Set theory
Set theory in mathematical education


Relation with equivalence, congruence, and isomorphism


Equality (mathematics)
Equality as predicate


Equality (mathematics)
Approximate equality


Equality (mathematics)
Logical definitions


Equality (mathematics)
Equality in set theory

Set equality based on first-order logic with equality

Set equality based on first-order logic without equality


Set equality based on first-order logic with equality


Set equality based on first-order logic without equality


Examples
Paradoxes
Classes in formal set theories
Notes
References
External links


Class (set theory)
Classes in formal set theories


Statements and proofs
Discussion and consequences
Historical notes
References
External links


Cantor's paradox
Discussion and consequences


Countably infinite sets
Uncountably infinite sets


Infinite set
Examples
Countably infinite sets


Infinite set
Examples
Uncountably infinite sets


History
Permutations without repetitions
Definition
Notations
Other uses of the term permutation
Properties
Permutations of totally ordered sets
Permutations in computing


Permutation
Permutations without repetitions


Two-line notation
One-line notation
Cycle notation
Canonical cycle notation
Composition of permutations


Permutation
Notations
Canonical cycle notation


Permutation
Notations
Composition of permutations


Permutation
Other uses of the term permutation

k-permutations of n
Permutations with repetition
Permutations of multisets
Circular permutations


Permutation
Other uses of the term permutation
k-permutations of n


Permutation
Other uses of the term permutation
Permutations with repetition


Permutation
Other uses of the term permutation
Permutations of multisets


Permutation
Other uses of the term permutation
Circular permutations


Cycle type
Conjugating permutations
Permutation order
Parity of a permutation
Matrix representation


Permutation
Properties
Conjugating permutations


Permutation
Properties
Permutation order


Permutation
Properties
Parity of a permutation


Permutation
Properties
Matrix representation


Permutation
Permutations of totally ordered sets

Ascents, descents, runs and exceedances
Foata's transition lemma
Inversions


Permutation
Permutations of totally ordered sets
Ascents, descents, runs and exceedances


Permutation
Permutations of totally ordered sets
Foata's transition lemma


Permutation
Permutations of totally ordered sets
Inversions


Numbering permutations
Algorithms to generate permutations
Applications


Permutation
Permutations in computing
Numbering permutations


Permutation
Permutations in computing
Algorithms to generate permutations

Random generation of permutations
Generation in lexicographic order
Generation with minimal changes
Meandric permutations


Permutation
Permutations in computing
Algorithms to generate permutations
Random generation of permutations


Permutation
Permutations in computing
Algorithms to generate permutations
Generation in lexicographic order


Permutation
Permutations in computing
Algorithms to generate permutations
Generation with minimal changes


Permutation
Permutations in computing
Algorithms to generate permutations
Meandric permutations


Permutation
Permutations in computing
Applications


Definition
Examples
Properties
Space of surjections
Gallery


Surjections as right invertible functions

Cardinality of the domain of a surjection

Induced surjection and induced bijection


Surjections as right invertible functions


Surjective function
Properties
Surjections as epimorphisms


Surjective function
Properties
Surjections as binary relations


Cardinality of the domain of a surjection


Surjective function
Properties
Composition and decomposition


Induced surjection and induced bijection


Surjective function
Space of surjections


Restrictions, extensions, and variations

Automated theorem proving and formal methods


Alphabet
Formation rules
Free and bound variables
Example: ordered abelian groups


First-order logic
Syntax
Alphabet
Logical symbols


First-order logic
Syntax
Alphabet
Non-logical symbols


First-order logic
Syntax
Formation rules


First-order logic
Syntax
Formation rules
Terms


First-order logic
Syntax
Formation rules
Formulas


First-order logic
Syntax
Formation rules
Notational conventions


First-order logic
Syntax
Free and bound variables


First-order logic
Syntax
Example: ordered abelian groups


Validity, satisfiability, and logical consequence

First-order theories, models, and elementary classes


First-order logic
Semantics
First-order structures


First-order logic
Semantics
Evaluation of truth values


Validity, satisfiability, and logical consequence


First-order logic
Semantics
Algebraizations


First-order theories, models, and elementary classes


First-order logic
Semantics
Empty domains


Hilbert-style systems and natural deduction


First-order logic
Deductive systems
Rules of inference


First-order logic
Deductive systems
Sequent calculus


First-order logic
Deductive systems
Tableaux method


First-order logic
Deductive systems
Resolution


First-order logic
Deductive systems
Provable identities


First-order logic
Equality and its axioms

First-order logic without equality
Defining equality within a theory


First-order logic
Equality and its axioms
First-order logic without equality


First-order logic
Equality and its axioms
Defining equality within a theory


First-order logic
Metalogical properties

Completeness and undecidability
The Löwenheim–Skolem theorem
The compactness theorem
Lindström's theorem


