD"],"downwards":[]}
{"upwards":["Boolean algebra","Applications","Boolean operations","Boolean searches"],"downwards":[]}
{"upwards":["Mathematical logic"],"downwards":["Subfields and scope","History","Formal logical systems","Set theory","Model theory","Recursion theory","Proof theory and constructive mathematics","Applications","Connections with computer science","Foundations of mathematics"]}
{"upwards":["Mathematical logic","Subfields and scope"],"downwards":[]}
{"upwards":["Mathematical logic","History"],"downwards":["Early history","19th century","20th century"]}
{"upwards":["Mathematical logic","History","Early history"],"downwards":[]}
{"upwards":["Mathematical logic","History","19th century"],"downwards":["Foundational theories"]}
{"upwards":["Mathematical logic","History","19th century","Foundational theories"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century"],"downwards":["Set theory and paradoxes","Symbolic logic","Beginnings of the other branches"]}
{"upwards":["Mathematical logic","History","20th century","Set theory and paradoxes"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Symbolic logic"],"downwards":[]}
{"upwards":["Mathematical logic","History","20th century","Beginnings of the other branches"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems"],"downwards":["First-order logic","Other classical logics","Nonclassical and modal logic","Algebraic logic"]}
{"upwards":["Mathematical logic","Formal logical systems","First-order logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Other classical logics"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Nonclassical and modal logic"],"downwards":[]}
{"upwards":["Mathematical logic","Formal logical systems","Algebraic logic"],"downwards":[]}
{"upwards":["Mathematical logic","Set theory"],"downwards":[]}
{"upwards":["Mathematical logic","Model theory"],"downwards":[]}
{"upwards":["Mathematical logic","Recursion theory"],"downwards":["Algorithmically unsolvable problems"]}
{"upwards":["Mathematical logic","Recursion theory","Algorithmically unsolvable problems"],"downwards":[]}
{"upwards":["Mathematical logic","Proof theory and constructive mathematics"],"downwards":[]}
{"upwards":["Mathematical logic","Applications"],"downwards":[]}
{"upwards":["Mathematical logic","Connections with computer science"],"downwards":[]}
{"upwards":["Mathematical logic","Foundations of mathematics"],"downwards":[]}
{"upwards":["Discrete mathematics"],"downwards":["Grand challenges, past and present","Topics in discrete mathematics"]}
{"upwards":["Discrete mathematics","Grand challenges, past and present"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics"],"downwards":["Theoretical computer science","Information theory","Logic","Set theory","Combinatorics","Graph theory","Number theory","Algebraic structures","Discrete analogues of continuous mathematics"]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Theoretical computer science"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Information theory"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Logic"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Set theory"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Combinatorics"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Graph theory"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Number theory"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Algebraic structures"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Discrete analogues of continuous mathematics"],"downwards":["Calculus of finite differences, discrete analysis, and discrete calculus","Discrete geometry","Discrete modelling"]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Discrete analogues of continuous mathematics","Calculus of finite differences, discrete analysis, and discrete calculus"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Discrete analogues of continuous mathematics","Discrete geometry"],"downwards":[]}
{"upwards":["Discrete mathematics","Topics in discrete mathematics","Discrete analogues of continuous mathematics","Discrete modelling"],"downwards":[]}
{"upwards":["Natural language"],"downwards":["Defining natural language","Controlled languages","International constructed languages"]}
{"upwards":["Natural language","Defining natural language"],"downwards":[]}
{"upwards":["Natural language","Controlled languages"],"downwards":[]}
{"upwards":["Natural language","International constructed languages"],"downwards":[]}
{"upwards":["Set theory"],"downwards":["History","Basic concepts and notation","Ontology","Formalized set theory","Applications","Areas of study","Objections to set theory","Set theory in mathematical education"]}
{"upwards":["Set theory","History"],"downwards":[]}
{"upwards":["Set theory","Basic concepts and notation"],"downwards":[]}
{"upwards":["Set theory","Ontology"],"downwards":[]}
{"upwards":["Set theory","Formalized set theory"],"downwards":[]}
{"upwards":["Set theory","Applications"],"downwards":[]}
{"upwards":["Set theory","Areas of study"],"downwards":["Combinatorial set theory","Descriptive set theory","Fuzzy set theory","Inner model theory","Large cardinals","Determinacy","Forcing","Cardinal invariants","Set-theoretic topology"]}
{"upwards":["Set theory","Areas of study","Combinatorial set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Descriptive set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Fuzzy set theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Inner model theory"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Large cardinals"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Determinacy"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Forcing"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Cardinal invariants"],"downwards":[]}
{"upwards":["Set theory","Areas of study","Set-theoretic topology"],"downwards":[]}
{"upwards":["Set theory","Objections to set theory"],"downwards":[]}
{"upwards":["Set theory","Set theory in mathematical education"],"downwards":[]}
{"upwards":["Gottlob Frege"],"downwards":["Life","Work as a logician","Philosopher","Sense and reference","1924 diary","Personality","Important dates","Important works"]}
{"upwards":["Gottlob Frege","Life"],"downwards":["Childhood (1848\u201369)","Studies at University (1869\u201374)"]}
{"upwards":["Gottlob Frege","Life","Childhood (1848\u201369)"],"downwards":[]}
{"upwards":["Gottlob Frege","Life","Studies at University (1869\u201374)"],"downwards":[]}
{"upwards":["Gottlob Frege","Work as a logician"],"downwards":[]}
{"upwards":["Gottlob Frege","Philosopher"],"downwards":[]}
{"upwards":["Gottlob Frege","Sense and reference"],"downwards":[]}
{"upwards":["Gottlob Frege","1924 diary"],"downwards":[]}
{"upwards":["Gottlob Frege","Personality"],"downwards":[]}
{"upwards":["Gottlob Frege","Important dates"],"downwards":[]}
{"upwards":["Gottlob Frege","Important works"],"downwards":["Logic, foundation of arithmetic","Philosophical studies","Articles on geometry"]}
{"upwards":["Gottlob Frege","Important works","Logic, foundation of arithmetic"],"downwards":[]}
{"upwards":["Gottlob Frege","Important works","Philosophical studies"],"downwards":[]}
{"upwards":["Gottlob Frege","Important works","Articles on geometry"],"downwards":[]}
{"upwards":["If and only if"],"downwards":["Definition","Usage","Distinction from \"if\" and \"only if\"","In terms of Euler diagrams","More general usage"]}
{"upwards":["If and only if","Definition"],"downwards":[]}
{"upwards":["If and only if","Usage"],"downwards":["Notation","Proofs","Origin of iff and pronunciation","Usage in definitions"]}
{"upwards":["If and only if","Usage","Notation"],"downwards":[]}
{"upwards":["If and only if","Usage","Proofs"],"downwards":[]}
{"upwards":["If and only if","Usage","Origin of iff and pronunciation"],"downwards":[]}
{"upwards":["If and only if","Usage","Usage in definitions"],"downwards":[]}
{"upwards":["If and only if","Distinction from \"if\" and \"only if\""],"downwards":[]}
{"upwards":["If and only if","In terms of Euler diagrams"],"downwards":[]}
{"upwards":["If and only if","More general usage"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski"],"downwards":["Biography and studies","Legacy","Research","Publications"]}
{"upwards":["Kazimierz Kuratowski","Biography and studies"],"downwards":["Doctoral thesis","Academic career until World War II","During and after the war"]}
{"upwards":["Kazimierz Kuratowski","Biography and studies","Doctoral thesis"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Biography and studies","Academic career until World War II"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Biography and studies","During and after the war"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Legacy"],"downwards":["Kazimierz Kuratowski Prize"]}
{"upwards":["Kazimierz Kuratowski","Legacy","Kazimierz Kuratowski Prize"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Research"],"downwards":[]}
{"upwards":["Kazimierz Kuratowski","Publications"],"downwards":[]}
{"upwards":["Functional programming"],"downwards":["History","Concepts","Comparison to imperative programming","Applications"]}
{"upwards":["Functional programming","History"],"downwards":[]}
{"upwards":["Functional programming","Concepts"],"downwards":["First-class and higher-order functions","Pure functions","Recursion","Strict versus non-strict evaluation","Type systems","Referential transparency","Data structures"]}
{"upwards":["Functional programming","Concepts","First-class and higher-order functions"],"downwards":[]}
{"upwards":["Functional programming","Concepts","Pure functions"],"downwards":[]}
{"upwards":["Functional programming","Concepts","Recursion"],"downwards":[]}
{"upwards":["Functional programming","Concepts","Strict versus non-strict evaluation"],"downwards":[]}
{"upwards":["Functional programming","Concepts","Type systems"],"downwards":[]}
{"upwards":["Functional programming","Concepts","Referential transparency"],"downwards":[]}
{"upwards":["Functional programming","Concepts","Data structures"],"downwards":[]}
{"upwards":["Functional programming","Comparison to imperative programming"],"downwards":["Imperative vs. functional programming","Simulating state","Efficiency issues","Functional programming in non-functional languages"]}
{"upwards":["Functional programming","Comparison to imperative programming","Imperative vs. functional programming"],"downwards":[]}
{"upwards":["Functional programming","Comparison to imperative programming","Simulating state"],"downwards":[]}
{"upwards":["Functional programming","Comparison to imperative programming","Efficiency issues"],"downwards":[]}
{"upwards":["Functional programming","Comparison to imperative programming","Functional programming in non-functional languages"],"downwards":[]}
{"upwards":["Functional programming","Applications"],"downwards":["Spreadsheets","Academia","Industry","Education"]}
{"upwards":["Functional programming","Applications","Spreadsheets"],"downwards":[]}
{"upwards":["Functional programming","Applications","Academia"],"downwards":[]}
{"upwards":["Functional programming","Applications","Industry"],"downwards":[]}
{"upwards":["Functional programming","Applications","Education"],"downwards":[]}
{"upwards":["Multiplicity (mathematics)"],"downwards":["Multiplicity of a prime factor","Multiplicity of a root of a polynomial","Multiplicity of a solution of a nonlinear system of equations","Intersection multiplicity","In complex analysis","References"]}
{"upwards":["Multiplicity (mathematics)","Multiplicity of a prime factor"],"downwards":[]}
{"upwards":["Multiplicity (mathematics)","Multiplicity of a root of a polynomial"],"downwards":["Behavior of a polynomial function near a multiple root"]}
{"upwards":["Multiplicity (mathematics)","Multiplicity of a root of a polynomial","Behavior of a polynomial function near a multiple root"],"downwards":[]}
{"upwards":["Multiplicity (mathematics)","Multiplicity of a solution of a nonlinear system of equations"],"downwards":[]}
{"upwards":["Multiplicity (mathematics)","Intersection multiplicity"],"downwards":[]}
{"upwards":["Multiplicity (mathematics)","In complex analysis"],"downwards":[]}
{"upwards":["Multiplicity (mathematics)","References"],"downwards":[]}
{"upwards":["Parity (mathematics)"],"downwards":["Definition","Properties","History","Higher mathematics","Additional applications"]}
{"upwards":["Parity (mathematics)","Definition"],"downwards":[]}
{"upwards":["Parity (mathematics)","Properties"],"downwards":["Addition and subtraction","Multiplication","Division"]}
{"upwards":["Parity (mathematics)","Properties","Addition and subtraction"],"downwards":[]}
{"upwards":["Parity (mathematics)","Properties","Multiplication"],"downwards":[]}
{"upwards":["Parity (mathematics)","Properties","Division"],"downwards":[]}
{"upwards":["Parity (mathematics)","History"],"downwards":[]}
{"upwards":["Parity (mathematics)","Higher mathematics"],"downwards":["Higher dimensions and more general classes of numbers","Number theory","Group theory","Analysis","Combinatorial game theory"]}
{"upwards":["Parity (mathematics)","Higher mathematics","Higher dimensions and more general classes of numbers"],"downwards":[]}
{"upwards":["Parity (mathematics)","Higher mathematics","Number theory"],"downwards":[]}
{"upwards":["Parity (mathematics)","Higher mathematics","Group theory"],"downwards":[]}
{"upwards":["Parity (mathematics)","Higher mathematics","Analysis"],"downwards":[]}
{"upwards":["Parity (mathematics)","Higher mathematics","Combinatorial game theory"],"downwards":[]}
{"upwards":["Parity (mathematics)","Additional applications"],"downwards":[]}
{"upwards":["President of the United States"],"downwards":["History and development","Legislative powers","Executive powers","Leadership roles","Selection process","Incumbency","Post-presidency","Political affiliation","Timeline of presidents"]}
{"upwards":["President of the United States","History and development"],"downwards":["Origins","1789\u20131933","Imperial presidency","Critics of presidency's evolution"]}
{"upwards":["President of the United States","History and development","Origins"],"downwards":[]}
{"upwards":["President of the United States","History and development","1789\u20131933"],"downwards":[]}
{"upwards":["President of the United States","History and development","Imperial presidency"],"downwards":[]}
{"upwards":["President of the United States","History and development","Critics of presidency's evolution"],"downwards":[]}
{"upwards":["President of the United States","Legislative powers"],"downwards":["Signing and vetoing bills","Setting the agenda","Promulgating regulations","Convening and adjourning Congress"]}
{"upwards":["President of the United States","Legislative powers","Signing and vetoing bills"],"downwards":[]}
{"upwards":["President of the United States","Legislative powers","Setting the agenda"],"downwards":[]}
{"upwards":["President of the United States","Legislative powers","Promulgating regulations"],"downwards":[]}
{"upwards":["President of the United States","Legislative powers","Convening and adjourning Congress"],"downwards":[]}
{"upwards":["President of the United States","Executive powers"],"downwards":["Administrative powers","Foreign affairs","Commander-in-chief","Juridical powers and privileges"]}
{"upwards":["President of the United States","Executive powers","Administrative powers"],"downwards":[]}
{"upwards":["President of the United States","Executive powers","Foreign affairs"],"downwards":[]}
{"upwards":["President of the United States","Executive powers","Commander-in-chief"],"downwards":[]}
{"upwards":["President of the United States","Executive powers","Juridical powers and privileges"],"downwards":[]}
{"upwards":["President of the United States","Leadership roles"],"downwards":["Head of state","Head of party","Global leader"]}
{"upwards":["President of the United States","Leadership roles","Head of state"],"downwards":[]}
{"upwards":["President of the United States","Leadership roles","Head of party"],"downwards":[]}
{"upwards":["President of the United States","Leadership roles","Global leader"],"downwards":[]}
{"upwards":["President of the United States","Selection process"],"downwards":["Eligibility","Campaigns and nomination","Election","Inauguration"]}
{"upwards":["President of the United States","Selection process","Eligibility"],"downwards":[]}
{"upwards":["President of the United States","Selection process","Campaigns and nomination"],"downwards":[]}
{"upwards":["President of the United States","Selection process","Election"],"downwards":[]}
{"upwards":["President of the United States","Selection process","Inauguration"],"downwards":[]}
{"upwards":["President of the United States","Incumbency"],"downwards":["Term limit","Vacancies and succession","Declarations of inability","Removal","Circumvention of authority","Compensation","Residence","Travel","Protection"]}
{"upwards":["President of the United States","Incumbency","Term limit"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Vacancies and succession"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Declarations of inability"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Removal"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Circumvention of authority"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Compensation"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Residence"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Travel"],"downwards":[]}
{"upwards":["President of the United States","Incumbency","Protection"],"downwards":[]}
{"upwards":["President of the United States","Post-presidency"],"downwards":["Activities","Pension and other benefits","Presidential libraries"]}
{"upwards":["President of the United States","Post-presidency","Activities"],"downwards":[]}
{"upwards":["President of the United States","Post-presidency","Pension and other benefits"],"downwards":[]}
{"upwards":["President of the United States","Post-presidency","Presidential libraries"],"downwards":[]}
{"upwards":["President of the United States","Political affiliation"],"downwards":[]}
{"upwards":["President of the United States","Timeline of presidents"],"downwards":[]}
{"upwards":["Oldest people"],"downwards":["Ten oldest verified people ever","Ten oldest living people","Chronological list of the oldest known living person since 1955","References"]}
{"upwards":["Oldest people","Ten oldest verified people ever"],"downwards":["Oldest people (all women)","Oldest men"]}
{"upwards":["Oldest people","Ten oldest verified people ever","Oldest people (all women)"],"downwards":[]}
{"upwards":["Oldest people","Ten oldest verified people ever","Oldest men"],"downwards":[]}
{"upwards":["Oldest people","Ten oldest living people"],"downwards":["Oldest living people","Oldest living men"]}
{"upwards":["Oldest people","Ten oldest living people","Oldest living people"],"downwards":[]}
{"upwards":["Oldest people","Ten oldest living people","Oldest living men"],"downwards":[]}
{"upwards":["Oldest people","Chronological list of the oldest known living person since 1955"],"downwards":["Chronological list of the oldest living man since 1973"]}
{"upwards":["Oldest people","Chronological list of the oldest known living person since 1955","Chronological list of the oldest living man since 1973"],"downwards":[]}
{"upwards":["Oldest people","References"],"downwards":[]}
{"upwards":["Vacuous truth"],"downwards":["Scope of the concept","In computer programming","Examples"]}
{"upwards":["Vacuous truth","Scope of the concept"],"downwards":[]}
{"upwards":["Vacuous truth","In computer programming"],"downwards":[]}
{"upwards":["Vacuous truth","Examples"],"downwards":[]}
{"upwards":["Axiom schema of specification"],"downwards":["Statement","Relation to the axiom schema of replacement","Unrestricted comprehension","In NBG class theory","In higher-order settings","In Quine's New Foundations","References"]}
{"upwards":["Axiom schema of specification","Statement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Relation to the axiom schema of replacement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Unrestricted comprehension"],"downwards":[]}
{"upwards":["Axiom schema of specification","In NBG class theory"],"downwards":[]}
{"upwards":["Axiom schema of specification","In higher-order settings"],"downwards":[]}
{"upwards":["Axiom schema of specification","In Quine's New Foundations"],"downwards":[]}
{"upwards":["Axiom schema of specification","References"],"downwards":["Citations"]}
{"upwards":["Axiom schema of specification","References","Citations"],"downwards":[]}
{"upwards":["Axiom of empty set"],"downwards":["Formal statement","Interpretation","References","Sources"]}
{"upwards":["Axiom of empty set","Formal statement"],"downwards":[]}
{"upwards":["Axiom of empty set","Interpretation"],"downwards":[]}
{"upwards":["Axiom of empty set","References"],"downwards":[]}
{"upwards":["Axiom of empty set","Sources"],"downwards":[]}
{"upwards":["Script typeface"],"downwards":["Styles","Technology","Unicode"]}
{"upwards":["Script typeface","Styles"],"downwards":["Formal scripts","Casual scripts"]}
{"upwards":["Script typeface","Styles","Formal scripts"],"downwards":[]}
{"upwards":["Script typeface","Styles","Casual scripts"],"downwards":[]}
{"upwards":["Script typeface","Technology"],"downwards":[]}
{"upwards":["Script typeface","Unicode"],"downwards":[]}
{"upwards":["Axiom schema of replacement"],"downwards":["Statement","Applications","Relation to other axiom schemas","History","References"]}
{"upwards":["Axiom schema of replacement","Statement"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Applications"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas"],"downwards":["Collection","Separation"]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas","Collection"],"downwards":[]}
{"upwards":["Axiom schema of replacement","Relation to other axiom schemas","Separation"],"downwards":[]}
{"upwards":["Axiom schema of replacement","History"],"downwards":[]}
{"upwards":["Axiom schema of replacement","References"],"downwards":[]}
{"upwards":["Boolean algebra (structure)"],"downwards":["History","Definition","Examples","Homomorphisms and isomorphisms","Boolean rings","Ideals and filters","Representations","Axiomatics","Generalizations"]}
{"upwards":["Boolean algebra (structure)","History"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Definition"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Examples"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Homomorphisms and isomorphisms"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Boolean rings"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Ideals and filters"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Representations"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Axiomatics"],"downwards":[]}
{"upwards":["Boolean algebra (structure)","Generalizations"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes"],"downwards":["Life","Philosophical work","Historical impact","Bibliography"]}
{"upwards":["Ren\u00e9 Descartes","Life"],"downwards":["Early life","Career","Death"]}
{"upwards":["Ren\u00e9 Descartes","Life","Early life"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Life","Career"],"downwards":["France","Netherlands","Sweden"]}
{"upwards":["Ren\u00e9 Descartes","Life","Career","France"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Life","Career","Netherlands"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Life","Career","Sweden"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Life","Death"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work"],"downwards":["Mind\u2013body dualism","Physiology and psychology","Moral philosophy","Religion","Natural science","Meteorology"]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Mind\u2013body dualism"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Physiology and psychology"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Moral philosophy"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Religion"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Natural science"],"downwards":["On animals"]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Natural science","On animals"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Philosophical work","Meteorology"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Historical impact"],"downwards":["Emancipation from Church doctrine","Mathematical legacy","Contemporary reception","Purported Rosicrucianism"]}
{"upwards":["Ren\u00e9 Descartes","Historical impact","Emancipation from Church doctrine"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Historical impact","Mathematical legacy"],"downwards":["Influence on Newton's mathematics"]}
{"upwards":["Ren\u00e9 Descartes","Historical impact","Mathematical legacy","Influence on Newton's mathematics"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Historical impact","Contemporary reception"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Historical impact","Purported Rosicrucianism"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Bibliography"],"downwards":["Writings","Collected editions","Early editions of specific works","Collected English translations","Translation of single works"]}
{"upwards":["Ren\u00e9 Descartes","Bibliography","Writings"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Bibliography","Collected editions"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Bibliography","Early editions of specific works"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Bibliography","Collected English translations"],"downwards":[]}
{"upwards":["Ren\u00e9 Descartes","Bibliography","Translation of single works"],"downwards":[]}
{"upwards":["Total order"],"downwards":["Strict and non-strict total orders","Examples","Chains","Further concepts","Orders on the Cartesian product of totally ordered sets","Related structures"]}