First-order logic
Metalogical properties
Completeness and undecidability


First-order logic
Metalogical properties
The Löwenheim–Skolem theorem


First-order logic
Metalogical properties
The compactness theorem


First-order logic
Metalogical properties
Lindström's theorem


Expressiveness
Formalizing natural languages


First-order logic
Limitations
Expressiveness


First-order logic
Limitations
Formalizing natural languages


Restrictions, extensions, and variations

Restricted languages
Many-sorted logic
Additional quantifiers
Infinitary logics
Non-classical and modal logics
Fixpoint logic
Higher-order logics


Automated theorem proving and formal methods


Minimal model of set theory is countable


History
Notation
Properties
Generalizations
Formal definitions


Natural number
History
Modern definitions


Relationship between addition and multiplication

Algebraic properties satisfied by the natural numbers


Natural number
Properties
Multiplication


Relationship between addition and multiplication


Algebraic properties satisfied by the natural numbers


Natural number
Formal definitions
Peano axioms


Natural number
Formal definitions
Set-theoretic definition


History
Usage in English
Usage in other languages
Mathematics and logic
Computing


Syntactical-deductive
Syntactical-descriptive
Appositive
Segmental
Use of capitals
Spacing and parentheses


Colon (punctuation)
Usage in English
Syntactical-deductive


Colon (punctuation)
Usage in English
Syntactical-descriptive


Colon (punctuation)
Usage in English
Appositive


Colon (punctuation)
Usage in English
Segmental


Colon (punctuation)
Usage in English
Use of capitals


Colon (punctuation)
Usage in English
Spacing and parentheses


Colon (punctuation)
Usage in other languages

Suffix separator
Abbreviation mark
Word separator
End of sentence or verse
Score divider


Colon (punctuation)
Usage in other languages
Suffix separator


Colon (punctuation)
Usage in other languages
Abbreviation mark


Colon (punctuation)
Usage in other languages
Word separator


Colon (punctuation)
Usage in other languages
End of sentence or verse


Colon (punctuation)
Usage in other languages
Score divider


Colon (punctuation)
Mathematics and logic


Programming languages
Addresses
Text markup


Colon (punctuation)
Computing
Programming languages


Colon (punctuation)
Computing
Text markup


History
Comparing sets
Cardinal numbers
Finite, countable and uncountable sets
Infinite sets
Examples and properties
Union and intersection


Definition 1: |A| = |B|
Definition 2: |A| ≤ |B|
Definition 3: |A| < |B|


Cardinality
Comparing sets
Definition 1: |A| = |B|


Cardinality
Comparing sets
Definition 2: |A| ≤ |B|


Cardinality
Comparing sets
Definition 3: |A| < |B|


Cardinality
Finite, countable and uncountable sets


Cardinality
Infinite sets
Cardinality of the continuum


More complex expressions on the left side of the notation


Set-builder notation
Sets defined by enumeration


Set-builder notation
Sets defined by a predicate


Set-builder notation
Sets defined by a predicate
Specifying the domain


Set-builder notation
Sets defined by a predicate
Examples


Set-builder notation
Equivalent predicates yield equal sets


Set-builder notation
Set existence axiom


Set-builder notation
In programming languages


History
Algebraic properties
Order-theoretic properties
Construction
Computer science
Cardinality


Traditional development
Equivalence classes of ordered pairs
Other approaches


Integer
Construction
Traditional development


Integer
Construction
Equivalence classes of ordered pairs


Intensional definition
Extensional definition
History


Extensional and intensional definitions
Intensional definition


Extensional and intensional definitions
Extensional definition


Extensional and intensional definitions
History


Properties
Definition
Other representations
Generalizations of the Euclidean line
Extensions


In higher dimensions
Types
In axiomatic systems


Line (geometry)
Properties
In higher dimensions


Line (geometry)
Properties
In higher dimensions
Collinear points


Line (geometry)
Properties
In axiomatic systems


Linear equation
Parametric equation
Hesse normal form


Line (geometry)
Definition
Linear equation


Line (geometry)
Definition
Parametric equation


Line (geometry)
Definition
Hesse normal form


Line (geometry)
Other representations
Vectors


Line (geometry)
Other representations
Polar coordinates


Line (geometry)
Generalizations of the Euclidean line


Line (geometry)
Generalizations of the Euclidean line
Projective geometry


Characterizations
Properties
Examples
Without the axiom of choice


Uncountable set
Without the axiom of choice


Definition
Representation of relations
Properties of relations
Operations on relations
Examples
Generalizations


Relation (mathematics)
Representation of relations


Relation (mathematics)
Properties of relations


Relation (mathematics)
Properties of relations
Combinations of properties


Relation (mathematics)
Operations on relations


Integer
Construction
Traditional development


Integer
Construction
Equivalence classes of ordered pairs


Terminology
Arithmetic
Continued fraction representation
Other representations
Formal construction
Properties
Real numbers and topological properties
p-adic numbers


Irreducible fraction
Embedding of integers
Equality
Ordering
Addition
Subtraction
Multiplication
Inverse
Division
Exponentiation to integer power