{"upwards":["Total order","Strict and non-strict total orders"],"downwards":[]}
{"upwards":["Total order","Examples"],"downwards":[]}
{"upwards":["Total order","Chains"],"downwards":[]}
{"upwards":["Total order","Further concepts"],"downwards":["Lattice theory","Finite total orders","Category theory","Order topology","Completeness","Sums of orders","Decidability"]}
{"upwards":["Total order","Further concepts","Lattice theory"],"downwards":[]}
{"upwards":["Total order","Further concepts","Finite total orders"],"downwards":[]}
{"upwards":["Total order","Further concepts","Category theory"],"downwards":[]}
{"upwards":["Total order","Further concepts","Order topology"],"downwards":[]}
{"upwards":["Total order","Further concepts","Completeness"],"downwards":[]}
{"upwards":["Total order","Further concepts","Sums of orders"],"downwards":[]}
{"upwards":["Total order","Further concepts","Decidability"],"downwards":[]}
{"upwards":["Total order","Orders on the Cartesian product of totally ordered sets"],"downwards":[]}
{"upwards":["Total order","Related structures"],"downwards":[]}
{"upwards":["nth root"],"downwards":["History","Definition and notation","Identities and properties","Simplified form of a radical expression","Infinite series","Computing principal roots","Geometric constructibility","Complex roots","Solving polynomials","Proof of irrationality for non-perfect nth power x"]}
{"upwards":["nth root","History"],"downwards":[]}
{"upwards":["nth root","Definition and notation"],"downwards":["Square roots","Cube roots"]}
{"upwards":["nth root","Definition and notation","Square roots"],"downwards":[]}
{"upwards":["nth root","Definition and notation","Cube roots"],"downwards":[]}
{"upwards":["nth root","Identities and properties"],"downwards":[]}
{"upwards":["nth root","Simplified form of a radical expression"],"downwards":[]}
{"upwards":["nth root","Infinite series"],"downwards":[]}
{"upwards":["nth root","Computing principal roots"],"downwards":["Using Newton's method","Digit-by-digit calculation of principal roots of decimal (base 10) numbers","Logarithmic calculation"]}
{"upwards":["nth root","Computing principal roots","Using Newton's method"],"downwards":[]}
{"upwards":["nth root","Computing principal roots","Digit-by-digit calculation of principal roots of decimal (base 10) numbers"],"downwards":["Examples"]}
{"upwards":["nth root","Computing principal roots","Digit-by-digit calculation of principal roots of decimal (base 10) numbers","Examples"],"downwards":[]}
{"upwards":["nth root","Computing principal roots","Logarithmic calculation"],"downwards":[]}
{"upwards":["nth root","Geometric constructibility"],"downwards":[]}
{"upwards":["nth root","Complex roots"],"downwards":["Square roots","Roots of unity","nth roots"]}
{"upwards":["nth root","Complex roots","Square roots"],"downwards":[]}
{"upwards":["nth root","Complex roots","Roots of unity"],"downwards":[]}
{"upwards":["nth root","Complex roots","nth roots"],"downwards":[]}
{"upwards":["nth root","Solving polynomials"],"downwards":[]}
{"upwards":["nth root","Proof of irrationality for non-perfect nth power x"],"downwards":[]}
{"upwards":["Transcendental number"],"downwards":["History","Properties","Numbers proven to be transcendental","Possible transcendental numbers","Sketch of a proof that e is transcendental"]}
{"upwards":["Transcendental number","History"],"downwards":[]}
{"upwards":["Transcendental number","Properties"],"downwards":[]}
{"upwards":["Transcendental number","Numbers proven to be transcendental"],"downwards":[]}
{"upwards":["Transcendental number","Possible transcendental numbers"],"downwards":[]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental"],"downwards":["The transcendence of \u03c0"]}
{"upwards":["Transcendental number","Sketch of a proof that e is transcendental","The transcendence of \u03c0"],"downwards":[]}
{"upwards":["Algebraic closure"],"downwards":["Examples","Existence of an algebraic closure and splitting fields","Separable closure"]}
{"upwards":["Algebraic closure","Examples"],"downwards":[]}
{"upwards":["Algebraic closure","Existence of an algebraic closure and splitting fields"],"downwards":[]}
{"upwards":["Algebraic closure","Separable closure"],"downwards":[]}
{"upwards":["Polynomial"],"downwards":["Etymology","Notation and terminology","Definition","Classification","Operations","Polynomial functions","Equations","Polynomial expressions","Related concepts","Polynomial ring","Applications","History"]}
{"upwards":["Polynomial","Etymology"],"downwards":[]}
{"upwards":["Polynomial","Notation and terminology"],"downwards":[]}
{"upwards":["Polynomial","Definition"],"downwards":[]}
{"upwards":["Polynomial","Classification"],"downwards":[]}
{"upwards":["Polynomial","Operations"],"downwards":["Addition and subtraction","Multiplication","Composition","Division","Factoring","Calculus"]}
{"upwards":["Polynomial","Operations","Addition and subtraction"],"downwards":[]}
{"upwards":["Polynomial","Operations","Multiplication"],"downwards":[]}
{"upwards":["Polynomial","Operations","Composition"],"downwards":[]}
{"upwards":["Polynomial","Operations","Division"],"downwards":[]}
{"upwards":["Polynomial","Operations","Factoring"],"downwards":[]}
{"upwards":["Polynomial","Operations","Calculus"],"downwards":[]}
{"upwards":["Polynomial","Polynomial functions"],"downwards":["Graphs"]}
{"upwards":["Polynomial","Polynomial functions","Graphs"],"downwards":[]}
{"upwards":["Polynomial","Equations"],"downwards":["Solving equations"]}
{"upwards":["Polynomial","Equations","Solving equations"],"downwards":[]}
{"upwards":["Polynomial","Polynomial expressions"],"downwards":["Trigonometric polynomials","Matrix polynomials","Exponential polynomials"]}
{"upwards":["Polynomial","Polynomial expressions","Trigonometric polynomials"],"downwards":[]}
{"upwards":["Polynomial","Polynomial expressions","Matrix polynomials"],"downwards":[]}
{"upwards":["Polynomial","Polynomial expressions","Exponential polynomials"],"downwards":[]}
{"upwards":["Polynomial","Related concepts"],"downwards":["Rational functions","Laurent polynomials","Power series"]}
{"upwards":["Polynomial","Related concepts","Rational functions"],"downwards":[]}
{"upwards":["Polynomial","Related concepts","Laurent polynomials"],"downwards":[]}
{"upwards":["Polynomial","Related concepts","Power series"],"downwards":[]}
{"upwards":["Polynomial","Polynomial ring"],"downwards":["Divisibility"]}
{"upwards":["Polynomial","Polynomial ring","Divisibility"],"downwards":[]}
{"upwards":["Polynomial","Applications"],"downwards":["Positional notation","Interpolation and approximation","Other applications"]}
{"upwards":["Polynomial","Applications","Positional notation"],"downwards":[]}
{"upwards":["Polynomial","Applications","Interpolation and approximation"],"downwards":[]}
{"upwards":["Polynomial","Applications","Other applications"],"downwards":[]}
{"upwards":["Polynomial","History"],"downwards":["History of the notation"]}
{"upwards":["Polynomial","History","History of the notation"],"downwards":[]}
{"upwards":["Quotient"],"downwards":["Notation","Integer part definition","Quotient of two integers","More general quotients"]}
{"upwards":["Quotient","Notation"],"downwards":[]}
{"upwards":["Quotient","Integer part definition"],"downwards":[]}
{"upwards":["Quotient","Quotient of two integers"],"downwards":[]}
{"upwards":["Quotient","More general quotients"],"downwards":[]}
{"upwards":["Axiom schema of specification"],"downwards":["Statement","Relation to the axiom schema of replacement","Unrestricted comprehension","In NBG class theory","In higher-order settings","In Quine's New Foundations","References"]}
{"upwards":["Axiom schema of specification","Statement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Relation to the axiom schema of replacement"],"downwards":[]}
{"upwards":["Axiom schema of specification","Unrestricted comprehension"],"downwards":[]}
{"upwards":["Axiom schema of specification","In NBG class theory"],"downwards":[]}
{"upwards":["Axiom schema of specification","In higher-order settings"],"downwards":[]}
{"upwards":["Axiom schema of specification","In Quine's New Foundations"],"downwards":[]}
{"upwards":["Axiom schema of specification","References"],"downwards":["Citations"]}
{"upwards":["Axiom schema of specification","References","Citations"],"downwards":[]}
{"upwards":["Paradox"],"downwards":["Introduction","Quine's classification","Ramsey's classification","In philosophy","In medicine"]}
{"upwards":["Paradox","Introduction"],"downwards":[]}
{"upwards":["Paradox","Quine's classification"],"downwards":[]}
{"upwards":["Paradox","Ramsey's classification"],"downwards":[]}
{"upwards":["Paradox","In philosophy"],"downwards":[]}
{"upwards":["Paradox","In medicine"],"downwards":[]}
{"upwards":["Domain of discourse"],"downwards":["Overview","Examples","Universe of discourse","Boole\u2019s 1854 definition"]}
{"upwards":["Domain of discourse","Overview"],"downwards":[]}
{"upwards":["Domain of discourse","Examples"],"downwards":[]}
{"upwards":["Domain of discourse","Universe of discourse"],"downwards":[]}
{"upwards":["Domain of discourse","Boole\u2019s 1854 definition"],"downwards":[]}
{"upwards":["David Hilbert"],"downwards":["Life","Contributions to mathematics and physics","Works"]}
{"upwards":["David Hilbert","Life"],"downwards":["Early life and education","Career","G\u00f6ttingen school","Personal life","Later years","Death"]}
{"upwards":["David Hilbert","Life","Early life and education"],"downwards":[]}
{"upwards":["David Hilbert","Life","Career"],"downwards":[]}
{"upwards":["David Hilbert","Life","G\u00f6ttingen school"],"downwards":[]}
{"upwards":["David Hilbert","Life","Personal life"],"downwards":[]}
{"upwards":["David Hilbert","Life","Later years"],"downwards":[]}
{"upwards":["David Hilbert","Life","Death"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics"],"downwards":["Hilbert solves Gordan's Problem","Axiomatization of geometry","The 23 problems","Formalism","Functional analysis","Physics","Number theory"]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Hilbert solves Gordan's Problem"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Axiomatization of geometry"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","The 23 problems"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Formalism"],"downwards":["Hilbert's program","G\u00f6del's work"]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Formalism","Hilbert's program"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Formalism","G\u00f6del's work"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Functional analysis"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Physics"],"downwards":[]}
{"upwards":["David Hilbert","Contributions to mathematics and physics","Number theory"],"downwards":[]}
{"upwards":["David Hilbert","Works"],"downwards":[]}
{"upwards":["Internal set theory"],"downwards":["Intuitive justification","Formal axioms for IST","Formal justification for the axioms","Related theories","Notes","References"]}
{"upwards":["Internal set theory","Intuitive justification"],"downwards":["Principles of the standard predicate"]}
{"upwards":["Internal set theory","Intuitive justification","Principles of the standard predicate"],"downwards":[]}
{"upwards":["Internal set theory","Formal axioms for IST"],"downwards":["I: Idealisation","S: Standardisation","T: Transfer"]}
{"upwards":["Internal set theory","Formal axioms for IST","I: Idealisation"],"downwards":["Applied to the relation \u2260","Applied to the relation <","Applied to the  relation \u2208"]}
{"upwards":["Internal set theory","Formal axioms for IST","I: Idealisation","Applied to the relation \u2260"],"downwards":[]}
{"upwards":["Internal set theory","Formal axioms for IST","I: Idealisation","Applied to the relation <"],"downwards":[]}
{"upwards":["Internal set theory","Formal axioms for IST","I: Idealisation","Applied to the  relation \u2208"],"downwards":[]}
{"upwards":["Internal set theory","Formal axioms for IST","S: Standardisation"],"downwards":[]}
{"upwards":["Internal set theory","Formal axioms for IST","T: Transfer"],"downwards":[]}
{"upwards":["Internal set theory","Formal justification for the axioms"],"downwards":[]}
{"upwards":["Internal set theory","Related theories"],"downwards":[]}
{"upwards":["Internal set theory","Notes"],"downwards":[]}
{"upwards":["Internal set theory","References"],"downwards":[]}
{"upwards":["Semantic Scholar"],"downwards":["Technology","Number of users and publications"]}
{"upwards":["Semantic Scholar","Technology"],"downwards":[]}
{"upwards":["Semantic Scholar","Number of users and publications"],"downwards":[]}
{"upwards":["Partially ordered set"],"downwards":["Partial order relations","Notation","Alternative definitions","Examples","Derived notions","Mappings between partially ordered sets","Number of partial orders","Linear extension","In category theory","Partial orders in topological spaces","Intervals"]}
{"upwards":["Partially ordered set","Partial order relations"],"downwards":["Partial orders","Strict partial orders","Correspondence of strict and non-strict partial order relations","Dual orders"]}
{"upwards":["Partially ordered set","Partial order relations","Partial orders"],"downwards":[]}
{"upwards":["Partially ordered set","Partial order relations","Strict partial orders"],"downwards":[]}
{"upwards":["Partially ordered set","Partial order relations","Correspondence of strict and non-strict partial order relations"],"downwards":[]}
{"upwards":["Partially ordered set","Partial order relations","Dual orders"],"downwards":[]}
{"upwards":["Partially ordered set","Notation"],"downwards":[]}
{"upwards":["Partially ordered set","Alternative definitions"],"downwards":[]}
{"upwards":["Partially ordered set","Examples"],"downwards":["Orders on the Cartesian product of partially ordered sets","Sums of partially ordered sets"]}
{"upwards":["Partially ordered set","Examples","Orders on the Cartesian product of partially ordered sets"],"downwards":[]}
{"upwards":["Partially ordered set","Examples","Sums of partially ordered sets"],"downwards":[]}
{"upwards":["Partially ordered set","Derived notions"],"downwards":["Extrema"]}
{"upwards":["Partially ordered set","Derived notions","Extrema"],"downwards":[]}
{"upwards":["Partially ordered set","Mappings between partially ordered sets"],"downwards":[]}
{"upwards":["Partially ordered set","Number of partial orders"],"downwards":[]}
{"upwards":["Partially ordered set","Linear extension"],"downwards":[]}
{"upwards":["Partially ordered set","In category theory"],"downwards":[]}
{"upwards":["Partially ordered set","Partial orders in topological spaces"],"downwards":[]}
{"upwards":["Partially ordered set","Intervals"],"downwards":[]}
{"upwards":["Mathematical Reviews"],"downwards":["Reviews","Online database","Mathematical citation quotient","Current Mathematical Publications"]}
{"upwards":["Mathematical Reviews","Reviews"],"downwards":[]}
{"upwards":["Mathematical Reviews","Online database"],"downwards":[]}
{"upwards":["Mathematical Reviews","Mathematical citation quotient"],"downwards":[]}
{"upwards":["Mathematical Reviews","Current Mathematical Publications"],"downwards":[]}
{"upwards":["Curry's paradox"],"downwards":["In natural language","Formal proof","Discussion","Resolution"]}
{"upwards":["Curry's paradox","In natural language"],"downwards":["Informal proof"]}
{"upwards":["Curry's paradox","In natural language","Informal proof"],"downwards":[]}
{"upwards":["Curry's paradox","Formal proof"],"downwards":["Sentential logic","Naive set theory","Lambda calculus","Combinatory logic"]}
{"upwards":["Curry's paradox","Formal proof","Sentential logic"],"downwards":[]}
{"upwards":["Curry's paradox","Formal proof","Naive set theory"],"downwards":[]}
{"upwards":["Curry's paradox","Formal proof","Lambda calculus"],"downwards":[]}
{"upwards":["Curry's paradox","Formal proof","Combinatory logic"],"downwards":[]}
{"upwards":["Curry's paradox","Discussion"],"downwards":["Consequences for some formal logic"]}
{"upwards":["Curry's paradox","Discussion","Consequences for some formal logic"],"downwards":[]}
{"upwards":["Curry's paradox","Resolution"],"downwards":["No resolution in lambda calculus","Resolution in unrestricted languages","Resolution in formal logic","Resolution in set theory"]}
{"upwards":["Curry's paradox","Resolution","No resolution in lambda calculus"],"downwards":["Domain of lambda calculus terms"]}
{"upwards":["Curry's paradox","Resolution","No resolution in lambda calculus","Domain of lambda calculus terms"],"downwards":[]}
{"upwards":["Curry's paradox","Resolution","Resolution in unrestricted languages"],"downwards":[]}
{"upwards":["Curry's paradox","Resolution","Resolution in formal logic"],"downwards":[]}
{"upwards":["Curry's paradox","Resolution","Resolution in set theory"],"downwards":[]}
{"upwards":["Antinomy"],"downwards":["Kant's use","Marx's use"]}
{"upwards":["Antinomy","Kant's use"],"downwards":[]}
{"upwards":["Antinomy","Marx's use"],"downwards":[]}
{"upwards":["Lebesgue measure"],"downwards":["Definition","Examples","Properties","Null sets","Construction of the Lebesgue measure","Relation to other measures"]}
{"upwards":["Lebesgue measure","Definition"],"downwards":["Intuition"]}
{"upwards":["Lebesgue measure","Definition","Intuition"],"downwards":[]}
{"upwards":["Lebesgue measure","Examples"],"downwards":[]}
{"upwards":["Lebesgue measure","Properties"],"downwards":[]}
{"upwards":["Lebesgue measure","Null sets"],"downwards":[]}
{"upwards":["Lebesgue measure","Construction of the Lebesgue measure"],"downwards":[]}
{"upwards":["Lebesgue measure","Relation to other measures"],"downwards":[]}
{"upwards":["Keith Devlin"],"downwards":["Biography","Career","Awards","Bibliography","References","External links"]}
{"upwards":["Keith Devlin","Biography"],"downwards":[]}
{"upwards":["Keith Devlin","Career"],"downwards":[]}
{"upwards":["Keith Devlin","Awards"],"downwards":[]}
{"upwards":["Keith Devlin","Bibliography"],"downwards":[]}
{"upwards":["Keith Devlin","References"],"downwards":[]}
{"upwards":["Keith Devlin","External links"],"downwards":[]}
{"upwards":["Zero of a function"],"downwards":["Solution of an equation","Polynomial roots","Computing roots","Zero set"]}
{"upwards":["Zero of a function","Solution of an equation"],"downwards":[]}
{"upwards":["Zero of a function","Polynomial roots"],"downwards":["Fundamental theorem of algebra"]}
{"upwards":["Zero of a function","Polynomial roots","Fundamental theorem of algebra"],"downwards":[]}
{"upwards":["Zero of a function","Computing roots"],"downwards":[]}
{"upwards":["Zero of a function","Zero set"],"downwards":["Applications"]}
{"upwards":["Zero of a function","Zero set","Applications"],"downwards":[]}
{"upwards":["Crelle's Journal"],"downwards":["History","Past editors","References","External links"]}
{"upwards":["Crelle's Journal","History"],"downwards":[]}
{"upwards":["Crelle's Journal","Past editors"],"downwards":[]}
{"upwards":["Crelle's Journal","References"],"downwards":[]}
{"upwards":["Crelle's Journal","External links"],"downwards":[]}
{"upwards":["Gottlob Frege"],"downwards":["Life","Work as a logician","Philosopher","Sense and reference","1924 diary","Personality","Important dates","Important works"]}
{"upwards":["Gottlob Frege","Life"],"downwards":["Childhood (1848\u201369)","Studies at University (1869\u201374)"]}
{"upwards":["Gottlob Frege","Life","Childhood (1848\u201369)"],"downwards":[]}
{"upwards":["Gottlob Frege","Life","Studies at University (1869\u201374)"],"downwards":[]}
{"upwards":["Gottlob Frege","Work as a logician"],"downwards":[]}
{"upwards":["Gottlob Frege","Philosopher"],"downwards":[]}
{"upwards":["Gottlob Frege","Sense and reference"],"downwards":[]}
{"upwards":["Gottlob Frege","1924 diary"],"downwards":[]}
{"upwards":["Gottlob Frege","Personality"],"downwards":[]}
{"upwards":["Gottlob Frege","Important dates"],"downwards":[]}
{"upwards":["Gottlob Frege","Important works"],"downwards":["Logic, foundation of arithmetic","Philosophical studies","Articles on geometry"]}
{"upwards":["Gottlob Frege","Important works","Logic, foundation of arithmetic"],"downwards":[]}
{"upwards":["Gottlob Frege","Important works","Philosophical studies"],"downwards":[]}
{"upwards":["Gottlob Frege","Important works","Articles on geometry"],"downwards":[]}
{"upwards":["Michoac\u00e1n"],"downwards":["History","Geography","Economy","Infrastructure","Education","Tourism","Crime","Demographics","Municipalities"]}
{"upwards":["Michoac\u00e1n","History"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Geography"],"downwards":["Flora","Fauna","Protected areas"]}
{"upwards":["Michoac\u00e1n","Geography","Flora"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Geography","Fauna"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Geography","Protected areas"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Economy"],"downwards":["Agriculture","Forestry","Fishing","Mining and minerals","Manufacturing"]}
{"upwards":["Michoac\u00e1n","Economy","Agriculture"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Economy","Forestry"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Economy","Fishing"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Economy","Mining and minerals"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Economy","Manufacturing"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Infrastructure"],"downwards":["Transport","Media"]}
{"upwards":["Michoac\u00e1n","Infrastructure","Transport"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Infrastructure","Media"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Education"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Tourism"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Crime"],"downwards":["Killings","Kidnappings and Ransoms","Drug Cartels","Police and Militia"]}
{"upwards":["Michoac\u00e1n","Crime","Killings"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Crime","Kidnappings and Ransoms"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Crime","Drug Cartels"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Crime","Police and Militia"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Demographics"],"downwards":["Largest cities"]}
{"upwards":["Michoac\u00e1n","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Michoac\u00e1n","Municipalities"],"downwards":["Major communities"]}
{"upwards":["Michoac\u00e1n","Municipalities","Major communities"],"downwards":[]}
{"upwards":["Numerical digit"],"downwards":["Overview","History","Modern digital systems","Digits in mathematics","History of ancient numbers","History of modern numbers"]}
{"upwards":["Numerical digit","Overview"],"downwards":["Digital values","Computation of place values"]}
{"upwards":["Numerical digit","Overview","Digital values"],"downwards":[]}
{"upwards":["Numerical digit","Overview","Computation of place values"],"downwards":[]}
{"upwards":["Numerical digit","History"],"downwards":["Other historical numeral systems using digits"]}
{"upwards":["Numerical digit","History","Other historical numeral systems using digits"],"downwards":[]}
{"upwards":["Numerical digit","Modern digital systems"],"downwards":["In computer science","Unusual systems"]}
{"upwards":["Numerical digit","Modern digital systems","In computer science"],"downwards":[]}
{"upwards":["Numerical digit","Modern digital systems","Unusual systems"],"downwards":[]}
{"upwards":["Numerical digit","Digits in mathematics"],"downwards":["Digital roots","Casting out nines","Repunits and repdigits","Palindromic numbers and Lychrel numbers"]}
{"upwards":["Numerical digit","Digits in mathematics","Digital roots"],"downwards":[]}
{"upwards":["Numerical digit","Digits in mathematics","Casting out nines"],"downwards":[]}
{"upwards":["Numerical digit","Digits in mathematics","Repunits and repdigits"],"downwards":[]}
{"upwards":["Numerical digit","Digits in mathematics","Palindromic numbers and Lychrel numbers"],"downwards":[]}
{"upwards":["Numerical digit","History of ancient numbers"],"downwards":[]}
{"upwards":["Numerical digit","History of modern numbers"],"downwards":["Numerals in most popular systems","Additional numerals"]}
{"upwards":["Numerical digit","History of modern numbers","Numerals in most popular systems"],"downwards":[]}
{"upwards":["Numerical digit","History of modern numbers","Additional numerals"],"downwards":[]}
{"upwards":["Decimal"],"downwards":["Origin","Decimal notation","Decimal fractions","Real number approximation","Infinite decimal expansion","Decimal computation","History"]}
{"upwards":["Decimal","Origin"],"downwards":[]}
{"upwards":["Decimal","Decimal notation"],"downwards":[]}
{"upwards":["Decimal","Decimal fractions"],"downwards":[]}
{"upwards":["Decimal","Real number approximation"],"downwards":[]}
{"upwards":["Decimal","Infinite decimal expansion"],"downwards":["Rational numbers"]}
{"upwards":["Decimal","Infinite decimal expansion","Rational numbers"],"downwards":[]}
{"upwards":["Decimal","Decimal computation"],"downwards":[]}
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{"upwards":["Decimal","History","History of decimal fractions"],"downwards":[]}
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{"upwards":["Fraction","Arithmetic with fractions","Division"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions"],"downwards":["Converting repeating decimals to fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
{"upwards":["Fraction","Algebraic fractions"],"downwards":[]}
{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Repeating decimal"],"downwards":["Background","Table of values"]}
{"upwards":["Repeating decimal","Background"],"downwards":["Notation","Decimal expansion and recurrence sequence","Every rational number is either a terminating or repeating decimal","Every repeating or terminating decimal is a rational number"]}
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{"upwards":["Repeating decimal","Background","Decimal expansion and recurrence sequence"],"downwards":[]}
{"upwards":["Repeating decimal","Background","Every rational number is either a terminating or repeating decimal"],"downwards":[]}
{"upwards":["Repeating decimal","Background","Every repeating or terminating decimal is a rational number"],"downwards":[]}
{"upwards":["Repeating decimal","Table of values"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
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{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
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{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Decimal representation"],"downwards":["Integer and fractional parts","Finite decimal approximations","Non-uniqueness of decimal representation and notational conventions","Types","Conversion to fraction"]}
{"upwards":["Decimal representation","Integer and fractional parts"],"downwards":[]}
{"upwards":["Decimal representation","Finite decimal approximations"],"downwards":[]}
{"upwards":["Decimal representation","Non-uniqueness of decimal representation and notational conventions"],"downwards":[]}
{"upwards":["Decimal representation","Types"],"downwards":["Finite","Infinite"]}
{"upwards":["Decimal representation","Types","Finite"],"downwards":[]}
{"upwards":["Decimal representation","Types","Infinite"],"downwards":["Repeating decimal representations"]}
{"upwards":["Decimal representation","Types","Infinite","Repeating decimal representations"],"downwards":[]}
{"upwards":["Decimal representation","Conversion to fraction"],"downwards":[]}
{"upwards":["John L. Kelley"],"downwards":["Books by Kelley","References","External links"]}
{"upwards":["John L. Kelley","Books by Kelley"],"downwards":[]}
{"upwards":["John L. Kelley","References"],"downwards":[]}
{"upwards":["John L. Kelley","External links"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
{"upwards":["Fraction","Vocabulary"],"downwards":[]}
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{"upwards":["Fraction","Forms of fractions","Historical notions"],"downwards":["Egyptian fraction","Complex and compound fractions"]}
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{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
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{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Jean van Heijenoort"],"downwards":["Life","Selected works","References","Bibliography","External links"]}
{"upwards":["Jean van Heijenoort","Life"],"downwards":["Political views","Academic work","Personal life"]}
{"upwards":["Jean van Heijenoort","Life","Political views"],"downwards":[]}
{"upwards":["Jean van Heijenoort","Life","Academic work"],"downwards":[]}
{"upwards":["Jean van Heijenoort","Life","Personal life"],"downwards":[]}
{"upwards":["Jean van Heijenoort","Selected works"],"downwards":[]}
{"upwards":["Jean van Heijenoort","References"],"downwards":[]}
{"upwards":["Jean van Heijenoort","Bibliography"],"downwards":[]}
{"upwards":["Jean van Heijenoort","External links"],"downwards":[]}
{"upwards":["Mudflow"],"downwards":["Triggering of mudflows","Mudflows and landslides","Mudflow geography","Largest recorded mudflow","Areas at risk"]}
{"upwards":["Mudflow","Triggering of mudflows"],"downwards":[]}
{"upwards":["Mudflow","Mudflows and landslides"],"downwards":[]}
{"upwards":["Mudflow","Mudflow geography"],"downwards":[]}
{"upwards":["Mudflow","Largest recorded mudflow"],"downwards":[]}
{"upwards":["Mudflow","Areas at risk"],"downwards":[]}
{"upwards":["Quadroon"],"downwards":["Etymology","Racial classifications","Depiction in media"]}
{"upwards":["Quadroon","Etymology"],"downwards":[]}
{"upwards":["Quadroon","Racial classifications"],"downwards":[]}
{"upwards":["Quadroon","Depiction in media"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season"],"downwards":["Seasonal forecasts","Seasonal summary","Systems","Storm names","Season effects"]}
{"upwards":["2018 Pacific hurricane season","Seasonal forecasts"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Seasonal summary"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems"],"downwards":["Tropical Depression One-E","Hurricane Aletta","Hurricane Bud","Tropical Storm Carlotta","Tropical Storm Daniel","Tropical Storm Emilia","Hurricane Fabio","Tropical Storm Gilma","Tropical Depression Nine-E","Hurricane Hector","Tropical Storm Ileana","Hurricane John","Tropical Storm Kristy","Hurricane Lane","Hurricane Miriam","Hurricane Norman","Hurricane Olivia","Tropical Storm Paul","Tropical Depression Nineteen-E","Hurricane Rosa","Hurricane Sergio","Hurricane Walaka","Tropical Storm Tara","Tropical Storm Vicente","Hurricane Willa","Tropical Storm Xavier","Other systems"]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Depression One-E"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Aletta"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Bud"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Carlotta"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Daniel"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Emilia"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Fabio"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Gilma"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Depression Nine-E"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Hector"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Ileana"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane John"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Kristy"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Lane"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Miriam"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Norman"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Olivia"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Paul"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Depression Nineteen-E"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Rosa"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Sergio"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Walaka"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Tara"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Vicente"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Hurricane Willa"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Tropical Storm Xavier"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Systems","Other systems"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Storm names"],"downwards":[]}
{"upwards":["2018 Pacific hurricane season","Season effects"],"downwards":[]}
{"upwards":["Tropical wave"],"downwards":["Characteristics","Atlantic","East Pacific","Screaming eagle waves"]}
{"upwards":["Tropical wave","Characteristics"],"downwards":[]}
{"upwards":["Tropical wave","Atlantic"],"downwards":[]}
{"upwards":["Tropical wave","East Pacific"],"downwards":[]}
{"upwards":["Tropical wave","Screaming eagle waves"],"downwards":[]}
{"upwards":["Plessy v. Ferguson"],"downwards":["Background","Decision","Aftermath","Significance"]}
{"upwards":["Plessy v. Ferguson","Background"],"downwards":["Legal background and incident","Trial","State appeal","Supreme Court appeal"]}
{"upwards":["Plessy v. Ferguson","Background","Legal background and incident"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Background","Trial"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Background","State appeal"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Background","Supreme Court appeal"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Decision"],"downwards":["Opinion of the Court","Harlan's dissent"]}
{"upwards":["Plessy v. Ferguson","Decision","Opinion of the Court"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Decision","Harlan's dissent"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Aftermath"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Significance"],"downwards":["Plessy and Ferguson Foundation","Pardon"]}
{"upwards":["Plessy v. Ferguson","Significance","Plessy and Ferguson Foundation"],"downwards":[]}
{"upwards":["Plessy v. Ferguson","Significance","Pardon"],"downwards":[]}
{"upwards":["Bar (unit)"],"downwards":["Definition and conversion","Origin","Usage"]}
{"upwards":["Bar (unit)","Definition and conversion"],"downwards":[]}
{"upwards":["Bar (unit)","Origin"],"downwards":[]}
{"upwards":["Bar (unit)","Usage"],"downwards":[]}
{"upwards":["1981"],"downwards":["Events","Births and deaths","Nobel Prizes","References","External links"]}
{"upwards":["1981","Events"],"downwards":["January","February","March","April","May","June","July","August","September","October","November","December","Date unknown"]}
{"upwards":["1981","Events","January"],"downwards":[]}
{"upwards":["1981","Events","February"],"downwards":[]}
{"upwards":["1981","Events","March"],"downwards":[]}
{"upwards":["1981","Events","April"],"downwards":[]}
{"upwards":["1981","Events","May"],"downwards":[]}
{"upwards":["1981","Events","June"],"downwards":[]}
{"upwards":["1981","Events","July"],"downwards":[]}
{"upwards":["1981","Events","August"],"downwards":[]}
{"upwards":["1981","Events","September"],"downwards":[]}
{"upwards":["1981","Events","October"],"downwards":[]}
{"upwards":["1981","Events","November"],"downwards":[]}
{"upwards":["1981","Events","December"],"downwards":[]}
{"upwards":["1981","Events","Date unknown"],"downwards":[]}
{"upwards":["1981","Births and deaths"],"downwards":[]}
{"upwards":["1981","Nobel Prizes"],"downwards":[]}
{"upwards":["1981","References"],"downwards":[]}
{"upwards":["1981","External links"],"downwards":[]}
{"upwards":["Veracruz"],"downwards":["Etymology","Geography","History","Economy","Culture","Education","Demographics","Tourism","Archeological sites","Government","Infrastructure","Major communities","Notable People"]}
{"upwards":["Veracruz","Etymology"],"downwards":[]}
{"upwards":["Veracruz","Geography"],"downwards":["Political geography","Natural geography","Climate","Ecosystems"]}
{"upwards":["Veracruz","Geography","Political geography"],"downwards":[]}
{"upwards":["Veracruz","Geography","Natural geography"],"downwards":[]}
{"upwards":["Veracruz","Geography","Climate"],"downwards":[]}
{"upwards":["Veracruz","Geography","Ecosystems"],"downwards":[]}
{"upwards":["Veracruz","History"],"downwards":["Pre-Columbian","Colonial period, 1519\u20131821","Independence","20th century to the present"]}
{"upwards":["Veracruz","History","Pre-Columbian"],"downwards":[]}
{"upwards":["Veracruz","History","Colonial period, 1519\u20131821"],"downwards":[]}
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{"upwards":["Veracruz","History","20th century to the present"],"downwards":[]}
{"upwards":["Veracruz","Economy"],"downwards":["Agriculture","Natural resources","Golden Lane Oil Fields","Industry, transportation and commerce","Handcrafts"]}
{"upwards":["Veracruz","Economy","Agriculture"],"downwards":[]}
{"upwards":["Veracruz","Economy","Natural resources"],"downwards":[]}
{"upwards":["Veracruz","Economy","Golden Lane Oil Fields"],"downwards":[]}
{"upwards":["Veracruz","Economy","Industry, transportation and commerce"],"downwards":[]}
{"upwards":["Veracruz","Economy","Handcrafts"],"downwards":[]}
{"upwards":["Veracruz","Culture"],"downwards":["Gastronomy","Museums","Fairs and festivals","Dance and music","Art and architecture","Literature","Religion"]}
{"upwards":["Veracruz","Culture","Gastronomy"],"downwards":[]}
{"upwards":["Veracruz","Culture","Museums"],"downwards":[]}
{"upwards":["Veracruz","Culture","Fairs and festivals"],"downwards":[]}
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{"upwards":["Veracruz","Culture","Religion"],"downwards":[]}
{"upwards":["Veracruz","Education"],"downwards":[]}
{"upwards":["Veracruz","Demographics"],"downwards":["Largest cities"]}
{"upwards":["Veracruz","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Veracruz","Tourism"],"downwards":[]}
{"upwards":["Veracruz","Archeological sites"],"downwards":[]}
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{"upwards":["Veracruz","Infrastructure","Transport"],"downwards":[]}
{"upwards":["Veracruz","Infrastructure","Media"],"downwards":[]}
{"upwards":["Veracruz","Major communities"],"downwards":[]}
{"upwards":["Veracruz","Notable People"],"downwards":[]}
{"upwards":["Israeli Air Force"],"downwards":["Mission statement","Insignia","History","Organization","Aircraft","Pilot selection and training","List of IAF commanders"]}
{"upwards":["Israeli Air Force","Mission statement"],"downwards":[]}
{"upwards":["Israeli Air Force","Insignia"],"downwards":[]}
{"upwards":["Israeli Air Force","History"],"downwards":["Early years (1948\u20131967)","Suez Crisis (1956)","Six-Day War (1967)","War of Attrition","Yom Kippur War (1973)","Expansion (1973\u20131982)","1982 Lebanon War and aftermath","1990s and beyond","2006 Lebanon War","2007 Operation Outside the Box","Against Hamas","Incidents during the Syrian Civil War"]}
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{"upwards":["Israeli Air Force","History","Suez Crisis (1956)"],"downwards":[]}
{"upwards":["Israeli Air Force","History","Six-Day War (1967)"],"downwards":[]}
{"upwards":["Israeli Air Force","History","War of Attrition"],"downwards":[]}
{"upwards":["Israeli Air Force","History","Yom Kippur War (1973)"],"downwards":[]}
{"upwards":["Israeli Air Force","History","Expansion (1973\u20131982)"],"downwards":[]}
{"upwards":["Israeli Air Force","History","1982 Lebanon War and aftermath"],"downwards":[]}
{"upwards":["Israeli Air Force","History","1990s and beyond"],"downwards":[]}
{"upwards":["Israeli Air Force","History","2006 Lebanon War"],"downwards":[]}
{"upwards":["Israeli Air Force","History","2007 Operation Outside the Box"],"downwards":[]}
{"upwards":["Israeli Air Force","History","Against Hamas"],"downwards":[]}
{"upwards":["Israeli Air Force","History","Incidents during the Syrian Civil War"],"downwards":[]}
{"upwards":["Israeli Air Force","Organization"],"downwards":["Administrative organization","Operational organization"]}
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{"upwards":["Israeli Air Force","Organization","Operational organization"],"downwards":[]}
{"upwards":["Israeli Air Force","Aircraft"],"downwards":["Current inventory","Historic","Munitions and space systems"]}
{"upwards":["Israeli Air Force","Aircraft","Current inventory"],"downwards":[]}
{"upwards":["Israeli Air Force","Aircraft","Historic"],"downwards":[]}
{"upwards":["Israeli Air Force","Aircraft","Munitions and space systems"],"downwards":[]}
{"upwards":["Israeli Air Force","Pilot selection and training"],"downwards":["Ranks"]}
{"upwards":["Israeli Air Force","Pilot selection and training","Ranks"],"downwards":["Officers","Enlisted"]}
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{"upwards":["Israeli Air Force","Pilot selection and training","Ranks","Enlisted"],"downwards":[]}
{"upwards":["Israeli Air Force","List of IAF commanders"],"downwards":[]}
{"upwards":["Jalisco"],"downwards":["Geography","Demographics","Government and regions","History","Tourism","Culture","Economy","Education","Communication","Infrastructure","Sports","Twinning and covenants"]}
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{"upwards":["Jalisco","Demographics"],"downwards":["Largest cities"]}
{"upwards":["Jalisco","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Jalisco","Government and regions"],"downwards":[]}
{"upwards":["Jalisco","History"],"downwards":["Nomenclature and seal","Pre-Hispanic period","Colonial period","Independence","Mexican Revolution to present"]}
{"upwards":["Jalisco","History","Nomenclature and seal"],"downwards":[]}
{"upwards":["Jalisco","History","Pre-Hispanic period"],"downwards":[]}
{"upwards":["Jalisco","History","Colonial period"],"downwards":[]}
{"upwards":["Jalisco","History","Independence"],"downwards":[]}
{"upwards":["Jalisco","History","Mexican Revolution to present"],"downwards":[]}
{"upwards":["Jalisco","Tourism"],"downwards":[]}
{"upwards":["Jalisco","Culture"],"downwards":["\"Jalisco is Mexico\"","Mariachi and other music","Traditional clothing and dance","Tequila","Cuisine","Catholic faith"]}
{"upwards":["Jalisco","Culture","\"Jalisco is Mexico\""],"downwards":[]}
{"upwards":["Jalisco","Culture","Mariachi and other music"],"downwards":[]}
{"upwards":["Jalisco","Culture","Traditional clothing and dance"],"downwards":["Dresses"]}
{"upwards":["Jalisco","Culture","Traditional clothing and dance","Dresses"],"downwards":[]}
{"upwards":["Jalisco","Culture","Tequila"],"downwards":[]}
{"upwards":["Jalisco","Culture","Cuisine"],"downwards":[]}
{"upwards":["Jalisco","Culture","Catholic faith"],"downwards":[]}
{"upwards":["Jalisco","Economy"],"downwards":[]}
{"upwards":["Jalisco","Education"],"downwards":[]}
{"upwards":["Jalisco","Communication"],"downwards":[]}
{"upwards":["Jalisco","Infrastructure"],"downwards":["Media","Transport"]}
{"upwards":["Jalisco","Infrastructure","Media"],"downwards":[]}
{"upwards":["Jalisco","Infrastructure","Transport"],"downwards":[]}
{"upwards":["Jalisco","Sports"],"downwards":[]}
{"upwards":["Jalisco","Twinning and covenants"],"downwards":[]}
{"upwards":["Oaxaca"],"downwards":["History","Geography","Demographics","Government and political geography","Economy","Transport","Media","Culture","Education","Health","Sports","Notable people from Oaxaca"]}
{"upwards":["Oaxaca","History"],"downwards":["Name","Prehistoric and pre-Hispanic period","Spanish colonization","Independence","Late 19th century to present"]}
{"upwards":["Oaxaca","History","Name"],"downwards":[]}
{"upwards":["Oaxaca","History","Prehistoric and pre-Hispanic period"],"downwards":[]}
{"upwards":["Oaxaca","History","Spanish colonization"],"downwards":[]}
{"upwards":["Oaxaca","History","Independence"],"downwards":[]}
{"upwards":["Oaxaca","History","Late 19th century to present"],"downwards":[]}
{"upwards":["Oaxaca","Geography"],"downwards":["Regions, districts and major communities","Climate","Nature and conservation"]}
{"upwards":["Oaxaca","Geography","Regions, districts and major communities"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Climate"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation"],"downwards":["Lagunas de Chacahua National Park","Benito Ju\u00e1rez National Park","Huatulco National Park","Tehuac\u00e1n-Cuicatl\u00e1n Biosphere Reserve"]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Lagunas de Chacahua National Park"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Benito Ju\u00e1rez National Park"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Huatulco National Park"],"downwards":[]}
{"upwards":["Oaxaca","Geography","Nature and conservation","Tehuac\u00e1n-Cuicatl\u00e1n Biosphere Reserve"],"downwards":[]}
{"upwards":["Oaxaca","Demographics"],"downwards":["Largest cities","Overview","Indigenous peoples","Afro-Mexicans","Religion"]}
{"upwards":["Oaxaca","Demographics","Largest cities"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Overview"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples"],"downwards":["Zapotecs","Mixtecs","Mazatecos","Mixe","Other"]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Zapotecs"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Mixtecs"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Mazatecos"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Mixe"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Indigenous peoples","Other"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Afro-Mexicans"],"downwards":[]}
{"upwards":["Oaxaca","Demographics","Religion"],"downwards":[]}
{"upwards":["Oaxaca","Government and political geography"],"downwards":[]}
{"upwards":["Oaxaca","Economy"],"downwards":["Migration","Agriculture","Mining and industry","Commerce"]}
{"upwards":["Oaxaca","Economy","Migration"],"downwards":[]}
{"upwards":["Oaxaca","Economy","Agriculture"],"downwards":[]}
{"upwards":["Oaxaca","Economy","Mining and industry"],"downwards":[]}
{"upwards":["Oaxaca","Economy","Commerce"],"downwards":[]}
{"upwards":["Oaxaca","Transport"],"downwards":["Road, rail and sea","Air","Local transportation services"]}
{"upwards":["Oaxaca","Transport","Road, rail and sea"],"downwards":[]}
{"upwards":["Oaxaca","Transport","Air"],"downwards":[]}
{"upwards":["Oaxaca","Transport","Local transportation services"],"downwards":[]}
{"upwards":["Oaxaca","Media"],"downwards":[]}
{"upwards":["Oaxaca","Culture"],"downwards":["Arts","Food and drink","Landmarks and tourist attractions","Handicrafts","Symbols"]}
{"upwards":["Oaxaca","Culture","Arts"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Food and drink"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Landmarks and tourist attractions"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Handicrafts"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Symbols"],"downwards":["Flag","Coat of Arms","State emblem"]}
{"upwards":["Oaxaca","Culture","Symbols","Flag"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Symbols","Coat of Arms"],"downwards":[]}
{"upwards":["Oaxaca","Culture","Symbols","State emblem"],"downwards":[]}
{"upwards":["Oaxaca","Education"],"downwards":["Primary education","Higher education"]}
{"upwards":["Oaxaca","Education","Primary education"],"downwards":[]}
{"upwards":["Oaxaca","Education","Higher education"],"downwards":[]}
{"upwards":["Oaxaca","Health"],"downwards":[]}
{"upwards":["Oaxaca","Sports"],"downwards":[]}
{"upwards":["Oaxaca","Notable people from Oaxaca"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam"],"downwards":["Background","Destruction","Attribution","Consequences","Reactions"]}
{"upwards":["Destruction of the Kakhovka Dam","Background"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Destruction"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Attribution"],"downwards":["Russia","Ukraine"]}
{"upwards":["Destruction of the Kakhovka Dam","Attribution","Russia"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Attribution","Ukraine"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences"],"downwards":["Flooding and evacuations","Animals and environment","Counteroffensive","Water supply"]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Flooding and evacuations"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Animals and environment"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Counteroffensive"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Consequences","Water supply"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions"],"downwards":["Russia","Ukraine","International community"]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions","Russia"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions","Ukraine"],"downwards":[]}
{"upwards":["Destruction of the Kakhovka Dam","Reactions","International community"],"downwards":[]}
{"upwards":["Governor (India)"],"downwards":["Selection process","Appointment","Powers and functions","Contingency situation","Emoluments","Removal","Legal immunity","Analysis of role in Government"]}
{"upwards":["Governor (India)","Selection process"],"downwards":["Qualifications"]}
{"upwards":["Governor (India)","Selection process","Qualifications"],"downwards":[]}
{"upwards":["Governor (India)","Appointment"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions"],"downwards":["Executive powers","Legislative powers","Financial powers","Discretionary powers"]}
{"upwards":["Governor (India)","Powers and functions","Executive powers"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions","Legislative powers"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions","Financial powers"],"downwards":[]}
{"upwards":["Governor (India)","Powers and functions","Discretionary powers"],"downwards":[]}
{"upwards":["Governor (India)","Contingency situation"],"downwards":[]}
{"upwards":["Governor (India)","Emoluments"],"downwards":[]}
{"upwards":["Governor (India)","Removal"],"downwards":[]}
{"upwards":["Governor (India)","Legal immunity"],"downwards":[]}
{"upwards":["Governor (India)","Analysis of role in Government"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction"],"downwards":["History","Public perception"]}
{"upwards":["Iraq and weapons of mass destruction","History"],"downwards":["Program development 1960s\u20131980s","Iran\u2013Iraq War","1991 Gulf War","UN inspections","2003 Iraq War","2009 Declaration"]}
{"upwards":["Iraq and weapons of mass destruction","History","Program development 1960s\u20131980s"],"downwards":["Western help with Iraq's WMD program","United States","United Kingdom"]}
{"upwards":["Iraq and weapons of mass destruction","History","Program development 1960s\u20131980s","Western help with Iraq's WMD program"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","Program development 1960s\u20131980s","United States"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","Program development 1960s\u20131980s","United Kingdom"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","Iran\u2013Iraq War"],"downwards":["Chemical weapon attacks"]}
{"upwards":["Iraq and weapons of mass destruction","History","Iran\u2013Iraq War","Chemical weapon attacks"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","1991 Gulf War"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","UN inspections"],"downwards":["UNSCOM inspections 1991\u20131998","Between inspections: 1998\u20132003"]}
{"upwards":["Iraq and weapons of mass destruction","History","UN inspections","UNSCOM inspections 1991\u20131998"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","UN inspections","Between inspections: 1998\u20132003"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War"],"downwards":["Prelude","Legal justification","Coalition expanded intelligence","Iraq Survey Group","Captured documents","Post-war discoveries and incidents","2005: Operation Avarice","2006: House Armed Services Committee Hearing","The New York Times investigative report"]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","Prelude"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","Legal justification"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","Coalition expanded intelligence"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","Iraq Survey Group"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","Captured documents"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","Post-war discoveries and incidents"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","2005: Operation Avarice"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","2006: House Armed Services Committee Hearing"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2003 Iraq War","The New York Times investigative report"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","History","2009 Declaration"],"downwards":[]}
{"upwards":["Iraq and weapons of mass destruction","Public perception"],"downwards":[]}
{"upwards":["Hidalgo (state)"],"downwards":["History","Demographics","Culture","Native gastronomy","Tourism","Geography","Ecology","Economy","Education","Infrastructure","Major communities"]}
{"upwards":["Hidalgo (state)","History"],"downwards":[]}
{"upwards":["Hidalgo (state)","Demographics"],"downwards":[]}
{"upwards":["Hidalgo (state)","Culture"],"downwards":[]}
{"upwards":["Hidalgo (state)","Native gastronomy"],"downwards":[]}
{"upwards":["Hidalgo (state)","Tourism"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography"],"downwards":["Mountains","Valleys","Water","Climate"]}
{"upwards":["Hidalgo (state)","Geography","Mountains"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography","Valleys"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography","Water"],"downwards":[]}
{"upwards":["Hidalgo (state)","Geography","Climate"],"downwards":[]}
{"upwards":["Hidalgo (state)","Ecology"],"downwards":[]}
{"upwards":["Hidalgo (state)","Economy"],"downwards":[]}
{"upwards":["Hidalgo (state)","Education"],"downwards":[]}
{"upwards":["Hidalgo (state)","Infrastructure"],"downwards":[]}
{"upwards":["Hidalgo (state)","Major communities"],"downwards":[]}
{"upwards":["Operation Opera"],"downwards":["Iraq's nuclear program","Strategy and diplomacy","Attack","International political reactions","Assessment"]}
{"upwards":["Operation Opera","Iraq's nuclear program"],"downwards":[]}
{"upwards":["Operation Opera","Strategy and diplomacy"],"downwards":["Iranian attack","Operational planning","Preliminary Israeli\/Iranian actions"]}
{"upwards":["Operation Opera","Strategy and diplomacy","Iranian attack"],"downwards":[]}
{"upwards":["Operation Opera","Strategy and diplomacy","Operational planning"],"downwards":[]}
{"upwards":["Operation Opera","Strategy and diplomacy","Preliminary Israeli\/Iranian actions"],"downwards":[]}
{"upwards":["Operation Opera","Attack"],"downwards":[]}
{"upwards":["Operation Opera","International political reactions"],"downwards":["Aftermath"]}
{"upwards":["Operation Opera","International political reactions","Aftermath"],"downwards":[]}
{"upwards":["Operation Opera","Assessment"],"downwards":[]}
{"upwards":["Balasore"],"downwards":["History","Geography","Demographics","Government and politics","Economy","Culture\/Cityscape","Transport","Education","Sports and entertainment","Notable people"]}
{"upwards":["Balasore","History"],"downwards":[]}
{"upwards":["Balasore","Geography"],"downwards":["Climate"]}
{"upwards":["Balasore","Geography","Climate"],"downwards":[]}
{"upwards":["Balasore","Demographics"],"downwards":["Languages"]}
{"upwards":["Balasore","Demographics","Languages"],"downwards":[]}
{"upwards":["Balasore","Government and politics"],"downwards":["Civic Utility \/ Amenities \/ Services"]}
{"upwards":["Balasore","Government and politics","Civic Utility \/ Amenities \/ Services"],"downwards":["Healthcare"]}
{"upwards":["Balasore","Government and politics","Civic Utility \/ Amenities \/ Services","Healthcare"],"downwards":[]}
{"upwards":["Balasore","Economy"],"downwards":["Industry"]}
{"upwards":["Balasore","Economy","Industry"],"downwards":[]}
{"upwards":["Balasore","Culture\/Cityscape"],"downwards":["Festivals","Places of interest"]}
{"upwards":["Balasore","Culture\/Cityscape","Festivals"],"downwards":[]}
{"upwards":["Balasore","Culture\/Cityscape","Places of interest"],"downwards":[]}
{"upwards":["Balasore","Transport"],"downwards":["Air","Rail","Road"]}
{"upwards":["Balasore","Transport","Air"],"downwards":[]}
{"upwards":["Balasore","Transport","Rail"],"downwards":[]}
{"upwards":["Balasore","Transport","Road"],"downwards":[]}
{"upwards":["Balasore","Education"],"downwards":["Technical colleges\/institutions","Universities\/colleges","Schools"]}
{"upwards":["Balasore","Education","Technical colleges\/institutions"],"downwards":[]}
{"upwards":["Balasore","Education","Universities\/colleges"],"downwards":[]}
{"upwards":["Balasore","Education","Schools"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment"],"downwards":["Permit Field","Reserved Police Line Ground","Zilla School Stadium","ITI Field","Entertainment"]}
{"upwards":["Balasore","Sports and entertainment","Permit Field"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","Reserved Police Line Ground"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","Zilla School Stadium"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","ITI Field"],"downwards":[]}
{"upwards":["Balasore","Sports and entertainment","Entertainment"],"downwards":[]}
{"upwards":["Balasore","Notable people"],"downwards":[]}
{"upwards":["Pickup truck"],"downwards":["History","International markets","Design and features","Safety","Uses","Gallery"]}
{"upwards":["Pickup truck","History"],"downwards":[]}
{"upwards":["Pickup truck","International markets"],"downwards":[]}
{"upwards":["Pickup truck","Design and features"],"downwards":[]}
{"upwards":["Pickup truck","Safety"],"downwards":[]}
{"upwards":["Pickup truck","Uses"],"downwards":[]}
{"upwards":["Pickup truck","Gallery"],"downwards":[]}
{"upwards":["Geoff Bent"],"downwards":["Early life","Manchester United career","Munich air disaster","Legacy","Notes, citations and sources","External links"]}
{"upwards":["Geoff Bent","Early life"],"downwards":[]}
{"upwards":["Geoff Bent","Manchester United career"],"downwards":[]}
{"upwards":["Geoff Bent","Munich air disaster"],"downwards":[]}
{"upwards":["Geoff Bent","Legacy"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources"],"downwards":["Notes","Citations","Sources"]}
{"upwards":["Geoff Bent","Notes, citations and sources","Notes"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Citations"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources"],"downwards":["Books","Newspapers","Online"]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources","Books"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources","Newspapers"],"downwards":[]}
{"upwards":["Geoff Bent","Notes, citations and sources","Sources","Online"],"downwards":[]}
{"upwards":["Geoff Bent","External links"],"downwards":[]}
{"upwards":["1998"],"downwards":["Events","Births and deaths","Nobel Prizes","Fields Medal","References","External links"]}
{"upwards":["1998","Events"],"downwards":["January","February","March","April","May","June","July","August","September","October","November","December","Date unknown"]}
{"upwards":["1998","Events","January"],"downwards":[]}
{"upwards":["1998","Events","February"],"downwards":[]}
{"upwards":["1998","Events","March"],"downwards":[]}
{"upwards":["1998","Events","April"],"downwards":[]}
{"upwards":["1998","Events","May"],"downwards":[]}
{"upwards":["1998","Events","June"],"downwards":[]}
{"upwards":["1998","Events","July"],"downwards":[]}
{"upwards":["1998","Events","August"],"downwards":[]}
{"upwards":["1998","Events","September"],"downwards":[]}
{"upwards":["1998","Events","October"],"downwards":[]}
{"upwards":["1998","Events","November"],"downwards":[]}
{"upwards":["1998","Events","December"],"downwards":[]}
{"upwards":["1998","Events","Date unknown"],"downwards":[]}
{"upwards":["1998","Births and deaths"],"downwards":[]}
{"upwards":["1998","Nobel Prizes"],"downwards":[]}
{"upwards":["1998","Fields Medal"],"downwards":[]}
{"upwards":["1998","References"],"downwards":[]}
{"upwards":["1998","External links"],"downwards":[]}
{"upwards":["Colima"],"downwards":["Geography","Demographics","Principal communities","Economy","Tourist attractions","Culture","Communications and transportation","History","Archaeology","Education","Sister cities"]}
{"upwards":["Colima","Geography"],"downwards":[]}
{"upwards":["Colima","Demographics"],"downwards":[]}
{"upwards":["Colima","Principal communities"],"downwards":[]}
{"upwards":["Colima","Economy"],"downwards":[]}
{"upwards":["Colima","Tourist attractions"],"downwards":[]}
{"upwards":["Colima","Culture"],"downwards":[]}
{"upwards":["Colima","Communications and transportation"],"downwards":["Transport","Media"]}
{"upwards":["Colima","Communications and transportation","Transport"],"downwards":[]}
{"upwards":["Colima","Communications and transportation","Media"],"downwards":[]}
{"upwards":["Colima","History"],"downwards":["Name and seal","Pre-Hispanic period","Colonial era","Independence to the present"]}
{"upwards":["Colima","History","Name and seal"],"downwards":[]}
{"upwards":["Colima","History","Pre-Hispanic period"],"downwards":[]}
{"upwards":["Colima","History","Colonial era"],"downwards":[]}
{"upwards":["Colima","History","Independence to the present"],"downwards":[]}
{"upwards":["Colima","Archaeology"],"downwards":[]}
{"upwards":["Colima","Education"],"downwards":[]}
{"upwards":["Colima","Sister cities"],"downwards":[]}
{"upwards":["Khartoum"],"downwards":["Etymology","History","Geography","Climate","Demographics","Economy","Education","Transportation","Architecture","Culture","References","Bibliography","External links"]}
{"upwards":["Khartoum","Etymology"],"downwards":[]}
{"upwards":["Khartoum","History"],"downwards":["19th century","20th century","21st century"]}
{"upwards":["Khartoum","History","19th century"],"downwards":[]}
{"upwards":["Khartoum","History","20th century"],"downwards":[]}
{"upwards":["Khartoum","History","21st century"],"downwards":[]}
{"upwards":["Khartoum","Geography"],"downwards":["Location"]}
{"upwards":["Khartoum","Geography","Location"],"downwards":[]}
{"upwards":["Khartoum","Climate"],"downwards":[]}
{"upwards":["Khartoum","Demographics"],"downwards":[]}
{"upwards":["Khartoum","Economy"],"downwards":["Retailing"]}
{"upwards":["Khartoum","Economy","Retailing"],"downwards":[]}
{"upwards":["Khartoum","Education"],"downwards":[]}
{"upwards":["Khartoum","Transportation"],"downwards":[]}
{"upwards":["Khartoum","Architecture"],"downwards":[]}
{"upwards":["Khartoum","Culture"],"downwards":["Museums","Botanical gardens","Clubs","Places of worship"]}
{"upwards":["Khartoum","Culture","Museums"],"downwards":[]}
{"upwards":["Khartoum","Culture","Botanical gardens"],"downwards":[]}
{"upwards":["Khartoum","Culture","Clubs"],"downwards":[]}
{"upwards":["Khartoum","Culture","Places of worship"],"downwards":[]}
{"upwards":["Khartoum","References"],"downwards":[]}
{"upwards":["Khartoum","Bibliography"],"downwards":[]}
{"upwards":["Khartoum","External links"],"downwards":[]}
{"upwards":["Busby Babes"],"downwards":["History"]}
{"upwards":["Busby Babes","History"],"downwards":[]}
{"upwards":["President of India"],"downwards":["Origin","Powers and duties","Selection process","Time of election","Emoluments","Impeachment","Succession","President vs Chief Justice","President vs Prime minister","Important presidential interventions in the past"]}
{"upwards":["President of India","Origin"],"downwards":[]}
{"upwards":["President of India","Powers and duties"],"downwards":["Duty","Legislative powers","Executive powers","Judicial powers","Appointment powers","Financial powers","Diplomatic powers","Military powers","Pardoning powers","Emergency powers"]}
{"upwards":["President of India","Powers and duties","Duty"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Legislative powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Executive powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Judicial powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Appointment powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Financial powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Diplomatic powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Military powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Pardoning powers"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Emergency powers"],"downwards":["National emergency","State emergency","Financial emergency"]}
{"upwards":["President of India","Powers and duties","Emergency powers","National emergency"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Emergency powers","State emergency"],"downwards":[]}
{"upwards":["President of India","Powers and duties","Emergency powers","Financial emergency"],"downwards":[]}
{"upwards":["President of India","Selection process"],"downwards":["Eligibility"]}
{"upwards":["President of India","Selection process","Eligibility"],"downwards":[]}
{"upwards":["President of India","Time of election"],"downwards":["Conditions for the presidency","Election process","Oath or affirmation"]}
{"upwards":["President of India","Time of election","Conditions for the presidency"],"downwards":[]}
{"upwards":["President of India","Time of election","Election process"],"downwards":[]}
{"upwards":["President of India","Time of election","Oath or affirmation"],"downwards":[]}
{"upwards":["President of India","Emoluments"],"downwards":[]}
{"upwards":["President of India","Impeachment"],"downwards":[]}
{"upwards":["President of India","Succession"],"downwards":[]}
{"upwards":["President of India","President vs Chief Justice"],"downwards":[]}
{"upwards":["President of India","President vs Prime minister"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past"],"downwards":["Proving majority in the parliament","Proof of Majority to form a Government","Vetoing of a Bill","Rashtrapati Bhavan Communiqu\u00e9s","Return of a Bill","Sacking state governors"]}
{"upwards":["President of India","Important presidential interventions in the past","Proving majority in the parliament"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Proof of Majority to form a Government"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Vetoing of a Bill"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Rashtrapati Bhavan Communiqu\u00e9s"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Return of a Bill"],"downwards":[]}
{"upwards":["President of India","Important presidential interventions in the past","Sacking state governors"],"downwards":[]}
{"upwards":["White supremacy"],"downwards":["History","Ideologies and movements","Academic use of the term"]}
{"upwards":["White supremacy","History"],"downwards":["United States","British Commonwealth","Germany","Australia and New Zealand"]}
{"upwards":["White supremacy","History","United States"],"downwards":["Early history","20th century","21st century","Patterns of influence"]}
{"upwards":["White supremacy","History","United States","Early history"],"downwards":[]}
{"upwards":["White supremacy","History","United States","20th century"],"downwards":[]}
{"upwards":["White supremacy","History","United States","21st century"],"downwards":[]}
{"upwards":["White supremacy","History","United States","Patterns of influence"],"downwards":[]}
{"upwards":["White supremacy","History","British Commonwealth"],"downwards":["South Africa","Rhodesia"]}
{"upwards":["White supremacy","History","British Commonwealth","South Africa"],"downwards":[]}
{"upwards":["White supremacy","History","British Commonwealth","Rhodesia"],"downwards":[]}
{"upwards":["White supremacy","History","Germany"],"downwards":[]}
{"upwards":["White supremacy","History","Australia and New Zealand"],"downwards":[]}
{"upwards":["White supremacy","Ideologies and movements"],"downwards":["White nationalism","White separatism"]}
{"upwards":["White supremacy","Ideologies and movements","White nationalism"],"downwards":[]}
{"upwards":["White supremacy","Ideologies and movements","White separatism"],"downwards":["Notable white separatists","Aligned organizations and philosophies"]}
{"upwards":["White supremacy","Ideologies and movements","White separatism","Notable white separatists"],"downwards":[]}
{"upwards":["White supremacy","Ideologies and movements","White separatism","Aligned organizations and philosophies"],"downwards":[]}
{"upwards":["White supremacy","Academic use of the term"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr."],"downwards":["Background","Murder","Perpetrators","Reactions"]}
{"upwards":["Murder of James Byrd Jr.","Background"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr.","Murder"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr.","Perpetrators"],"downwards":["Shawn Berry","Lawrence Brewer","John King"]}
{"upwards":["Murder of James Byrd Jr.","Perpetrators","Shawn Berry"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr.","Perpetrators","Lawrence Brewer"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr.","Perpetrators","John King"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr.","Reactions"],"downwards":["Impact on US politics","Musical and poetry tributes"]}
{"upwards":["Murder of James Byrd Jr.","Reactions","Impact on US politics"],"downwards":[]}
{"upwards":["Murder of James Byrd Jr.","Reactions","Musical and poetry tributes"],"downwards":[]}
{"upwards":["Jasper, Texas"],"downwards":["History","Geography","Climate","Demographics","Economy","Government","Parks and recreation","Education","Media","Notable people","References","External links"]}
{"upwards":["Jasper, Texas","History"],"downwards":["19th century","20th century","21st century"]}
{"upwards":["Jasper, Texas","History","19th century"],"downwards":[]}
{"upwards":["Jasper, Texas","History","20th century"],"downwards":[]}
{"upwards":["Jasper, Texas","History","21st century"],"downwards":[]}
{"upwards":["Jasper, Texas","Geography"],"downwards":[]}
{"upwards":["Jasper, Texas","Climate"],"downwards":[]}
{"upwards":["Jasper, Texas","Demographics"],"downwards":[]}
{"upwards":["Jasper, Texas","Economy"],"downwards":[]}
{"upwards":["Jasper, Texas","Government"],"downwards":[]}
{"upwards":["Jasper, Texas","Parks and recreation"],"downwards":[]}
{"upwards":["Jasper, Texas","Education"],"downwards":[]}
{"upwards":["Jasper, Texas","Media"],"downwards":[]}
{"upwards":["Jasper, Texas","Notable people"],"downwards":[]}
{"upwards":["Jasper, Texas","References"],"downwards":[]}
{"upwards":["Jasper, Texas","External links"],"downwards":[]}
{"upwards":["Gujarat Titans"],"downwards":["Franchise history","Team history","Team identity","Home ground","Kit manufacturers and sponsors","Seasons","Current squad","Administration and support staff","Statistics","References","External links"]}
{"upwards":["Gujarat Titans","Franchise history"],"downwards":[]}
{"upwards":["Gujarat Titans","Team history"],"downwards":["2022 IPL season"]}
{"upwards":["Gujarat Titans","Team history","2022 IPL season"],"downwards":[]}
{"upwards":["Gujarat Titans","Team identity"],"downwards":[]}
{"upwards":["Gujarat Titans","Home ground"],"downwards":[]}
{"upwards":["Gujarat Titans","Kit manufacturers and sponsors"],"downwards":[]}
{"upwards":["Gujarat Titans","Seasons"],"downwards":["Indian Premier League"]}
{"upwards":["Gujarat Titans","Seasons","Indian Premier League"],"downwards":[]}
{"upwards":["Gujarat Titans","Current squad"],"downwards":[]}
{"upwards":["Gujarat Titans","Administration and support staff"],"downwards":[]}
{"upwards":["Gujarat Titans","Statistics"],"downwards":["Overall results in the IPL","By opposition"]}
{"upwards":["Gujarat Titans","Statistics","Overall results in the IPL"],"downwards":[]}
{"upwards":["Gujarat Titans","Statistics","By opposition"],"downwards":[]}
{"upwards":["Gujarat Titans","References"],"downwards":[]}
{"upwards":["Gujarat Titans","External links"],"downwards":[]}
{"upwards":["Hyperboloid"],"downwards":["Parametric representations","Properties","In more than three dimensions","Hyperboloid structures","Relation to the sphere"]}
{"upwards":["Hyperboloid","Parametric representations"],"downwards":["Generalised equations"]}
{"upwards":["Hyperboloid","Parametric representations","Generalised equations"],"downwards":[]}
{"upwards":["Hyperboloid","Properties"],"downwards":["Hyperboloid of one sheet","Hyperboloid of two sheets","Other properties"]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of one sheet"],"downwards":["Lines on the surface","Plane sections"]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of one sheet","Lines on the surface"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of one sheet","Plane sections"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Hyperboloid of two sheets"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Other properties"],"downwards":["Symmetries","Curvature"]}
{"upwards":["Hyperboloid","Properties","Other properties","Symmetries"],"downwards":[]}
{"upwards":["Hyperboloid","Properties","Other properties","Curvature"],"downwards":[]}
{"upwards":["Hyperboloid","In more than three dimensions"],"downwards":[]}
{"upwards":["Hyperboloid","Hyperboloid structures"],"downwards":[]}
{"upwards":["Hyperboloid","Relation to the sphere"],"downwards":[]}
{"upwards":["Topology"],"downwards":["Motivation","History","Concepts","Topics","Applications"]}
{"upwards":["Topology","Motivation"],"downwards":[]}
{"upwards":["Topology","History"],"downwards":[]}
{"upwards":["Topology","Concepts"],"downwards":["Topologies on sets","Continuous functions and homeomorphisms","Manifolds"]}
{"upwards":["Topology","Concepts","Topologies on sets"],"downwards":[]}
{"upwards":["Topology","Concepts","Continuous functions and homeomorphisms"],"downwards":[]}
{"upwards":["Topology","Concepts","Manifolds"],"downwards":[]}
{"upwards":["Topology","Topics"],"downwards":["General topology","Algebraic topology","Differential topology","Geometric topology","Generalizations"]}
{"upwards":["Topology","Topics","General topology"],"downwards":[]}
{"upwards":["Topology","Topics","Algebraic topology"],"downwards":[]}
{"upwards":["Topology","Topics","Differential topology"],"downwards":[]}
{"upwards":["Topology","Topics","Geometric topology"],"downwards":[]}
{"upwards":["Topology","Topics","Generalizations"],"downwards":[]}
{"upwards":["Topology","Applications"],"downwards":["Biology","Computer science","Physics","Robotics","Games and puzzles","Fiber art"]}
{"upwards":["Topology","Applications","Biology"],"downwards":[]}
{"upwards":["Topology","Applications","Computer science"],"downwards":[]}
{"upwards":["Topology","Applications","Physics"],"downwards":[]}
{"upwards":["Topology","Applications","Robotics"],"downwards":[]}
{"upwards":["Topology","Applications","Games and puzzles"],"downwards":[]}
{"upwards":["Topology","Applications","Fiber art"],"downwards":[]}
{"upwards":["Scalar (mathematics)"],"downwards":["Etymology","Definitions and properties"]}
{"upwards":["Scalar (mathematics)","Etymology"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties"],"downwards":["Scalars of vector spaces","Scalars as vector components","Scalars in normed vector spaces","Scalars in modules","Scaling transformation"]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars of vector spaces"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars as vector components"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars in normed vector spaces"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scalars in modules"],"downwards":[]}
{"upwards":["Scalar (mathematics)","Definitions and properties","Scaling transformation"],"downwards":[]}
{"upwards":["Linear algebra"],"downwards":["History","Vector spaces","Matrices","Linear systems","Endomorphisms and square matrices","Duality","Relationship with geometry","Usage and applications","Extensions and generalizations"]}
{"upwards":["Linear algebra","History"],"downwards":[]}
{"upwards":["Linear algebra","Vector spaces"],"downwards":["Linear maps","Subspaces, span, and basis"]}
{"upwards":["Linear algebra","Vector spaces","Linear maps"],"downwards":[]}
{"upwards":["Linear algebra","Vector spaces","Subspaces, span, and basis"],"downwards":[]}
{"upwards":["Linear algebra","Matrices"],"downwards":[]}
{"upwards":["Linear algebra","Linear systems"],"downwards":[]}
{"upwards":["Linear algebra","Endomorphisms and square matrices"],"downwards":["Determinant","Eigenvalues and eigenvectors"]}
{"upwards":["Linear algebra","Endomorphisms and square matrices","Determinant"],"downwards":[]}
{"upwards":["Linear algebra","Endomorphisms and square matrices","Eigenvalues and eigenvectors"],"downwards":[]}
{"upwards":["Linear algebra","Duality"],"downwards":["Dual map","Inner-product spaces"]}
{"upwards":["Linear algebra","Duality","Dual map"],"downwards":[]}
{"upwards":["Linear algebra","Duality","Inner-product spaces"],"downwards":[]}
{"upwards":["Linear algebra","Relationship with geometry"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications"],"downwards":["Geometry of ambient space","Functional analysis","Study of complex systems","Scientific computation"]}
{"upwards":["Linear algebra","Usage and applications","Geometry of ambient space"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications","Functional analysis"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications","Study of complex systems"],"downwards":[]}
{"upwards":["Linear algebra","Usage and applications","Scientific computation"],"downwards":[]}
{"upwards":["Linear algebra","Extensions and generalizations"],"downwards":["Module theory","Multilinear algebra and tensors","Topological vector spaces"]}
{"upwards":["Linear algebra","Extensions and generalizations","Module theory"],"downwards":[]}
{"upwards":["Linear algebra","Extensions and generalizations","Multilinear algebra and tensors"],"downwards":[]}
{"upwards":["Linear algebra","Extensions and generalizations","Topological vector spaces"],"downwards":[]}
{"upwards":["Cylinder"],"downwards":["Types","Properties","Cylindrical surfaces","Projective geometry","Prisms"]}
{"upwards":["Cylinder","Types"],"downwards":["Right circular cylinders"]}
{"upwards":["Cylinder","Types","Right circular cylinders"],"downwards":[]}
{"upwards":["Cylinder","Properties"],"downwards":["Cylindric sections","Volume","Surface area","Right circular hollow cylinder (cylindrical shell)","On the Sphere and Cylinder"]}
{"upwards":["Cylinder","Properties","Cylindric sections"],"downwards":[]}
{"upwards":["Cylinder","Properties","Volume"],"downwards":[]}
{"upwards":["Cylinder","Properties","Surface area"],"downwards":[]}
{"upwards":["Cylinder","Properties","Right circular hollow cylinder (cylindrical shell)"],"downwards":[]}
{"upwards":["Cylinder","Properties","On the Sphere and Cylinder"],"downwards":[]}
{"upwards":["Cylinder","Cylindrical surfaces"],"downwards":["Elliptic cylinder","Hyperbolic cylinder","Parabolic cylinder"]}
{"upwards":["Cylinder","Cylindrical surfaces","Elliptic cylinder"],"downwards":[]}
{"upwards":["Cylinder","Cylindrical surfaces","Hyperbolic cylinder"],"downwards":[]}
{"upwards":["Cylinder","Cylindrical surfaces","Parabolic cylinder"],"downwards":[]}
{"upwards":["Cylinder","Projective geometry"],"downwards":[]}
{"upwards":["Cylinder","Prisms"],"downwards":[]}
{"upwards":["Tree (graph theory)"],"downwards":["Definitions","Properties","Enumeration","Types of trees"]}
{"upwards":["Tree (graph theory)","Definitions"],"downwards":["Tree","Forest","Polytree","Polyforest","Rooted tree","Ordered tree"]}
{"upwards":["Tree (graph theory)","Definitions","Tree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Forest"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Polytree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Polyforest"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Rooted tree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Definitions","Ordered tree"],"downwards":[]}
{"upwards":["Tree (graph theory)","Properties"],"downwards":[]}
{"upwards":["Tree (graph theory)","Enumeration"],"downwards":["Labeled trees","Unlabeled trees"]}
{"upwards":["Tree (graph theory)","Enumeration","Labeled trees"],"downwards":[]}
{"upwards":["Tree (graph theory)","Enumeration","Unlabeled trees"],"downwards":[]}
{"upwards":["Tree (graph theory)","Types of trees"],"downwards":[]}
{"upwards":["Topological space"],"downwards":["History","Definitions","Comparison of topologies","Continuous functions","Examples of topological spaces","Topological constructions","Classification of topological spaces","Topological spaces with algebraic structure","Topological spaces with order structure"]}
{"upwards":["Topological space","History"],"downwards":[]}
{"upwards":["Topological space","Definitions"],"downwards":["Definition via neighbourhoods","Definition via open sets","Definition via closed sets","Other definitions"]}
{"upwards":["Topological space","Definitions","Definition via neighbourhoods"],"downwards":[]}
{"upwards":["Topological space","Definitions","Definition via open sets"],"downwards":["Examples of topologies"]}
{"upwards":["Topological space","Definitions","Definition via open sets","Examples of topologies"],"downwards":[]}
{"upwards":["Topological space","Definitions","Definition via closed sets"],"downwards":[]}
{"upwards":["Topological space","Definitions","Other definitions"],"downwards":[]}
{"upwards":["Topological space","Comparison of topologies"],"downwards":[]}
{"upwards":["Topological space","Continuous functions"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces"],"downwards":["Metric spaces","Proximity spaces","Uniform spaces","Function spaces","Cauchy spaces","Convergence spaces","Grothendieck sites","Other spaces"]}
{"upwards":["Topological space","Examples of topological spaces","Metric spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Proximity spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Uniform spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Function spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Cauchy spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Convergence spaces"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Grothendieck sites"],"downwards":[]}
{"upwards":["Topological space","Examples of topological spaces","Other spaces"],"downwards":[]}
{"upwards":["Topological space","Topological constructions"],"downwards":[]}
{"upwards":["Topological space","Classification of topological spaces"],"downwards":[]}
{"upwards":["Topological space","Topological spaces with algebraic structure"],"downwards":[]}
{"upwards":["Topological space","Topological spaces with order structure"],"downwards":[]}
{"upwards":["Glossary of graph theory"],"downwards":["Symbols","A","B","C","D","E","F","G","H","I","K","L","M","N","O","P","Q","R","S","T","U","V","W"]}
{"upwards":["Glossary of graph theory","Symbols"],"downwards":[]}
{"upwards":["Glossary of graph theory","A"],"downwards":[]}
{"upwards":["Glossary of graph theory","B"],"downwards":[]}
{"upwards":["Glossary of graph theory","C"],"downwards":[]}
{"upwards":["Glossary of graph theory","D"],"downwards":[]}
{"upwards":["Glossary of graph theory","E"],"downwards":[]}
{"upwards":["Glossary of graph theory","F"],"downwards":[]}
{"upwards":["Glossary of graph theory","G"],"downwards":[]}
{"upwards":["Glossary of graph theory","H"],"downwards":[]}
{"upwards":["Glossary of graph theory","I"],"downwards":[]}
{"upwards":["Glossary of graph theory","K"],"downwards":[]}
{"upwards":["Glossary of graph theory","L"],"downwards":[]}
{"upwards":["Glossary of graph theory","M"],"downwards":[]}
{"upwards":["Glossary of graph theory","N"],"downwards":[]}
{"upwards":["Glossary of graph theory","O"],"downwards":[]}
{"upwards":["Glossary of graph theory","P"],"downwards":[]}
{"upwards":["Glossary of graph theory","Q"],"downwards":[]}
{"upwards":["Glossary of graph theory","R"],"downwards":[]}
{"upwards":["Glossary of graph theory","S"],"downwards":[]}
{"upwards":["Glossary of graph theory","T"],"downwards":[]}
{"upwards":["Glossary of graph theory","U"],"downwards":[]}
{"upwards":["Glossary of graph theory","V"],"downwards":[]}
{"upwards":["Glossary of graph theory","W"],"downwards":[]}
{"upwards":["Manifold"],"downwards":["Motivating examples","Mathematical definition","Charts, atlases, and transition maps","Manifold with boundary","Construction","History","Additional structure","Classification and invariants","Surfaces","Maps of manifolds","Generalizations of manifolds"]}
{"upwards":["Manifold","Motivating examples"],"downwards":["Circle","Sphere","Other curves"]}
{"upwards":["Manifold","Motivating examples","Circle"],"downwards":[]}
{"upwards":["Manifold","Motivating examples","Sphere"],"downwards":[]}
{"upwards":["Manifold","Motivating examples","Other curves"],"downwards":[]}
{"upwards":["Manifold","Mathematical definition"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps"],"downwards":["Charts","Atlases","Transition maps","Additional structure"]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Charts"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Atlases"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Transition maps"],"downwards":[]}
{"upwards":["Manifold","Charts, atlases, and transition maps","Additional structure"],"downwards":[]}
{"upwards":["Manifold","Manifold with boundary"],"downwards":["Boundary and interior"]}
{"upwards":["Manifold","Manifold with boundary","Boundary and interior"],"downwards":[]}
{"upwards":["Manifold","Construction"],"downwards":["Charts","Patchwork","Identifying points of a manifold","Gluing along boundaries","Cartesian products"]}
{"upwards":["Manifold","Construction","Charts"],"downwards":["Sphere with charts"]}
{"upwards":["Manifold","Construction","Charts","Sphere with charts"],"downwards":[]}
{"upwards":["Manifold","Construction","Patchwork"],"downwards":["Intrinsic and extrinsic view","n-Sphere as a patchwork"]}
{"upwards":["Manifold","Construction","Patchwork","Intrinsic and extrinsic view"],"downwards":[]}
{"upwards":["Manifold","Construction","Patchwork","n-Sphere as a patchwork"],"downwards":[]}
{"upwards":["Manifold","Construction","Identifying points of a manifold"],"downwards":[]}
{"upwards":["Manifold","Construction","Gluing along boundaries"],"downwards":[]}
{"upwards":["Manifold","Construction","Cartesian products"],"downwards":[]}
{"upwards":["Manifold","History"],"downwards":["Early development","Synthesis","Poincar\u00e9's definition","Topology of manifolds: highlights"]}
{"upwards":["Manifold","History","Early development"],"downwards":[]}
{"upwards":["Manifold","History","Synthesis"],"downwards":[]}
{"upwards":["Manifold","History","Poincar\u00e9's definition"],"downwards":[]}
{"upwards":["Manifold","History","Topology of manifolds: highlights"],"downwards":[]}
{"upwards":["Manifold","Additional structure"],"downwards":["Topological manifolds","Differentiable manifolds","Riemannian manifolds","Finsler manifolds","Lie groups","Other types of manifolds"]}
{"upwards":["Manifold","Additional structure","Topological manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Differentiable manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Riemannian manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Finsler manifolds"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Lie groups"],"downwards":[]}
{"upwards":["Manifold","Additional structure","Other types of manifolds"],"downwards":[]}
{"upwards":["Manifold","Classification and invariants"],"downwards":[]}
{"upwards":["Manifold","Surfaces"],"downwards":["Orientability","Genus and the Euler characteristic"]}
{"upwards":["Manifold","Surfaces","Orientability"],"downwards":["M\u00f6bius strip","Klein bottle","Real projective plane"]}
{"upwards":["Manifold","Surfaces","Orientability","M\u00f6bius strip"],"downwards":[]}
{"upwards":["Manifold","Surfaces","Orientability","Klein bottle"],"downwards":[]}
{"upwards":["Manifold","Surfaces","Orientability","Real projective plane"],"downwards":[]}
{"upwards":["Manifold","Surfaces","Genus and the Euler characteristic"],"downwards":[]}
{"upwards":["Manifold","Maps of manifolds"],"downwards":["Scalar-valued functions"]}
{"upwards":["Manifold","Maps of manifolds","Scalar-valued functions"],"downwards":[]}
{"upwards":["Manifold","Generalizations of manifolds"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)"],"downwards":["Vectors in Euclidean geometry","Vector spaces","Vectors in algebra","Data represented by vectors"]}
{"upwards":["Vector (mathematics and physics)","Vectors in Euclidean geometry"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vector spaces"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Vectors in algebra"],"downwards":[]}
{"upwards":["Vector (mathematics and physics)","Data represented by vectors"],"downwards":[]}
{"upwards":["Graph theory"],"downwards":["Definitions","Applications","History","Representation","Problems"]}
{"upwards":["Graph theory","Definitions"],"downwards":["Graph","Directed graph"]}
{"upwards":["Graph theory","Definitions","Graph"],"downwards":[]}
{"upwards":["Graph theory","Definitions","Directed graph"],"downwards":[]}
{"upwards":["Graph theory","Applications"],"downwards":["Computer science","Linguistics","Physics and chemistry","Social sciences","Biology","Mathematics","Other topics"]}
{"upwards":["Graph theory","Applications","Computer science"],"downwards":[]}
{"upwards":["Graph theory","Applications","Linguistics"],"downwards":[]}
{"upwards":["Graph theory","Applications","Physics and chemistry"],"downwards":[]}
{"upwards":["Graph theory","Applications","Social sciences"],"downwards":[]}
{"upwards":["Graph theory","Applications","Biology"],"downwards":[]}
{"upwards":["Graph theory","Applications","Mathematics"],"downwards":[]}
{"upwards":["Graph theory","Applications","Other topics"],"downwards":[]}
{"upwards":["Graph theory","History"],"downwards":[]}
{"upwards":["Graph theory","Representation"],"downwards":["Visual: Graph drawing","Tabular: Graph data structures"]}
{"upwards":["Graph theory","Representation","Visual: Graph drawing"],"downwards":[]}
{"upwards":["Graph theory","Representation","Tabular: Graph data structures"],"downwards":[]}
{"upwards":["Graph theory","Problems"],"downwards":["Enumeration","Subgraphs, induced subgraphs, and minors","Graph coloring","Subsumption and unification","Route problems","Network flow","Visibility problems","Covering problems","Decomposition problems","Graph classes"]}
{"upwards":["Graph theory","Problems","Enumeration"],"downwards":[]}
{"upwards":["Graph theory","Problems","Subgraphs, induced subgraphs, and minors"],"downwards":[]}
{"upwards":["Graph theory","Problems","Graph coloring"],"downwards":[]}
{"upwards":["Graph theory","Problems","Subsumption and unification"],"downwards":[]}
{"upwards":["Graph theory","Problems","Route problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Network flow"],"downwards":[]}
{"upwards":["Graph theory","Problems","Visibility problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Covering problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Decomposition problems"],"downwards":[]}
{"upwards":["Graph theory","Problems","Graph classes"],"downwards":[]}
{"upwards":["Radix"],"downwards":["Etymology","In numeral systems"]}
{"upwards":["Radix","Etymology"],"downwards":[]}
{"upwards":["Radix","In numeral systems"],"downwards":[]}
{"upwards":["Hexadecimal"],"downwards":["Representation","Conversion","Elementary arithmetic","Real numbers","Cultural history","Base16 (transfer encoding)"]}
{"upwards":["Hexadecimal","Representation"],"downwards":["Written representation","Other symbols for 10\u201315 and mostly different symbol sets","Verbal and digital representations","Signs","Hexadecimal exponential notation"]}
{"upwards":["Hexadecimal","Representation","Written representation"],"downwards":["Distinguishing from decimal"]}
{"upwards":["Hexadecimal","Representation","Written representation","Distinguishing from decimal"],"downwards":[]}
{"upwards":["Hexadecimal","Representation","Other symbols for 10\u201315 and mostly different symbol sets"],"downwards":[]}
{"upwards":["Hexadecimal","Representation","Verbal and digital representations"],"downwards":[]}
{"upwards":["Hexadecimal","Representation","Signs"],"downwards":[]}
{"upwards":["Hexadecimal","Representation","Hexadecimal exponential notation"],"downwards":[]}
{"upwards":["Hexadecimal","Conversion"],"downwards":["Binary conversion","Other simple conversions","Division-remainder in source base","Conversion through addition and multiplication","Tools for conversion"]}
{"upwards":["Hexadecimal","Conversion","Binary conversion"],"downwards":[]}
{"upwards":["Hexadecimal","Conversion","Other simple conversions"],"downwards":[]}
{"upwards":["Hexadecimal","Conversion","Division-remainder in source base"],"downwards":[]}
{"upwards":["Hexadecimal","Conversion","Conversion through addition and multiplication"],"downwards":[]}
{"upwards":["Hexadecimal","Conversion","Tools for conversion"],"downwards":[]}
{"upwards":["Hexadecimal","Elementary arithmetic"],"downwards":[]}
{"upwards":["Hexadecimal","Real numbers"],"downwards":["Rational numbers","Irrational numbers","Powers"]}
{"upwards":["Hexadecimal","Real numbers","Rational numbers"],"downwards":[]}
{"upwards":["Hexadecimal","Real numbers","Irrational numbers"],"downwards":[]}
{"upwards":["Hexadecimal","Real numbers","Powers"],"downwards":[]}
{"upwards":["Hexadecimal","Cultural history"],"downwards":[]}
{"upwards":["Hexadecimal","Base16 (transfer encoding)"],"downwards":[]}
{"upwards":["Binary number"],"downwards":["History","Representation","Counting in binary","Fractions","Binary arithmetic","Bitwise operations","Conversion to and from other numeral systems","Representing real numbers"]}
{"upwards":["Binary number","History"],"downwards":["Egypt","China","India","Other cultures","Western predecessors to Leibniz","Leibniz and the I Ching","Later developments"]}
{"upwards":["Binary number","History","Egypt"],"downwards":[]}
{"upwards":["Binary number","History","China"],"downwards":[]}
{"upwards":["Binary number","History","India"],"downwards":[]}
{"upwards":["Binary number","History","Other cultures"],"downwards":[]}
{"upwards":["Binary number","History","Western predecessors to Leibniz"],"downwards":[]}
{"upwards":["Binary number","History","Leibniz and the I Ching"],"downwards":[]}
{"upwards":["Binary number","History","Later developments"],"downwards":[]}
{"upwards":["Binary number","Representation"],"downwards":[]}
{"upwards":["Binary number","Counting in binary"],"downwards":["Decimal counting","Binary counting"]}
{"upwards":["Binary number","Counting in binary","Decimal counting"],"downwards":[]}
{"upwards":["Binary number","Counting in binary","Binary counting"],"downwards":[]}
{"upwards":["Binary number","Fractions"],"downwards":[]}
{"upwards":["Binary number","Binary arithmetic"],"downwards":["Addition","Subtraction","Multiplication","Division","Square root"]}
{"upwards":["Binary number","Binary arithmetic","Addition"],"downwards":["Long carry method","Addition table"]}
{"upwards":["Binary number","Binary arithmetic","Addition","Long carry method"],"downwards":[]}
{"upwards":["Binary number","Binary arithmetic","Addition","Addition table"],"downwards":[]}
{"upwards":["Binary number","Binary arithmetic","Subtraction"],"downwards":[]}
{"upwards":["Binary number","Binary arithmetic","Multiplication"],"downwards":["Multiplication table"]}
{"upwards":["Binary number","Binary arithmetic","Multiplication","Multiplication table"],"downwards":[]}
{"upwards":["Binary number","Binary arithmetic","Division"],"downwards":[]}
{"upwards":["Binary number","Binary arithmetic","Square root"],"downwards":[]}
{"upwards":["Binary number","Bitwise operations"],"downwards":[]}
{"upwards":["Binary number","Conversion to and from other numeral systems"],"downwards":["Decimal to Binary","Binary to Decimal","Hexadecimal","Octal"]}
{"upwards":["Binary number","Conversion to and from other numeral systems","Decimal to Binary"],"downwards":[]}
{"upwards":["Binary number","Conversion to and from other numeral systems","Binary to Decimal"],"downwards":[]}
{"upwards":["Binary number","Conversion to and from other numeral systems","Hexadecimal"],"downwards":[]}
{"upwards":["Binary number","Conversion to and from other numeral systems","Octal"],"downwards":[]}
{"upwards":["Binary number","Representing real numbers"],"downwards":[]}
{"upwards":["2023 Indian Premier League"],"downwards":["Administration","Background","Participating teams","Personnel changes","Rules","Venues","Format","Points table","Match summary","League stage","Playoffs","Broadcasting","Criticism","Statistics and awards","Notes","References","External links"]}
{"upwards":["2023 Indian Premier League","Administration"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Background"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Participating teams"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Personnel changes"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Rules"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Venues"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Format"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Points table"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Match summary"],"downwards":[]}
{"upwards":["2023 Indian Premier League","League stage"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs"],"downwards":["Qualifier 1","Eliminator","Qualifier 2","Final"]}
{"upwards":["2023 Indian Premier League","Playoffs","Qualifier 1"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs","Eliminator"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs","Qualifier 2"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Playoffs","Final"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Broadcasting"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Criticism"],"downwards":["Slow over- rate","Impact player rule"]}
{"upwards":["2023 Indian Premier League","Criticism","Slow over- rate"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Criticism","Impact player rule"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Statistics and awards"],"downwards":["Most runs","Most wickets","End of the season awards"]}
{"upwards":["2023 Indian Premier League","Statistics and awards","Most runs"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Statistics and awards","Most wickets"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Statistics and awards","End of the season awards"],"downwards":[]}
{"upwards":["2023 Indian Premier League","Notes"],"downwards":[]}
{"upwards":["2023 Indian Premier League","References"],"downwards":[]}
{"upwards":["2023 Indian Premier League","External links"],"downwards":[]}
{"upwards":["Chennai Super Kings"],"downwards":["History","Home ground","Players","Identity","Kit and sponsors","Support and fan following","Brand value","Controversies","Result summary","Statistics","Rivalries","Seasons","Current squad","Administration and support staff","In popular culture"]}
{"upwards":["Chennai Super Kings","History"],"downwards":["2008\u20132009: First seasons","2010: IPL and CLT20 double","2011\u20132015: Success and suspension","2018\u2013present"]}
{"upwards":["Chennai Super Kings","History","2008\u20132009: First seasons"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2010: IPL and CLT20 double"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2011\u20132015: Success and suspension"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present"],"downwards":["2018","2019","2020","2021","2022","2023"]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2018"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2019"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2020"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2021"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2022"],"downwards":[]}
{"upwards":["Chennai Super Kings","History","2018\u2013present","2023"],"downwards":[]}
{"upwards":["Chennai Super Kings","Home ground"],"downwards":[]}
{"upwards":["Chennai Super Kings","Players"],"downwards":[]}
{"upwards":["Chennai Super Kings","Identity"],"downwards":["Name, logo, crest and colors","Theme song"]}
{"upwards":["Chennai Super Kings","Identity","Name, logo, crest and colors"],"downwards":[]}
{"upwards":["Chennai Super Kings","Identity","Theme song"],"downwards":[]}
{"upwards":["Chennai Super Kings","Kit and sponsors"],"downwards":[]}
{"upwards":["Chennai Super Kings","Support and fan following"],"downwards":[]}
{"upwards":["Chennai Super Kings","Brand value"],"downwards":[]}
{"upwards":["Chennai Super Kings","Controversies"],"downwards":["Ownership of N. Srinivasan","2013 betting scandal","IPL suspension"]}
{"upwards":["Chennai Super Kings","Controversies","Ownership of N. Srinivasan"],"downwards":[]}
{"upwards":["Chennai Super Kings","Controversies","2013 betting scandal"],"downwards":[]}
{"upwards":["Chennai Super Kings","Controversies","IPL suspension"],"downwards":[]}
{"upwards":["Chennai Super Kings","Result summary"],"downwards":["Overall results","By opposition","Overall results in the Champions League T20"]}
{"upwards":["Chennai Super Kings","Result summary","Overall results"],"downwards":[]}
{"upwards":["Chennai Super Kings","Result summary","By opposition"],"downwards":[]}
{"upwards":["Chennai Super Kings","Result summary","Overall results in the Champions League T20"],"downwards":[]}
{"upwards":["Chennai Super Kings","Statistics"],"downwards":["Most runs","Most wickets"]}
{"upwards":["Chennai Super Kings","Statistics","Most runs"],"downwards":[]}
{"upwards":["Chennai Super Kings","Statistics","Most wickets"],"downwards":[]}
{"upwards":["Chennai Super Kings","Rivalries"],"downwards":["Rivalry with Mumbai Indians","Rivalry with Royal Challengers Bangalore"]}
{"upwards":["Chennai Super Kings","Rivalries","Rivalry with Mumbai Indians"],"downwards":[]}
{"upwards":["Chennai Super Kings","Rivalries","Rivalry with Royal Challengers Bangalore"],"downwards":[]}
{"upwards":["Chennai Super Kings","Seasons"],"downwards":["Indian Premier League","Champions League T20"]}
{"upwards":["Chennai Super Kings","Seasons","Indian Premier League"],"downwards":[]}
{"upwards":["Chennai Super Kings","Seasons","Champions League T20"],"downwards":[]}
{"upwards":["Chennai Super Kings","Current squad"],"downwards":[]}
{"upwards":["Chennai Super Kings","Administration and support staff"],"downwards":[]}
{"upwards":["Chennai Super Kings","In popular culture"],"downwards":[]}
{"upwards":["2023 Turkish presidential election"],"downwards":["Background","Date","Electoral system","Candidates","Endorsements","Second round","Controversies","Opinion polls","Results","Aftermath"]}
{"upwards":["2023 Turkish presidential election","Background"],"downwards":["2018 elections"]}
{"upwards":["2023 Turkish presidential election","Background","2018 elections"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Date"],"downwards":["Discussion on snap elections","Discussion on postponing election date","Election calendar"]}
{"upwards":["2023 Turkish presidential election","Date","Discussion on snap elections"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Date","Discussion on postponing election date"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Date","Election calendar"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Electoral system"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Candidates"],"downwards":["Candidates","Nominations"]}
{"upwards":["2023 Turkish presidential election","Candidates","Candidates"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Candidates","Nominations"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Endorsements"],"downwards":["Parties and alliances"]}
{"upwards":["2023 Turkish presidential election","Endorsements","Parties and alliances"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round"],"downwards":["Negotiations by O\u011fan and \u00d6zda\u011f","Erdo\u011fan's strategy","K\u0131l\u0131\u00e7daro\u011flu's strategy","Potential debate"]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f"],"downwards":["Dissolution of ATA Alliance","Endorsement of Erdo\u011fan by O\u011fan","\u00d6zda\u011f's support for K\u0131l\u0131\u00e7daro\u011flu"]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f","Dissolution of ATA Alliance"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f","Endorsement of Erdo\u011fan by O\u011fan"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Negotiations by O\u011fan and \u00d6zda\u011f","\u00d6zda\u011f's support for K\u0131l\u0131\u00e7daro\u011flu"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Erdo\u011fan's strategy"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","K\u0131l\u0131\u00e7daro\u011flu's strategy"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Second round","Potential debate"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies"],"downwards":["Erdo\u011fan's right to seek re-election","Political violence","Erdo\u011fan's photo on ballot paper","Fake videos","Block of Ek\u015fi S\u00f6zl\u00fck","Reporting of the results","Azerbaijani and Russian interference allegations"]}
{"upwards":["2023 Turkish presidential election","Controversies","Erdo\u011fan's right to seek re-election"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Political violence"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Erdo\u011fan's photo on ballot paper"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Fake videos"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Block of Ek\u015fi S\u00f6zl\u00fck"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Reporting of the results"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Controversies","Azerbaijani and Russian interference allegations"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Opinion polls"],"downwards":["First round","Second round"]}
{"upwards":["2023 Turkish presidential election","Opinion polls","First round"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Opinion polls","Second round"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Results"],"downwards":["Analysis"]}
{"upwards":["2023 Turkish presidential election","Results","Analysis"],"downwards":[]}
{"upwards":["2023 Turkish presidential election","Aftermath"],"downwards":["Domestic"]}
{"upwards":["2023 Turkish presidential election","Aftermath","Domestic"],"downwards":["International"]}
{"upwards":["2023 Turkish presidential election","Aftermath","Domestic","International"],"downwards":[]}
{"upwards":["Robert Hanssen"],"downwards":["Early life","FBI career and first espionage activities (1976\u20131981)","FBI counterintelligence unit, further espionage activities (1985\u20131991)","Later FBI career, continued espionage activities (1992\u20132001)","Investigation and arrest","Guilty plea and imprisonment","Modus operandi","Personal life","In the media","Notes","References and further reading","External links"]}
{"upwards":["Robert Hanssen","Early life"],"downwards":[]}
{"upwards":["Robert Hanssen","FBI career and first espionage activities (1976\u20131981)"],"downwards":[]}
{"upwards":["Robert Hanssen","FBI counterintelligence unit, further espionage activities (1985\u20131991)"],"downwards":[]}
{"upwards":["Robert Hanssen","Later FBI career, continued espionage activities (1992\u20132001)"],"downwards":[]}
{"upwards":["Robert Hanssen","Investigation and arrest"],"downwards":[]}
{"upwards":["Robert Hanssen","Guilty plea and imprisonment"],"downwards":[]}
{"upwards":["Robert Hanssen","Modus operandi"],"downwards":[]}
{"upwards":["Robert Hanssen","Personal life"],"downwards":["Death"]}
{"upwards":["Robert Hanssen","Personal life","Death"],"downwards":[]}
{"upwards":["Robert Hanssen","In the media"],"downwards":[]}
{"upwards":["Robert Hanssen","Notes"],"downwards":[]}
{"upwards":["Robert Hanssen","References and further reading"],"downwards":[]}
{"upwards":["Robert Hanssen","External links"],"downwards":[]}
{"upwards":["Chief Seattle"],"downwards":["Biography","The speech or \"letter\"","Legacy"]}
{"upwards":["Chief Seattle","Biography"],"downwards":[]}
{"upwards":["Chief Seattle","The speech or \"letter\""],"downwards":[]}
{"upwards":["Chief Seattle","Legacy"],"downwards":[]}
{"upwards":["President of Turkey"],"downwards":["History","Leadership roles","Selection process","Incumbency","Duties and responsibilities","Acting President","Latest election","Timeline of presidents"]}
{"upwards":["President of Turkey","History"],"downwards":[]}
{"upwards":["President of Turkey","Leadership roles"],"downwards":["Head of state","Head of party"]}
{"upwards":["President of Turkey","Leadership roles","Head of state"],"downwards":[]}
{"upwards":["President of Turkey","Leadership roles","Head of party"],"downwards":[]}
{"upwards":["President of Turkey","Selection process"],"downwards":["Eligibility","Election","Inauguration"]}
{"upwards":["President of Turkey","Selection process","Eligibility"],"downwards":[]}
{"upwards":["President of Turkey","Selection process","Election"],"downwards":[]}
{"upwards":["President of Turkey","Selection process","Inauguration"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency"],"downwards":["Term limit","Vacancies and successions","Accountability and non-accountability","Residence"]}
{"upwards":["President of Turkey","Incumbency","Term limit"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Vacancies and successions"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Accountability and non-accountability"],"downwards":["After the 2017 constitutional amendment","Before the 2017 constitutional amendment"]}
{"upwards":["President of Turkey","Incumbency","Accountability and non-accountability","After the 2017 constitutional amendment"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Accountability and non-accountability","Before the 2017 constitutional amendment"],"downwards":[]}
{"upwards":["President of Turkey","Incumbency","Residence"],"downwards":[]}
{"upwards":["President of Turkey","Duties and responsibilities"],"downwards":[]}
{"upwards":["President of Turkey","Acting President"],"downwards":["After the 2017 constitutional referendum","Before the 2017 constitutional referendum"]}
{"upwards":["President of Turkey","Acting President","After the 2017 constitutional referendum"],"downwards":[]}
{"upwards":["President of Turkey","Acting President","Before the 2017 constitutional referendum"],"downwards":[]}
{"upwards":["President of Turkey","Latest election"],"downwards":[]}
{"upwards":["President of Turkey","Timeline of presidents"],"downwards":[]}
{"upwards":["2023 Indianapolis 500"],"downwards":["Race background","Race schedules","Entry list","Testing and rookie orientation","Practice","Time trials","Post-qualifying practice","Starting grid","Race report","Box score","Broadcasting","References","External links"]}
{"upwards":["2023 Indianapolis 500","Race background"],"downwards":["Sponsorship","Rule changes","2023 IndyCar Series"]}
{"upwards":["2023 Indianapolis 500","Race background","Sponsorship"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race background","Rule changes"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race background","2023 IndyCar Series"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race schedules"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Entry list"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Testing and rookie orientation"],"downwards":["Open Test\u2014Thursday April 20","Open test\u2014Friday April 21"]}
{"upwards":["2023 Indianapolis 500","Testing and rookie orientation","Open Test\u2014Thursday April 20"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Testing and rookie orientation","Open test\u2014Friday April 21"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice"],"downwards":["Opening day\u2014Tuesday May 16","Wednesday May 17","Thursday May 18","Fast Friday\u2014Friday May 19"]}
{"upwards":["2023 Indianapolis 500","Practice","Opening day\u2014Tuesday May 16"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice","Wednesday May 17"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice","Thursday May 18"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Practice","Fast Friday\u2014Friday May 19"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials"],"downwards":["Saturday, May 20","Sunday, May 21"]}
{"upwards":["2023 Indianapolis 500","Time trials","Saturday, May 20"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21"],"downwards":["Top 12 qualifying","Last Chance Qualifying","Fast Six Qualifying"]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21","Top 12 qualifying"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21","Last Chance Qualifying"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Time trials","Sunday, May 21","Fast Six Qualifying"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice"],"downwards":["Post-qualifying practice\u2014Monday, May 22","Special test\u2014Thursday, May 25","Carb Day\u2014Friday, May 26","Pit Stop Challenge"]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Post-qualifying practice\u2014Monday, May 22"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Special test\u2014Thursday, May 25"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Carb Day\u2014Friday, May 26"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Post-qualifying practice","Pit Stop Challenge"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Starting grid"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report"],"downwards":["First half","Second half"]}
{"upwards":["2023 Indianapolis 500","Race report","First half"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report","Second half"],"downwards":["First red flag","Second red flag","Third red flag and finish"]}
{"upwards":["2023 Indianapolis 500","Race report","Second half","First red flag"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report","Second half","Second red flag"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Race report","Second half","Third red flag and finish"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Box score"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Broadcasting"],"downwards":["Television","Radio"]}
{"upwards":["2023 Indianapolis 500","Broadcasting","Television"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","Broadcasting","Radio"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","References"],"downwards":[]}
{"upwards":["2023 Indianapolis 500","External links"],"downwards":[]}
{"upwards":["Homer Plessy"],"downwards":["Early life and historical context","Plessy v. Ferguson","Later life","Legacy","References","Further reading"]}
{"upwards":["Homer Plessy","Early life and historical context"],"downwards":[]}
{"upwards":["Homer Plessy","Plessy v. Ferguson"],"downwards":["Orchestrating a test case","Trial","Supreme Court appeal"]}
{"upwards":["Homer Plessy","Plessy v. Ferguson","Orchestrating a test case"],"downwards":[]}
{"upwards":["Homer Plessy","Plessy v. Ferguson","Trial"],"downwards":[]}
{"upwards":["Homer Plessy","Plessy v. Ferguson","Supreme Court appeal"],"downwards":[]}
{"upwards":["Homer Plessy","Later life"],"downwards":[]}
{"upwards":["Homer Plessy","Legacy"],"downwards":[]}
{"upwards":["Homer Plessy","References"],"downwards":["Notes","Citations","Sources"]}
{"upwards":["Homer Plessy","References","Notes"],"downwards":[]}
{"upwards":["Homer Plessy","References","Citations"],"downwards":[]}
{"upwards":["Homer Plessy","References","Sources"],"downwards":["Secondary sources","Tertiary sources"]}
{"upwards":["Homer Plessy","References","Sources","Secondary sources"],"downwards":[]}
{"upwards":["Homer Plessy","References","Sources","Tertiary sources"],"downwards":[]}
{"upwards":["Homer Plessy","Further reading"],"downwards":[]}
{"upwards":["Cynthia Weil"],"downwards":["Life and career","Songs written by Barry Mann and Cynthia Weil","Bibliography","References","External links"]}
{"upwards":["Cynthia Weil","Life and career"],"downwards":[]}
{"upwards":["Cynthia Weil","Songs written by Barry Mann and Cynthia Weil"],"downwards":[]}
{"upwards":["Cynthia Weil","Bibliography"],"downwards":[]}
{"upwards":["Cynthia Weil","References"],"downwards":[]}
{"upwards":["Cynthia Weil","External links"],"downwards":[]}
{"upwards":["Aelia Eudocia"],"downwards":["Early life","Life as an empress","Literary work","Legacy"]}
{"upwards":["Aelia Eudocia","Early life"],"downwards":[]}
{"upwards":["Aelia Eudocia","Life as an empress"],"downwards":["Marriage","Children","Pilgrimage to Jerusalem (438\u2013439)","Banishment","In Jerusalem (443\u2013460)"]}
{"upwards":["Aelia Eudocia","Life as an empress","Marriage"],"downwards":["Blending Christianity with classical culture"]}
{"upwards":["Aelia Eudocia","Life as an empress","Marriage","Blending Christianity with classical culture"],"downwards":[]}
{"upwards":["Aelia Eudocia","Life as an empress","Children"],"downwards":[]}
{"upwards":["Aelia Eudocia","Life as an empress","Pilgrimage to Jerusalem (438\u2013439)"],"downwards":[]}
{"upwards":["Aelia Eudocia","Life as an empress","Banishment"],"downwards":[]}
{"upwards":["Aelia Eudocia","Life as an empress","In Jerusalem (443\u2013460)"],"downwards":[]}
{"upwards":["Aelia Eudocia","Literary work"],"downwards":["Martyrdom of St. Cyprian","The Hamat Gader poem","Homeric centos"]}
{"upwards":["Aelia Eudocia","Literary work","Martyrdom of St. Cyprian"],"downwards":[]}
{"upwards":["Aelia Eudocia","Literary work","The Hamat Gader poem"],"downwards":[]}
{"upwards":["Aelia Eudocia","Literary work","Homeric centos"],"downwards":[]}
{"upwards":["Aelia Eudocia","Legacy"],"downwards":[]}
{"upwards":["Theodosius II"],"downwards":["Early life","Reign","Death"]}
{"upwards":["Theodosius II","Early life"],"downwards":[]}
{"upwards":["Theodosius II","Reign"],"downwards":["Early reign","University and Law Code","Famous \"apple incident\"","Wars with the Huns, Vandals, and Persians","Theological disputes"]}
{"upwards":["Theodosius II","Reign","Early reign"],"downwards":[]}
{"upwards":["Theodosius II","Reign","University and Law Code"],"downwards":[]}
{"upwards":["Theodosius II","Reign","Famous \"apple incident\""],"downwards":[]}
{"upwards":["Theodosius II","Reign","Wars with the Huns, Vandals, and Persians"],"downwards":[]}
{"upwards":["Theodosius II","Reign","Theological disputes"],"downwards":[]}
{"upwards":["Theodosius II","Death"],"downwards":[]}
{"upwards":["Thomas West, 3rd Baron De La Warr"],"downwards":["Early and family life","Career","Family","Notes","References","External links"]}
{"upwards":["Thomas West, 3rd Baron De La Warr","Early and family life"],"downwards":[]}
{"upwards":["Thomas West, 3rd Baron De La Warr","Career"],"downwards":[]}
{"upwards":["Thomas West, 3rd Baron De La Warr","Family"],"downwards":[]}
{"upwards":["Thomas West, 3rd Baron De La Warr","Notes"],"downwards":[]}
{"upwards":["Thomas West, 3rd Baron De La Warr","References"],"downwards":[]}
{"upwards":["Thomas West, 3rd Baron De La Warr","External links"],"downwards":[]}
{"upwards":["June 7"],"downwards":["Events","Births","Deaths","Holidays and observances","References","External links"]}
{"upwards":["June 7","Events"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 7","Events","Pre-1600"],"downwards":[]}
{"upwards":["June 7","Events","1601\u20131900"],"downwards":[]}
{"upwards":["June 7","Events","1901\u2013present"],"downwards":[]}
{"upwards":["June 7","Births"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 7","Births","Pre-1600"],"downwards":[]}
{"upwards":["June 7","Births","1601\u20131900"],"downwards":[]}
{"upwards":["June 7","Births","1901\u2013present"],"downwards":[]}
{"upwards":["June 7","Deaths"],"downwards":["Pre-1600","1601\u20131900","1901\u2013present"]}
{"upwards":["June 7","Deaths","Pre-1600"],"downwards":[]}
{"upwards":["June 7","Deaths","1601\u20131900"],"downwards":[]}
{"upwards":["June 7","Deaths","1901\u2013present"],"downwards":[]}
{"upwards":["June 7","Holidays and observances"],"downwards":[]}
{"upwards":["June 7","References"],"downwards":[]}
{"upwards":["June 7","External links"],"downwards":[]}
{"upwards":["Alice Gray"],"downwards":["Early life and education","American Museum of Natural History","Origami","Later life","References","External links"]}
{"upwards":["Alice Gray","Early life and education"],"downwards":[]}
{"upwards":["Alice Gray","American Museum of Natural History"],"downwards":[]}
{"upwards":["Alice Gray","Origami"],"downwards":[]}
{"upwards":["Alice Gray","Later life"],"downwards":[]}
{"upwards":["Alice Gray","References"],"downwards":[]}
{"upwards":["Alice Gray","External links"],"downwards":[]}
{"upwards":["Royal assent"],"downwards":["Usage","Development","Ceremony","Other countries","Notes","References","Further reading","External links"]}
{"upwards":["Royal assent","Usage"],"downwards":["United Kingdom","Other Commonwealth realms"]}
{"upwards":["Royal assent","Usage","United Kingdom"],"downwards":["Historical development","Relationship to royal consent"]}
{"upwards":["Royal assent","Usage","United Kingdom","Historical development"],"downwards":[]}
{"upwards":["Royal assent","Usage","United Kingdom","Relationship to royal consent"],"downwards":[]}
{"upwards":["Royal assent","Usage","Other Commonwealth realms"],"downwards":["Canada"]}
{"upwards":["Royal assent","Usage","Other Commonwealth realms","Canada"],"downwards":[]}
{"upwards":["Royal assent","Development"],"downwards":[]}
{"upwards":["Royal assent","Ceremony"],"downwards":["United Kingdom","Australia and New Zealand","Canada"]}
{"upwards":["Royal assent","Ceremony","United Kingdom"],"downwards":[]}
{"upwards":["Royal assent","Ceremony","Australia and New Zealand"],"downwards":[]}
{"upwards":["Royal assent","Ceremony","Canada"],"downwards":[]}
{"upwards":["Royal assent","Other countries"],"downwards":["Belgium","Japan","Jordan","Luxembourg","Norway","Spain","Tonga"]}
{"upwards":["Royal assent","Other countries","Belgium"],"downwards":[]}
{"upwards":["Royal assent","Other countries","Japan"],"downwards":[]}
{"upwards":["Royal assent","Other countries","Jordan"],"downwards":[]}
{"upwards":["Royal assent","Other countries","Luxembourg"],"downwards":[]}
{"upwards":["Royal assent","Other countries","Norway"],"downwards":[]}
{"upwards":["Royal assent","Other countries","Spain"],"downwards":[]}
{"upwards":["Royal assent","Other countries","Tonga"],"downwards":[]}
{"upwards":["Royal assent","Notes"],"downwards":[]}
{"upwards":["Royal assent","References"],"downwards":[]}
{"upwards":["Royal assent","Further reading"],"downwards":[]}
{"upwards":["Royal assent","External links"],"downwards":[]}
{"upwards":["1892"],"downwards":["Events","Births","Deaths","References"]}
{"upwards":["1892","Events"],"downwards":["January\u2013March","April\u2013June","July\u2013September","October\u2013December","Date unknown"]}
{"upwards":["1892","Events","January\u2013March"],"downwards":[]}
{"upwards":["1892","Events","April\u2013June"],"downwards":[]}
{"upwards":["1892","Events","July\u2013September"],"downwards":[]}
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{"upwards":["1892","Events","Date unknown"],"downwards":[]}
{"upwards":["1892","Births"],"downwards":["January","February","March","April","May","June","July","August","September","October","November","December","Date unknown"]}
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{"upwards":["1892","Births","May"],"downwards":[]}
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{"upwards":["1892","Births","July"],"downwards":[]}
{"upwards":["1892","Births","August"],"downwards":[]}
{"upwards":["1892","Births","September"],"downwards":[]}
{"upwards":["1892","Births","October"],"downwards":[]}
{"upwards":["1892","Births","November"],"downwards":[]}
{"upwards":["1892","Births","December"],"downwards":[]}
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{"upwards":["1892","Deaths","January\u2013June"],"downwards":[]}
{"upwards":["1892","Deaths","July\u2013December"],"downwards":[]}
{"upwards":["1892","References"],"downwards":[]}
{"upwards":["Knud Rasmussen"],"downwards":["Early years","Career","Honors","Bibliography","Notes","Further reading","External links"]}
{"upwards":["Knud Rasmussen","Early years"],"downwards":[]}
{"upwards":["Knud Rasmussen","Career"],"downwards":["The Thule expeditions"]}
{"upwards":["Knud Rasmussen","Career","The Thule expeditions"],"downwards":[]}
{"upwards":["Knud Rasmussen","Honors"],"downwards":[]}
{"upwards":["Knud Rasmussen","Bibliography"],"downwards":[]}
{"upwards":["Knud Rasmussen","Notes"],"downwards":[]}
{"upwards":["Knud Rasmussen","Further reading"],"downwards":[]}
{"upwards":["Knud Rasmussen","External links"],"downwards":[]}
{"upwards":["Reform Act 1832"],"downwards":["Unreformed House of Commons","Movement for reform","Passage of the Reform Act","Results","Assessment"]}
{"upwards":["Reform Act 1832","Unreformed House of Commons"],"downwards":["Composition","The franchise","Pocket boroughs, bribery"]}
{"upwards":["Reform Act 1832","Unreformed House of Commons","Composition"],"downwards":["Counties","Boroughs"]}
{"upwards":["Reform Act 1832","Unreformed House of Commons","Composition","Counties"],"downwards":[]}
{"upwards":["Reform Act 1832","Unreformed House of Commons","Composition","Boroughs"],"downwards":[]}
{"upwards":["Reform Act 1832","Unreformed House of Commons","The franchise"],"downwards":["Women's suffrage"]}
{"upwards":["Reform Act 1832","Unreformed House of Commons","The franchise","Women's suffrage"],"downwards":[]}
{"upwards":["Reform Act 1832","Unreformed House of Commons","Pocket boroughs, bribery"],"downwards":[]}
{"upwards":["Reform Act 1832","Movement for reform"],"downwards":["Early attempts at reform","After the French Revolution","Reform during the 1820s"]}
{"upwards":["Reform Act 1832","Movement for reform","Early attempts at reform"],"downwards":[]}
{"upwards":["Reform Act 1832","Movement for reform","After the French Revolution"],"downwards":[]}
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{"upwards":["Reform Act 1832","Passage of the Reform Act"],"downwards":["First Reform Bill","Second Reform Bill","Third Reform Bill"]}
{"upwards":["Reform Act 1832","Passage of the Reform Act","First Reform Bill"],"downwards":[]}
{"upwards":["Reform Act 1832","Passage of the Reform Act","Second Reform Bill"],"downwards":[]}
{"upwards":["Reform Act 1832","Passage of the Reform Act","Third Reform Bill"],"downwards":[]}
{"upwards":["Reform Act 1832","Results"],"downwards":["Provisions","Effects","Tenant voters","Limitations","Further reform"]}
{"upwards":["Reform Act 1832","Results","Provisions"],"downwards":["Abolition of seats","Creation of new seats","Extension of the franchise"]}
{"upwards":["Reform Act 1832","Results","Provisions","Abolition of seats"],"downwards":[]}
{"upwards":["Reform Act 1832","Results","Provisions","Creation of new seats"],"downwards":[]}
{"upwards":["Reform Act 1832","Results","Provisions","Extension of the franchise"],"downwards":[]}
{"upwards":["Reform Act 1832","Results","Effects"],"downwards":[]}
{"upwards":["Reform Act 1832","Results","Tenant voters"],"downwards":[]}
{"upwards":["Reform Act 1832","Results","Limitations"],"downwards":[]}
{"upwards":["Reform Act 1832","Results","Further reform"],"downwards":[]}
{"upwards":["Reform Act 1832","Assessment"],"downwards":[]}
{"upwards":["Ancient Greek religion"],"downwards":["Beliefs","Practices","Sanctuaries and temples","Role of women","Mystery religions","History"]}
{"upwards":["Ancient Greek religion","Beliefs"],"downwards":["Theology","Afterlife","Mythology","Morality","Sacred texts"]}
{"upwards":["Ancient Greek religion","Beliefs","Theology"],"downwards":[]}
{"upwards":["Ancient Greek religion","Beliefs","Afterlife"],"downwards":[]}
{"upwards":["Ancient Greek religion","Beliefs","Mythology"],"downwards":[]}
{"upwards":["Ancient Greek religion","Beliefs","Morality"],"downwards":[]}
{"upwards":["Ancient Greek religion","Beliefs","Sacred texts"],"downwards":[]}
{"upwards":["Ancient Greek religion","Practices"],"downwards":["Ceremonies","Sacrifice","Festivals","Rites of passage"]}
{"upwards":["Ancient Greek religion","Practices","Ceremonies"],"downwards":[]}
{"upwards":["Ancient Greek religion","Practices","Sacrifice"],"downwards":[]}
{"upwards":["Ancient Greek religion","Practices","Festivals"],"downwards":[]}
{"upwards":["Ancient Greek religion","Practices","Rites of passage"],"downwards":[]}
{"upwards":["Ancient Greek religion","Sanctuaries and temples"],"downwards":["Cult images"]}
{"upwards":["Ancient Greek religion","Sanctuaries and temples","Cult images"],"downwards":[]}
{"upwards":["Ancient Greek religion","Role of women"],"downwards":[]}
{"upwards":["Ancient Greek religion","Mystery religions"],"downwards":[]}
{"upwards":["Ancient Greek religion","History"],"downwards":["Origins","Archaic and classical periods","Hellenistic period","Roman Empire","Suppression and decline","Modern revivals"]}
{"upwards":["Ancient Greek religion","History","Origins"],"downwards":[]}
{"upwards":["Ancient Greek religion","History","Archaic and classical periods"],"downwards":[]}
{"upwards":["Ancient Greek religion","History","Hellenistic period"],"downwards":[]}
{"upwards":["Ancient Greek religion","History","Roman Empire"],"downwards":[]}
{"upwards":["Ancient Greek religion","History","Suppression and decline"],"downwards":[]}
{"upwards":["Ancient Greek religion","History","Modern revivals"],"downwards":[]}
{"upwards":["Square root of 2"],"downwards":["History","Decimal value","Proofs of irrationality","Multiplicative inverse","Properties","Representations","Applications"]}
{"upwards":["Square root of 2","History"],"downwards":["Ancient Roman architecture"]}
{"upwards":["Square root of 2","History","Ancient Roman architecture"],"downwards":[]}
{"upwards":["Square root of 2","Decimal value"],"downwards":["Computation algorithms","Rational approximations","Records in computation"]}
{"upwards":["Square root of 2","Decimal value","Computation algorithms"],"downwards":[]}
{"upwards":["Square root of 2","Decimal value","Rational approximations"],"downwards":[]}
{"upwards":["Square root of 2","Decimal value","Records in computation"],"downwards":[]}
{"upwards":["Square root of 2","Proofs of irrationality"],"downwards":["Proof by infinite descent","Proof by unique factorization","Geometric proof","Constructive proof","Proof by Pythagorean triples"]}
{"upwards":["Square root of 2","Proofs of irrationality","Proof by infinite descent"],"downwards":[]}
{"upwards":["Square root of 2","Proofs of irrationality","Proof by unique factorization"],"downwards":[]}
{"upwards":["Square root of 2","Proofs of irrationality","Geometric proof"],"downwards":[]}
{"upwards":["Square root of 2","Proofs of irrationality","Constructive proof"],"downwards":[]}
{"upwards":["Square root of 2","Proofs of irrationality","Proof by Pythagorean triples"],"downwards":[]}
{"upwards":["Square root of 2","Multiplicative inverse"],"downwards":[]}
{"upwards":["Square root of 2","Properties"],"downwards":[]}
{"upwards":["Square root of 2","Representations"],"downwards":["Series and product","Continued fraction","Nested square"]}
{"upwards":["Square root of 2","Representations","Series and product"],"downwards":[]}
{"upwards":["Square root of 2","Representations","Continued fraction"],"downwards":[]}
{"upwards":["Square root of 2","Representations","Nested square"],"downwards":[]}
{"upwards":["Square root of 2","Applications"],"downwards":["Paper size","Physical sciences"]}
{"upwards":["Square root of 2","Applications","Paper size"],"downwards":[]}
{"upwards":["Square root of 2","Applications","Physical sciences"],"downwards":[]}
{"upwards":["Repeating decimal"],"downwards":["Background","Table of values"]}
{"upwards":["Repeating decimal","Background"],"downwards":["Notation","Decimal expansion and recurrence sequence","Every rational number is either a terminating or repeating decimal","Every repeating or terminating decimal is a rational number"]}
{"upwards":["Repeating decimal","Background","Notation"],"downwards":[]}
{"upwards":["Repeating decimal","Background","Decimal expansion and recurrence sequence"],"downwards":[]}
{"upwards":["Repeating decimal","Background","Every rational number is either a terminating or repeating decimal"],"downwards":[]}
{"upwards":["Repeating decimal","Background","Every repeating or terminating decimal is a rational number"],"downwards":[]}
{"upwards":["Repeating decimal","Table of values"],"downwards":[]}
{"upwards":["Golden ratio"],"downwards":["Calculation","History","Mathematics","Applications and observations","Disputed observations"]}
{"upwards":["Golden ratio","Calculation"],"downwards":[]}
{"upwards":["Golden ratio","History"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics"],"downwards":["Irrationality","Minimal polynomial","Golden ratio conjugate and powers","Continued fraction and square root","Relationship to Fibonacci and Lucas numbers","Geometry","Other properties"]}
{"upwards":["Golden ratio","Mathematics","Irrationality"],"downwards":["Contradiction from an expression in lowest terms","By irrationality of \u221a5"]}
{"upwards":["Golden ratio","Mathematics","Irrationality","Contradiction from an expression in lowest terms"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Irrationality","By irrationality of \u221a5"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Minimal polynomial"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Golden ratio conjugate and powers"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Continued fraction and square root"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Relationship to Fibonacci and Lucas numbers"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Geometry"],"downwards":["Construction","Golden angle","Pentagonal symmetry system","In triangles and quadrilaterals","Golden spiral","In the dodecahedron and icosahedron"]}
{"upwards":["Golden ratio","Mathematics","Geometry","Construction"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Geometry","Golden angle"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Geometry","Pentagonal symmetry system"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Geometry","In triangles and quadrilaterals"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Geometry","Golden spiral"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Geometry","In the dodecahedron and icosahedron"],"downwards":[]}
{"upwards":["Golden ratio","Mathematics","Other properties"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations"],"downwards":["Architecture","Art","Books and design","Flags","Music","Nature","Physics","Optimization"]}
{"upwards":["Golden ratio","Applications and observations","Architecture"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Art"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Books and design"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Flags"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Music"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Nature"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Physics"],"downwards":[]}
{"upwards":["Golden ratio","Applications and observations","Optimization"],"downwards":[]}
{"upwards":["Golden ratio","Disputed observations"],"downwards":["Egyptian pyramids","The Parthenon","Modern art"]}
{"upwards":["Golden ratio","Disputed observations","Egyptian pyramids"],"downwards":[]}
{"upwards":["Golden ratio","Disputed observations","The Parthenon"],"downwards":[]}
{"upwards":["Golden ratio","Disputed observations","Modern art"],"downwards":[]}
{"upwards":["Multiplication"],"downwards":["Notation and terminology","Definitions","Computation","Products of measurements","Product of a sequence","Exponentiation","Properties","Axioms","Multiplication with set theory","Multiplication in group theory","Multiplication of different kinds of numbers"]}
{"upwards":["Multiplication","Notation and terminology"],"downwards":[]}
{"upwards":["Multiplication","Definitions"],"downwards":["Product of two natural numbers","Product of two integers","Product of two fractions","Product of two real numbers","Product of two complex numbers","Product of two quaternions"]}
{"upwards":["Multiplication","Definitions","Product of two natural numbers"],"downwards":[]}
{"upwards":["Multiplication","Definitions","Product of two integers"],"downwards":[]}
{"upwards":["Multiplication","Definitions","Product of two fractions"],"downwards":[]}
{"upwards":["Multiplication","Definitions","Product of two real numbers"],"downwards":[]}
{"upwards":["Multiplication","Definitions","Product of two complex numbers"],"downwards":[]}
{"upwards":["Multiplication","Definitions","Product of two quaternions"],"downwards":[]}
{"upwards":["Multiplication","Computation"],"downwards":["Historical algorithms","Modern methods","Computer algorithms"]}
{"upwards":["Multiplication","Computation","Historical algorithms"],"downwards":["Egyptians","Babylonians","Chinese"]}
{"upwards":["Multiplication","Computation","Historical algorithms","Egyptians"],"downwards":[]}
{"upwards":["Multiplication","Computation","Historical algorithms","Babylonians"],"downwards":[]}
{"upwards":["Multiplication","Computation","Historical algorithms","Chinese"],"downwards":[]}
{"upwards":["Multiplication","Computation","Modern methods"],"downwards":["Grid method"]}
{"upwards":["Multiplication","Computation","Modern methods","Grid method"],"downwards":[]}
{"upwards":["Multiplication","Computation","Computer algorithms"],"downwards":[]}
{"upwards":["Multiplication","Products of measurements"],"downwards":[]}
{"upwards":["Multiplication","Product of a sequence"],"downwards":["Capital pi notation","Infinite products"]}
{"upwards":["Multiplication","Product of a sequence","Capital pi notation"],"downwards":["Properties of capital pi notation"]}
{"upwards":["Multiplication","Product of a sequence","Capital pi notation","Properties of capital pi notation"],"downwards":[]}
{"upwards":["Multiplication","Product of a sequence","Infinite products"],"downwards":[]}
{"upwards":["Multiplication","Exponentiation"],"downwards":[]}
{"upwards":["Multiplication","Properties"],"downwards":[]}
{"upwards":["Multiplication","Axioms"],"downwards":[]}
{"upwards":["Multiplication","Multiplication with set theory"],"downwards":[]}
{"upwards":["Multiplication","Multiplication in group theory"],"downwards":[]}
{"upwards":["Multiplication","Multiplication of different kinds of numbers"],"downwards":[]}
{"upwards":["Field extension"],"downwards":["Subfield","Extension field","Caveats","Examples","Algebraic extension","Transcendental extension","Normal, separable and Galois extensions","Generalizations","Extension of scalars"]}
{"upwards":["Field extension","Subfield"],"downwards":[]}
{"upwards":["Field extension","Extension field"],"downwards":[]}
{"upwards":["Field extension","Caveats"],"downwards":[]}
{"upwards":["Field extension","Examples"],"downwards":[]}
{"upwards":["Field extension","Algebraic extension"],"downwards":[]}
{"upwards":["Field extension","Transcendental extension"],"downwards":[]}
{"upwards":["Field extension","Normal, separable and Galois extensions"],"downwards":[]}
{"upwards":["Field extension","Generalizations"],"downwards":[]}
{"upwards":["Field extension","Extension of scalars"],"downwards":[]}
{"upwards":["1832"],"downwards":["Events","Births","Deaths","References"]}
{"upwards":["1832","Events"],"downwards":["January\u2013March","April\u2013June","July\u2013September","October\u2013December","Date unknown"]}
{"upwards":["1832","Events","January\u2013March"],"downwards":[]}
{"upwards":["1832","Events","April\u2013June"],"downwards":[]}
{"upwards":["1832","Events","July\u2013September"],"downwards":[]}
{"upwards":["1832","Events","October\u2013December"],"downwards":[]}
{"upwards":["1832","Events","Date unknown"],"downwards":[]}
{"upwards":["1832","Births"],"downwards":["January\u2013June","July\u2013December","Date unknown"]}
{"upwards":["1832","Births","January\u2013June"],"downwards":[]}
{"upwards":["1832","Births","July\u2013December"],"downwards":[]}
{"upwards":["1832","Births","Date unknown"],"downwards":[]}
{"upwards":["1832","Deaths"],"downwards":["January\u2013June","July\u2013December"]}
{"upwards":["1832","Deaths","January\u2013June"],"downwards":[]}
{"upwards":["1832","Deaths","July\u2013December"],"downwards":[]}
{"upwards":["1832","References"],"downwards":[]}
{"upwards":["List of mathematical jargon"],"downwards":["Philosophy of mathematics","Descriptive informalities","Proof terminology","Proof techniques"]}
{"upwards":["List of mathematical jargon","Philosophy of mathematics"],"downwards":[]}
{"upwards":["List of mathematical jargon","Descriptive informalities"],"downwards":[]}
{"upwards":["List of mathematical jargon","Proof terminology"],"downwards":[]}
{"upwards":["List of mathematical jargon","Proof techniques"],"downwards":[]}
{"upwards":["Equivalence class"],"downwards":["Examples","Definition and notation","Properties","Graphical representation","Invariants","Quotient space in topology"]}
{"upwards":["Equivalence class","Examples"],"downwards":[]}
{"upwards":["Equivalence class","Definition and notation"],"downwards":[]}
{"upwards":["Equivalence class","Properties"],"downwards":[]}
{"upwards":["Equivalence class","Graphical representation"],"downwards":[]}
{"upwards":["Equivalence class","Invariants"],"downwards":[]}
{"upwards":["Equivalence class","Quotient space in topology"],"downwards":[]}
{"upwards":["Addition"],"downwards":["Notation and terminology","Interpretations","Properties","Performing addition","Addition of numbers","Generalizations","Related operations","In music"]}
{"upwards":["Addition","Notation and terminology"],"downwards":["Terms"]}
{"upwards":["Addition","Notation and terminology","Terms"],"downwards":[]}
{"upwards":["Addition","Interpretations"],"downwards":["Combining sets","Extending a length"]}
{"upwards":["Addition","Interpretations","Combining sets"],"downwards":[]}
{"upwards":["Addition","Interpretations","Extending a length"],"downwards":[]}
{"upwards":["Addition","Properties"],"downwards":["Commutativity","Associativity","Identity element","Successor","Units"]}
{"upwards":["Addition","Properties","Commutativity"],"downwards":[]}
{"upwards":["Addition","Properties","Associativity"],"downwards":[]}
{"upwards":["Addition","Properties","Identity element"],"downwards":[]}
{"upwards":["Addition","Properties","Successor"],"downwards":[]}
{"upwards":["Addition","Properties","Units"],"downwards":[]}
{"upwards":["Addition","Performing addition"],"downwards":["Innate ability","Childhood learning","Decimal system","Non-decimal","Computers"]}
{"upwards":["Addition","Performing addition","Innate ability"],"downwards":[]}
{"upwards":["Addition","Performing addition","Childhood learning"],"downwards":["Table"]}
{"upwards":["Addition","Performing addition","Childhood learning","Table"],"downwards":[]}
{"upwards":["Addition","Performing addition","Decimal system"],"downwards":["Carry","Decimal fractions","Scientific notation"]}
{"upwards":["Addition","Performing addition","Decimal system","Carry"],"downwards":[]}
{"upwards":["Addition","Performing addition","Decimal system","Decimal fractions"],"downwards":[]}
{"upwards":["Addition","Performing addition","Decimal system","Scientific notation"],"downwards":[]}
{"upwards":["Addition","Performing addition","Non-decimal"],"downwards":[]}
{"upwards":["Addition","Performing addition","Computers"],"downwards":[]}
{"upwards":["Addition","Addition of numbers"],"downwards":["Natural numbers","Integers","Rational numbers (fractions)","Real numbers","Complex numbers"]}
{"upwards":["Addition","Addition of numbers","Natural numbers"],"downwards":[]}
{"upwards":["Addition","Addition of numbers","Integers"],"downwards":[]}
{"upwards":["Addition","Addition of numbers","Rational numbers (fractions)"],"downwards":[]}
{"upwards":["Addition","Addition of numbers","Real numbers"],"downwards":[]}
{"upwards":["Addition","Addition of numbers","Complex numbers"],"downwards":[]}
{"upwards":["Addition","Generalizations"],"downwards":["Abstract algebra","Set theory and category theory"]}
{"upwards":["Addition","Generalizations","Abstract algebra"],"downwards":["Vectors","Matrices","Modular arithmetic","General theory"]}
{"upwards":["Addition","Generalizations","Abstract algebra","Vectors"],"downwards":[]}
{"upwards":["Addition","Generalizations","Abstract algebra","Matrices"],"downwards":[]}
{"upwards":["Addition","Generalizations","Abstract algebra","Modular arithmetic"],"downwards":[]}
{"upwards":["Addition","Generalizations","Abstract algebra","General theory"],"downwards":[]}
{"upwards":["Addition","Generalizations","Set theory and category theory"],"downwards":[]}
{"upwards":["Addition","Related operations"],"downwards":["Arithmetic","Ordering","Other ways to add"]}
{"upwards":["Addition","Related operations","Arithmetic"],"downwards":[]}
{"upwards":["Addition","Related operations","Ordering"],"downwards":[]}
{"upwards":["Addition","Related operations","Other ways to add"],"downwards":[]}
{"upwards":["Addition","In music"],"downwards":[]}
{"upwards":["Cauchy sequence"],"downwards":["In real numbers","In a metric space","Completeness","Generalizations"]}
{"upwards":["Cauchy sequence","In real numbers"],"downwards":["Modulus of Cauchy convergence"]}
{"upwards":["Cauchy sequence","In real numbers","Modulus of Cauchy convergence"],"downwards":[]}
{"upwards":["Cauchy sequence","In a metric space"],"downwards":[]}
{"upwards":["Cauchy sequence","Completeness"],"downwards":["Examples","Non-example: rational numbers","Non-example: open interval","Other properties"]}
{"upwards":["Cauchy sequence","Completeness","Examples"],"downwards":[]}
{"upwards":["Cauchy sequence","Completeness","Non-example: rational numbers"],"downwards":[]}
{"upwards":["Cauchy sequence","Completeness","Non-example: open interval"],"downwards":[]}
{"upwards":["Cauchy sequence","Completeness","Other properties"],"downwards":[]}
{"upwards":["Cauchy sequence","Generalizations"],"downwards":["In topological vector spaces","In topological groups","In groups","In a hyperreal continuum","Cauchy completion of categories"]}
{"upwards":["Cauchy sequence","Generalizations","In topological vector spaces"],"downwards":[]}
{"upwards":["Cauchy sequence","Generalizations","In topological groups"],"downwards":[]}
{"upwards":["Cauchy sequence","Generalizations","In groups"],"downwards":[]}
{"upwards":["Cauchy sequence","Generalizations","In a hyperreal continuum"],"downwards":[]}
{"upwards":["Cauchy sequence","Generalizations","Cauchy completion of categories"],"downwards":[]}
{"upwards":["Mathematical analysis"],"downwards":["History","Important concepts","Main branches","Other topics","Applications","Famous Textbooks"]}
{"upwards":["Mathematical analysis","History"],"downwards":["Ancient","Medieval","Modern"]}
{"upwards":["Mathematical analysis","History","Ancient"],"downwards":[]}
{"upwards":["Mathematical analysis","History","Medieval"],"downwards":[]}
{"upwards":["Mathematical analysis","History","Modern"],"downwards":["Foundations","Modernization"]}
{"upwards":["Mathematical analysis","History","Modern","Foundations"],"downwards":[]}
{"upwards":["Mathematical analysis","History","Modern","Modernization"],"downwards":[]}
{"upwards":["Mathematical analysis","Important concepts"],"downwards":["Metric spaces","Sequences and limits"]}
{"upwards":["Mathematical analysis","Important concepts","Metric spaces"],"downwards":[]}
{"upwards":["Mathematical analysis","Important concepts","Sequences and limits"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches"],"downwards":["Real analysis","Complex analysis","Functional analysis","Harmonic analysis","Differential equations","Measure theory","Numerical analysis","Vector analysis","Scalar analysis","Tensor analysis"]}
{"upwards":["Mathematical analysis","Main branches","Real analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Complex analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Functional analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Harmonic analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Differential equations"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Measure theory"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Numerical analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Vector analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Scalar analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Main branches","Tensor analysis"],"downwards":[]}
{"upwards":["Mathematical analysis","Other topics"],"downwards":[]}
{"upwards":["Mathematical analysis","Applications"],"downwards":["Physical sciences","Signal processing","Other areas of mathematics"]}
{"upwards":["Mathematical analysis","Applications","Physical sciences"],"downwards":[]}
{"upwards":["Mathematical analysis","Applications","Signal processing"],"downwards":[]}
{"upwards":["Mathematical analysis","Applications","Other areas of mathematics"],"downwards":[]}
{"upwards":["Mathematical analysis","Famous Textbooks"],"downwards":[]}
{"upwards":["Dedekind cut"],"downwards":["Definition","Representations","Ordering of cuts","Construction of the real numbers","Relation to interval arithmetic","Generalizations","Notes","References","External links"]}
{"upwards":["Dedekind cut","Definition"],"downwards":[]}
{"upwards":["Dedekind cut","Representations"],"downwards":[]}
{"upwards":["Dedekind cut","Ordering of cuts"],"downwards":[]}
{"upwards":["Dedekind cut","Construction of the real numbers"],"downwards":[]}
{"upwards":["Dedekind cut","Relation to interval arithmetic"],"downwards":[]}
{"upwards":["Dedekind cut","Generalizations"],"downwards":["Arbitrary linearly ordered sets","Surreal numbers","Partially ordered sets"]}
{"upwards":["Dedekind cut","Generalizations","Arbitrary linearly ordered sets"],"downwards":[]}
{"upwards":["Dedekind cut","Generalizations","Surreal numbers"],"downwards":[]}
{"upwards":["Dedekind cut","Generalizations","Partially ordered sets"],"downwards":[]}
{"upwards":["Dedekind cut","Notes"],"downwards":[]}
{"upwards":["Dedekind cut","References"],"downwards":[]}
{"upwards":["Dedekind cut","External links"],"downwards":[]}
{"upwards":["Algebraic number field"],"downwards":["Definition","Examples","Algebraicity, and ring of integers","Bases for number fields","Regular representation, trace and discriminant","Places","Ramification","Galois groups and Galois cohomology","Local-global principle"]}
{"upwards":["Algebraic number field","Definition"],"downwards":["Prerequisites","Definition"]}
{"upwards":["Algebraic number field","Definition","Prerequisites"],"downwards":[]}
{"upwards":["Algebraic number field","Definition","Definition"],"downwards":[]}
{"upwards":["Algebraic number field","Examples"],"downwards":["Non-examples"]}
{"upwards":["Algebraic number field","Examples","Non-examples"],"downwards":[]}
{"upwards":["Algebraic number field","Algebraicity, and ring of integers"],"downwards":["Unique factorization","Analytic objects: \u03b6-functions, L-functions, and class number formula"]}
{"upwards":["Algebraic number field","Algebraicity, and ring of integers","Unique factorization"],"downwards":[]}
{"upwards":["Algebraic number field","Algebraicity, and ring of integers","Analytic objects: \u03b6-functions, L-functions, and class number formula"],"downwards":[]}
{"upwards":["Algebraic number field","Bases for number fields"],"downwards":["Integral basis","Power basis"]}
{"upwards":["Algebraic number field","Bases for number fields","Integral basis"],"downwards":[]}
{"upwards":["Algebraic number field","Bases for number fields","Power basis"],"downwards":[]}
{"upwards":["Algebraic number field","Regular representation, trace and discriminant"],"downwards":["Example","Properties","Example with integral basis"]}
{"upwards":["Algebraic number field","Regular representation, trace and discriminant","Example"],"downwards":[]}
{"upwards":["Algebraic number field","Regular representation, trace and discriminant","Properties"],"downwards":[]}
{"upwards":["Algebraic number field","Regular representation, trace and discriminant","Example with integral basis"],"downwards":[]}
{"upwards":["Algebraic number field","Places"],"downwards":["Examples","Archimedean places","Non-Archimedean or ultrametric places","Prime ideals in OK"]}
{"upwards":["Algebraic number field","Places","Examples"],"downwards":[]}
{"upwards":["Algebraic number field","Places","Archimedean places"],"downwards":[]}
{"upwards":["Algebraic number field","Places","Non-Archimedean or ultrametric places"],"downwards":[]}
{"upwards":["Algebraic number field","Places","Prime ideals in OK"],"downwards":["Lying over theorem and places"]}
{"upwards":["Algebraic number field","Places","Prime ideals in OK","Lying over theorem and places"],"downwards":[]}
{"upwards":["Algebraic number field","Ramification"],"downwards":["An example","Dedekind discriminant theorem"]}
{"upwards":["Algebraic number field","Ramification","An example"],"downwards":[]}
{"upwards":["Algebraic number field","Ramification","Dedekind discriminant theorem"],"downwards":[]}
{"upwards":["Algebraic number field","Galois groups and Galois cohomology"],"downwards":[]}
{"upwards":["Algebraic number field","Local-global principle"],"downwards":["Local and global fields","Hasse principle","Adeles and ideles"]}
{"upwards":["Algebraic number field","Local-global principle","Local and global fields"],"downwards":[]}
{"upwards":["Algebraic number field","Local-global principle","Hasse principle"],"downwards":[]}
{"upwards":["Algebraic number field","Local-global principle","Adeles and ideles"],"downwards":[]}
{"upwards":["Coefficient"],"downwards":["Terminology and definition","Linear algebra"]}
{"upwards":["Coefficient","Terminology and definition"],"downwards":[]}
{"upwards":["Coefficient","Linear algebra"],"downwards":[]}
{"upwards":["Ratio"],"downwards":["Notation and terminology","History and etymology","Number of terms and use of fractions","Proportions and percentage ratios","Reduction","Irrational ratios","Odds","Units","Triangular coordinates"]}
{"upwards":["Ratio","Notation and terminology"],"downwards":[]}
{"upwards":["Ratio","History and etymology"],"downwards":["Euclid's definitions"]}
{"upwards":["Ratio","History and etymology","Euclid's definitions"],"downwards":[]}
{"upwards":["Ratio","Number of terms and use of fractions"],"downwards":[]}
{"upwards":["Ratio","Proportions and percentage ratios"],"downwards":[]}
{"upwards":["Ratio","Reduction"],"downwards":[]}
{"upwards":["Ratio","Irrational ratios"],"downwards":[]}
{"upwards":["Ratio","Odds"],"downwards":[]}
{"upwards":["Ratio","Units"],"downwards":[]}
{"upwards":["Ratio","Triangular coordinates"],"downwards":[]}
{"upwards":["Almost all"],"downwards":["Meanings in different areas of mathematics","Proofs"]}
{"upwards":["Almost all","Meanings in different areas of mathematics"],"downwards":["Prevalent meaning","Meaning in measure theory","Meaning in number theory","Meaning in graph theory","Meaning in topology","Meaning in algebra"]}
{"upwards":["Almost all","Meanings in different areas of mathematics","Prevalent meaning"],"downwards":[]}
{"upwards":["Almost all","Meanings in different areas of mathematics","Meaning in measure theory"],"downwards":[]}
{"upwards":["Almost all","Meanings in different areas of mathematics","Meaning in number theory"],"downwards":[]}
{"upwards":["Almost all","Meanings in different areas of mathematics","Meaning in graph theory"],"downwards":[]}
{"upwards":["Almost all","Meanings in different areas of mathematics","Meaning in topology"],"downwards":[]}
{"upwards":["Almost all","Meanings in different areas of mathematics","Meaning in algebra"],"downwards":[]}
{"upwards":["Almost all","Proofs"],"downwards":[]}
{"upwards":["Characteristic (algebra)"],"downwards":["Motivation","Equivalent characterizations","Case of rings","Case of fields","Notes","References","Sources"]}
{"upwards":["Characteristic (algebra)","Motivation"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Equivalent characterizations"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Case of rings"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Case of fields"],"downwards":["Fields of characteristic zero","Fields of prime characteristic"]}
{"upwards":["Characteristic (algebra)","Case of fields","Fields of characteristic zero"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Case of fields","Fields of prime characteristic"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Notes"],"downwards":[]}
{"upwards":["Characteristic (algebra)","References"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Sources"],"downwards":[]}
{"upwards":["Rational point"],"downwards":["Definition","Rational points on curves","Higher dimensions","Counting points over finite fields"]}
{"upwards":["Rational point","Definition"],"downwards":[]}
{"upwards":["Rational point","Rational points on curves"],"downwards":["Genus 0","Genus 1","Genus at least 2"]}
{"upwards":["Rational point","Rational points on curves","Genus 0"],"downwards":[]}
{"upwards":["Rational point","Rational points on curves","Genus 1"],"downwards":[]}
{"upwards":["Rational point","Rational points on curves","Genus at least 2"],"downwards":[]}
{"upwards":["Rational point","Higher dimensions"],"downwards":["Varieties with few rational points","Varieties with many rational points"]}
{"upwards":["Rational point","Higher dimensions","Varieties with few rational points"],"downwards":[]}
{"upwards":["Rational point","Higher dimensions","Varieties with many rational points"],"downwards":[]}
{"upwards":["Rational point","Counting points over finite fields"],"downwards":[]}
{"upwards":["Complete metric space"],"downwards":["Definition","Examples","Some theorems","Completion","Topologically complete spaces","Alternatives and generalizations"]}
{"upwards":["Complete metric space","Definition"],"downwards":[]}
{"upwards":["Complete metric space","Examples"],"downwards":[]}
{"upwards":["Complete metric space","Some theorems"],"downwards":[]}
{"upwards":["Complete metric space","Completion"],"downwards":[]}
{"upwards":["Complete metric space","Topologically complete spaces"],"downwards":[]}
{"upwards":["Complete metric space","Alternatives and generalizations"],"downwards":[]}
{"upwards":["Characteristic (algebra)"],"downwards":["Motivation","Equivalent characterizations","Case of rings","Case of fields","Notes","References","Sources"]}
{"upwards":["Characteristic (algebra)","Motivation"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Equivalent characterizations"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Case of rings"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Case of fields"],"downwards":["Fields of characteristic zero","Fields of prime characteristic"]}
{"upwards":["Characteristic (algebra)","Case of fields","Fields of characteristic zero"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Case of fields","Fields of prime characteristic"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Notes"],"downwards":[]}
{"upwards":["Characteristic (algebra)","References"],"downwards":[]}
{"upwards":["Characteristic (algebra)","Sources"],"downwards":[]}
{"upwards":["Coordinate system"],"downwards":["Common coordinate systems","Coordinates of geometric objects","Transformations","Coordinate lines\/curves and planes\/surfaces","Coordinate maps","Orientation-based coordinates","Geographic systems"]}
{"upwards":["Coordinate system","Common coordinate systems"],"downwards":["Number line","Cartesian coordinate system","Polar coordinate system","Cylindrical and spherical coordinate systems","Homogeneous coordinate system","Other commonly used systems"]}
{"upwards":["Coordinate system","Common coordinate systems","Number line"],"downwards":[]}
{"upwards":["Coordinate system","Common coordinate systems","Cartesian coordinate system"],"downwards":[]}
{"upwards":["Coordinate system","Common coordinate systems","Polar coordinate system"],"downwards":[]}
{"upwards":["Coordinate system","Common coordinate systems","Cylindrical and spherical coordinate systems"],"downwards":[]}
{"upwards":["Coordinate system","Common coordinate systems","Homogeneous coordinate system"],"downwards":[]}
{"upwards":["Coordinate system","Common coordinate systems","Other commonly used systems"],"downwards":[]}
{"upwards":["Coordinate system","Coordinates of geometric objects"],"downwards":[]}
{"upwards":["Coordinate system","Transformations"],"downwards":[]}
{"upwards":["Coordinate system","Coordinate lines\/curves and planes\/surfaces"],"downwards":[]}
{"upwards":["Coordinate system","Coordinate maps"],"downwards":[]}
{"upwards":["Coordinate system","Orientation-based coordinates"],"downwards":[]}
{"upwards":["Coordinate system","Geographic systems"],"downwards":[]}
{"upwards":["Rational function"],"downwards":["Definitions","Examples","Taylor series","Abstract algebra and geometric notion","Applications"]}
{"upwards":["Rational function","Definitions"],"downwards":["Degree"]}
{"upwards":["Rational function","Definitions","Degree"],"downwards":[]}
{"upwards":["Rational function","Examples"],"downwards":[]}
{"upwards":["Rational function","Taylor series"],"downwards":[]}
{"upwards":["Rational function","Abstract algebra and geometric notion"],"downwards":["Complex rational functions","Notion of a rational function on an algebraic variety"]}
{"upwards":["Rational function","Abstract algebra and geometric notion","Complex rational functions"],"downwards":[]}
{"upwards":["Rational function","Abstract algebra and geometric notion","Notion of a rational function on an algebraic variety"],"downwards":[]}
{"upwards":["Rational function","Applications"],"downwards":[]}
{"upwards":["Canonical form"],"downwards":["Definition","History","Examples"]}
{"upwards":["Canonical form","Definition"],"downwards":[]}
{"upwards":["Canonical form","History"],"downwards":[]}
{"upwards":["Canonical form","Examples"],"downwards":["Large number notation","Number theory","Linear algebra","Algebra","Geometry","Integrable systems","Dynamical systems","Three dimensional geometry","Functional analysis","Classical logic","Set theory","Game theory","Proof theory","Rewriting systems","Lambda calculus","Graph theory","Computing"]}
{"upwards":["Canonical form","Examples","Large number notation"],"downwards":[]}
{"upwards":["Canonical form","Examples","Number theory"],"downwards":[]}
{"upwards":["Canonical form","Examples","Linear algebra"],"downwards":[]}
{"upwards":["Canonical form","Examples","Algebra"],"downwards":[]}
{"upwards":["Canonical form","Examples","Geometry"],"downwards":[]}
{"upwards":["Canonical form","Examples","Integrable systems"],"downwards":[]}
{"upwards":["Canonical form","Examples","Dynamical systems"],"downwards":[]}
{"upwards":["Canonical form","Examples","Three dimensional geometry"],"downwards":[]}
{"upwards":["Canonical form","Examples","Functional analysis"],"downwards":[]}
{"upwards":["Canonical form","Examples","Classical logic"],"downwards":[]}
{"upwards":["Canonical form","Examples","Set theory"],"downwards":[]}
{"upwards":["Canonical form","Examples","Game theory"],"downwards":[]}
{"upwards":["Canonical form","Examples","Proof theory"],"downwards":[]}
{"upwards":["Canonical form","Examples","Rewriting systems"],"downwards":[]}
{"upwards":["Canonical form","Examples","Lambda calculus"],"downwards":[]}
{"upwards":["Canonical form","Examples","Graph theory"],"downwards":[]}
{"upwards":["Canonical form","Examples","Computing"],"downwards":[]}
{"upwards":["Coprime integers"],"downwards":["Notation and testing","Properties","Coprimality in sets","Coprimality in ring ideals","Probability of coprimality","Generating all coprime pairs","Applications","Generalizations"]}
{"upwards":["Coprime integers","Notation and testing"],"downwards":[]}
{"upwards":["Coprime integers","Properties"],"downwards":[]}
{"upwards":["Coprime integers","Coprimality in sets"],"downwards":[]}
{"upwards":["Coprime integers","Coprimality in ring ideals"],"downwards":[]}
{"upwards":["Coprime integers","Probability of coprimality"],"downwards":[]}
{"upwards":["Coprime integers","Generating all coprime pairs"],"downwards":[]}
{"upwards":["Coprime integers","Applications"],"downwards":[]}
{"upwards":["Coprime integers","Generalizations"],"downwards":[]}
{"upwards":["Irreducible fraction"],"downwards":["Examples","Uniqueness","Applications","Generalization"]}
{"upwards":["Irreducible fraction","Examples"],"downwards":[]}
{"upwards":["Irreducible fraction","Uniqueness"],"downwards":[]}
{"upwards":["Irreducible fraction","Applications"],"downwards":[]}
{"upwards":["Irreducible fraction","Generalization"],"downwards":[]}
{"upwards":["Greek mathematics"],"downwards":["Origins and etymology","Archaic and Classical periods","Hellenistic and Roman periods","Achievements","Transmission and the manuscript tradition"]}
{"upwards":["Greek mathematics","Origins and etymology"],"downwards":[]}
{"upwards":["Greek mathematics","Archaic and Classical periods"],"downwards":[]}
{"upwards":["Greek mathematics","Hellenistic and Roman periods"],"downwards":[]}
{"upwards":["Greek mathematics","Achievements"],"downwards":[]}
{"upwards":["Greek mathematics","Transmission and the manuscript tradition"],"downwards":[]}
{"upwards":["Imaginary number"],"downwards":["History","Geometric interpretation","Square roots of negative numbers"]}
{"upwards":["Imaginary number","History"],"downwards":[]}
{"upwards":["Imaginary number","Geometric interpretation"],"downwards":[]}
{"upwards":["Imaginary number","Square roots of negative numbers"],"downwards":[]}
{"upwards":["Fraction"],"downwards":["Vocabulary","Forms of fractions","Arithmetic with fractions","Fractions in abstract mathematics","Algebraic fractions","Radical expressions","Typographical variations","History","In formal education"]}
{"upwards":["Fraction","Vocabulary"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions"],"downwards":["Simple, common, or vulgar fractions","Proper and improper fractions","Reciprocals and the \"invisible denominator\"","Ratios","Decimal fractions and percentages","Mixed numbers","Historical notions"]}
{"upwards":["Fraction","Forms of fractions","Simple, common, or vulgar fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Proper and improper fractions"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Reciprocals and the \"invisible denominator\""],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Ratios"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Decimal fractions and percentages"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions"],"downwards":["Egyptian fraction","Complex and compound fractions"]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Egyptian fraction"],"downwards":[]}
{"upwards":["Fraction","Forms of fractions","Historical notions","Complex and compound fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions"],"downwards":["Equivalent fractions","Comparing fractions","Addition","Subtraction","Multiplication","Division","Converting between decimals and fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions"],"downwards":["Simplifying (reducing) fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Equivalent fractions","Simplifying (reducing) fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Comparing fractions"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Addition"],"downwards":["Adding unlike quantities"]}
{"upwards":["Fraction","Arithmetic with fractions","Addition","Adding unlike quantities"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Subtraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication"],"downwards":["Multiplying a fraction by another fraction","Multiplying a fraction by a whole number","Multiplying mixed numbers"]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by another fraction"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying a fraction by a whole number"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Multiplication","Multiplying mixed numbers"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Division"],"downwards":[]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions"],"downwards":["Converting repeating decimals to fractions"]}
{"upwards":["Fraction","Arithmetic with fractions","Converting between decimals and fractions","Converting repeating decimals to fractions"],"downwards":[]}
{"upwards":["Fraction","Fractions in abstract mathematics"],"downwards":[]}
{"upwards":["Fraction","Algebraic fractions"],"downwards":[]}
{"upwards":["Fraction","Radical expressions"],"downwards":[]}
{"upwards":["Fraction","Typographical variations"],"downwards":[]}
{"upwards":["Fraction","History"],"downwards":[]}
{"upwards":["Fraction","In formal education"],"downwards":["Pedagogical tools","Documents for teachers"]}
{"upwards":["Fraction","In formal education","Pedagogical tools"],"downwards":[]}
{"upwards":["Fraction","In formal education","Documents for teachers"],"downwards":[]}
{"upwards":["Greatest common divisor"],"downwards":["Overview","Applications","Calculation","Properties","Probabilities and expected value","In commutative rings"]}
{"upwards":["Greatest common divisor","Overview"],"downwards":["Definition","Example","Coprime numbers","A geometric view"]}
{"upwards":["Greatest common divisor","Overview","Definition"],"downwards":[]}
{"upwards":["Greatest common divisor","Overview","Example"],"downwards":[]}
{"upwards":["Greatest common divisor","Overview","Coprime numbers"],"downwards":[]}
{"upwards":["Greatest common divisor","Overview","A geometric view"],"downwards":[]}
{"upwards":["Greatest common divisor","Applications"],"downwards":["Reducing fractions","Least common multiple"]}
{"upwards":["Greatest common divisor","Applications","Reducing fractions"],"downwards":[]}
{"upwards":["Greatest common divisor","Applications","Least common multiple"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation"],"downwards":["Using prime factorizations","Euclid's algorithm","Euclidean algorithm","Lehmer's GCD algorithm","Binary GCD algorithm","Other methods","Complexity"]}
{"upwards":["Greatest common divisor","Calculation","Using prime factorizations"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation","Euclid's algorithm"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation","Euclidean algorithm"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation","Lehmer's GCD algorithm"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation","Binary GCD algorithm"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation","Other methods"],"downwards":[]}
{"upwards":["Greatest common divisor","Calculation","Complexity"],"downwards":[]}
{"upwards":["Greatest common divisor","Properties"],"downwards":[]}
{"upwards":["Greatest common divisor","Probabilities and expected value"],"downwards":[]}
{"upwards":["Greatest common divisor","In commutative rings"],"downwards":[]}
{"upwards":["Algebraic fraction"],"downwards":["Terminology","Rational fractions","Irrational fractions"]}
{"upwards":["Algebraic fraction","Terminology"],"downwards":[]}
{"upwards":["Algebraic fraction","Rational fractions"],"downwards":[]}
{"upwards":["Algebraic fraction","Irrational fractions"],"downwards":[]}
{"upwards":["New Orleans"],"downwards":["Etymology and nicknames","History","Geography","Demographics","Economy","Culture and contemporary life","Sports","National protected areas","Government","Crime","Education","Media","Transportation","Notable people","Sister cities"]}
{"upwards":["New Orleans","Etymology and nicknames"],"downwards":[]}
{"upwards":["New Orleans","History"],"downwards":["French\u2013Spanish colonial era","United States territorial era","Battle of New Orleans","Port","Civil War\u2013Reconstruction era","Jim Crow era","20th century","21st century"]}
{"upwards":["New Orleans","History","French\u2013Spanish colonial era"],"downwards":[]}
{"upwards":["New Orleans","History","United States territorial era"],"downwards":[]}
{"upwards":["New Orleans","History","Battle of New Orleans"],"downwards":[]}
{"upwards":["New Orleans","History","Port"],"downwards":[]}
{"upwards":["New Orleans","History","Civil War\u2013Reconstruction era"],"downwards":[]}
{"upwards":["New Orleans","History","Jim Crow era"],"downwards":[]}
{"upwards":["New Orleans","History","20th century"],"downwards":["Civil Rights movement","Drainage and flood control"]}
{"upwards":["New Orleans","History","20th century","Civil Rights movement"],"downwards":[]}
{"upwards":["New Orleans","History","20th century","Drainage and flood control"],"downwards":[]}
{"upwards":["New Orleans","History","21st century"],"downwards":["Hurricane Katrina","Hurricane Rita","Post-disaster recovery","Hurricane Ida"]}
{"upwards":["New Orleans","History","21st century","Hurricane Katrina"],"downwards":[]}
{"upwards":["New Orleans","History","21st century","Hurricane Rita"],"downwards":[]}
{"upwards":["New Orleans","History","21st century","Post-disaster recovery"],"downwards":[]}
{"upwards":["New Orleans","History","21st century","Hurricane Ida"],"downwards":[]}
{"upwards":["New Orleans","Geography"],"downwards":["Elevation","Cityscape","Climate","Threat from tropical cyclones"]}
{"upwards":["New Orleans","Geography","Elevation"],"downwards":[]}
{"upwards":["New Orleans","Geography","Cityscape"],"downwards":["Historic and residential architecture","Tallest buildings"]}
{"upwards":["New Orleans","Geography","Cityscape","Historic and residential architecture"],"downwards":[]}
{"upwards":["New Orleans","Geography","Cityscape","Tallest buildings"],"downwards":[]}
{"upwards":["New Orleans","Geography","Climate"],"downwards":[]}
{"upwards":["New Orleans","Geography","Threat from tropical cyclones"],"downwards":[]}
{"upwards":["New Orleans","Demographics"],"downwards":["Race and ethnicity","Sexual orientation and gender identity","Religion"]}
{"upwards":["New Orleans","Demographics","Race and ethnicity"],"downwards":[]}
{"upwards":["New Orleans","Demographics","Sexual orientation and gender identity"],"downwards":[]}
{"upwards":["New Orleans","Demographics","Religion"],"downwards":[]}
{"upwards":["New Orleans","Economy"],"downwards":["Port","Business","Tourist and convention business","Federal and military agencies"]}
{"upwards":["New Orleans","Economy","Port"],"downwards":[]}
{"upwards":["New Orleans","Economy","Business"],"downwards":[]}
{"upwards":["New Orleans","Economy","Tourist and convention business"],"downwards":[]}
{"upwards":["New Orleans","Economy","Federal and military agencies"],"downwards":[]}
{"upwards":["New Orleans","Culture and contemporary life"],"downwards":["Tourism","Entertainment and performing arts","Cuisine","Dialect"]}
{"upwards":["New Orleans","Culture and contemporary life","Tourism"],"downwards":[]}
{"upwards":["New Orleans","Culture and contemporary life","Entertainment and performing arts"],"downwards":[]}
{"upwards":["New Orleans","Culture and contemporary life","Cuisine"],"downwards":[]}
{"upwards":["New Orleans","Culture and contemporary life","Dialect"],"downwards":[]}
{"upwards":["New Orleans","Sports"],"downwards":[]}
{"upwards":["New Orleans","National protected areas"],"downwards":[]}
{"upwards":["New Orleans","Government"],"downwards":[]}
{"upwards":["New Orleans","Crime"],"downwards":[]}
{"upwards":["New Orleans","Education"],"downwards":["Colleges and universities","Primary and secondary schools","Libraries"]}
{"upwards":["New Orleans","Education","Colleges and universities"],"downwards":[]}
{"upwards":["New Orleans","Education","Primary and secondary schools"],"downwards":[]}
{"upwards":["New Orleans","Education","Libraries"],"downwards":[]}
{"upwards":["New Orleans","Media"],"downwards":[]}
{"upwards":["New Orleans","Transportation"],"downwards":["Public transportation","Bicycling","Roads","Taxi service","Airports","Rail","Modal characteristics"]}
{"upwards":["New Orleans","Transportation","Public transportation"],"downwards":["Streetcars","Buses","Ferries"]}
{"upwards":["New Orleans","Transportation","Public transportation","Streetcars"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Public transportation","Buses"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Public transportation","Ferries"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Bicycling"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Roads"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Taxi service"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Airports"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Rail"],"downwards":[]}
{"upwards":["New Orleans","Transportation","Modal characteristics"],"downwards":[]}
{"upwards":["New Orleans","Notable people"],"downwards":[]}
{"upwards":["New Orleans","Sister cities"],"downwards":[]}
{"upwards":["Algebraic curve"],"downwards":["In Euclidean geometry","Plane projective curves","Remarkable points of a plane curve","Analytic structure","Non-plane algebraic curves","Algebraic function fields","Complex curves and real surfaces","Singularities","Examples of curves"]}
{"upwards":["Algebraic curve","In Euclidean geometry"],"downwards":[]}
{"upwards":["Algebraic curve","Plane projective curves"],"downwards":[]}
{"upwards":["Algebraic curve","Remarkable points of a plane curve"],"downwards":["Intersection with a line","Tangent at a point","Asymptotes","Singular points"]}
{"upwards":["Algebraic curve","Remarkable points of a plane curve","Intersection with a line"],"downwards":[]}
{"upwards":["Algebraic curve","Remarkable points of a plane curve","Tangent at a point"],"downwards":[]}
{"upwards":["Algebraic curve","Remarkable points of a plane curve","Asymptotes"],"downwards":[]}
{"upwards":["Algebraic curve","Remarkable points of a plane curve","Singular points"],"downwards":[]}
{"upwards":["Algebraic curve","Analytic structure"],"downwards":[]}
{"upwards":["Algebraic curve","Non-plane algebraic curves"],"downwards":[]}
{"upwards":["Algebraic curve","Algebraic function fields"],"downwards":[]}
{"upwards":["Algebraic curve","Complex curves and real surfaces"],"downwards":["Compact Riemann surfaces"]}
{"upwards":["Algebraic curve","Complex curves and real surfaces","Compact Riemann surfaces"],"downwards":[]}
{"upwards":["Algebraic curve","Singularities"],"downwards":["Classification of singularities"]}
{"upwards":["Algebraic curve","Singularities","Classification of singularities"],"downwards":[]}
{"upwards":["Algebraic curve","Examples of curves"],"downwards":["Rational curves","Rational plane curves","Elliptic curves","Curves of genus greater than one"]}
{"upwards":["Algebraic curve","Examples of curves","Rational curves"],"downwards":[]}
{"upwards":["Algebraic curve","Examples of curves","Rational plane curves"],"downwards":[]}
{"upwards":["Algebraic curve","Examples of curves","Elliptic curves"],"downwards":[]}
{"upwards":["Algebraic curve","Examples of curves","Curves of genus greater than one"],"downwards":["Projective plane curves","Curves in product of projective lines"]}
{"upwards":["Algebraic curve","Examples of curves","Curves of genus greater than one","Projective plane curves"],"downwards":[]}
{"upwards":["Algebraic curve","Examples of curves","Curves of genus greater than one","Curves in product of projective lines"],"downwards":[]}
{"upwards":["Morphological derivation"],"downwards":["Derivational patterns","Derivation and inflection","Derivation and other types of word formation","Productivity"]}
{"upwards":["Morphological derivation","Derivational patterns"],"downwards":[]}
{"upwards":["Morphological derivation","Derivation and inflection"],"downwards":[]}
{"upwards":["Morphological derivation","Derivation and other types of word formation"],"downwards":[]}
{"upwards":["Morphological derivation","Productivity"],"downwards":[]}
{"upwards":["Plutonium"],"downwards":["Characteristics","Occurrence","History","Applications","Precautions","Transportation","In popular culture","Notes","References","External links"]}
{"upwards":["Plutonium","Characteristics"],"downwards":["Physical properties","Allotropes","Nuclear fission","Isotopes and nucleosynthesis","Decay heat and fission properties","Compounds and chemistry","Alloys"]}
{"upwards":["Plutonium","Characteristics","Physical properties"],"downwards":[]}
{"upwards":["Plutonium","Characteristics","Allotropes"],"downwards":[]}
{"upwards":["Plutonium","Characteristics","Nuclear fission"],"downwards":[]}
{"upwards":["Plutonium","Characteristics","Isotopes and nucleosynthesis"],"downwards":[]}
{"upwards":["Plutonium","Characteristics","Decay heat and fission properties"],"downwards":[]}
{"upwards":["Plutonium","Characteristics","Compounds and chemistry"],"downwards":["Electronic structure"]}
{"upwards":["Plutonium","Characteristics","Compounds and chemistry","Electronic structure"],"downwards":[]}
{"upwards":["Plutonium","Characteristics","Alloys"],"downwards":[]}
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