Rational number
Arithmetic
Irreducible fraction


Rational number
Arithmetic
Embedding of integers


Rational number
Arithmetic
Multiplication


Rational number
Arithmetic
Exponentiation to integer power


Rational number
Continued fraction representation


Rational number
Real numbers and topological properties


Definition
Notation
Visualization
History
Relations and operations
Properties
Formal construction
Complex analysis
Applications
Generalizations and related notions


Cartesian complex plane
Polar complex plane
Complex graphs


Complex number
Visualization
Cartesian complex plane


Complex number
Visualization
Polar complex plane


Complex number
Visualization
Polar complex plane
Modulus and argument


Complex number
Visualization
Complex graphs


Multiplication and division in polar form


Complex number
Relations and operations
Equality


Complex number
Relations and operations
Ordering


Complex number
Relations and operations
Conjugate


Complex number
Relations and operations
Addition and subtraction


Complex number
Relations and operations
Multiplication and square


Complex number
Relations and operations
Reciprocal and division


Complex number
Relations and operations
Square root


Complex number
Relations and operations
Exponential function


Complex number
Relations and operations
Exponential function
Functional equation


Complex number
Relations and operations
Exponential function
Euler's formula


Complex number
Relations and operations
Complex logarithm


Complex number
Relations and operations
Exponentiation


Complex number
Relations and operations
Exponentiation
Integer and fractional exponents


Field structure
Solutions of polynomial equations
Algebraic characterization
Characterization as a topological field


Complex number
Properties
Field structure


Complex number
Properties
Solutions of polynomial equations


Complex number
Properties
Algebraic characterization


Complex number
Properties
Characterization as a topological field


Matrix representation of complex numbers


Complex number
Formal construction
Construction as ordered pairs


Complex number
Formal construction
Construction as a quotient field


Complex exponential and related functions


Complex number
Complex analysis
Holomorphic functions


Geometry
Algebraic number theory
Analytic number theory
Improper integrals
Dynamic equations
Linear algebra
In applied mathematics
In physics


Complex number
Applications
Geometry
Shapes


Complex number
Applications
Geometry
Fractal geometry


Complex number
Applications
Geometry
Triangles


Complex number
Applications
Algebraic number theory


Complex number
Applications
Analytic number theory


Complex number
Applications
Improper integrals


Complex number
Applications
Dynamic equations


Complex number
Applications
Linear algebra


Complex number
Applications
In applied mathematics


Complex number
Applications
In applied mathematics
Control theory


Complex number
Applications
In applied mathematics
Signal analysis


Electromagnetism and electrical engineering


Complex number
Applications
In physics
Fluid dynamics


Complex number
Applications
In physics
Quantum mechanics


Complex number
Applications
In physics
Relativity


Complex number
Generalizations and related notions


Characterizing properties
Arithmetic
Dedekind completeness
Decimal representation
Topological completeness
Cardinality
Other properties
History
Formal definitions
Applications and connections
Vocabulary and notation
Generalizations and extensions


Auxiliary operations
Auxiliary order relations
Integers and fractions as real numbers


Real number
Arithmetic
Auxiliary operations


Real number
Arithmetic
Auxiliary order relations


Real number
Arithmetic
Integers and fractions as real numbers


Real number
Topological completeness
"The complete ordered field"


Axiomatic approach
Construction from the rational numbers


Real number
Formal definitions
Axiomatic approach


Real number
Formal definitions
Construction from the rational numbers


Real number
Applications and connections


Real number
Applications and connections
Physics


Real number
Applications and connections
Logic


Real number
Applications and connections
Computation


Real number
Applications and connections
Set theory


Real number
Generalizations and extensions


Elements of a mathematical model
Classifications
Construction
Significance in the natural sciences
Some applications
Examples


Mathematical model
Elements of a mathematical model


A priori information
Complexity
Training, tuning, and fitting
Evaluation and assessment


Mathematical model
Construction
A priori information


Mathematical model
Construction
A priori information
Subjective information


Mathematical model
Construction
Complexity


Mathematical model
Construction
Training, tuning, and fitting


Mathematical model
Construction
Evaluation and assessment

Prediction of empirical data
Scope of the model
Philosophical considerations


Mathematical model
Construction
Evaluation and assessment
Prediction of empirical data


Mathematical model
Construction
Evaluation and assessment
Scope of the model


Mathematical model
Construction
Evaluation and assessment
Philosophical considerations


Mathematical model
Significance in the natural sciences


History
Notation
Properties
Generalizations
Formal definitions


Natural number
History
Modern definitions


Relationship between addition and multiplication

Algebraic properties satisfied by the natural numbers


Natural number
Properties
Multiplication


Relationship between addition and multiplication


Algebraic properties satisfied by the natural numbers


Natural number
Formal definitions
Peano axioms


Natural number
Formal definitions
Set-theoretic definition


History
Basic concepts and notation
Ontology
Formalized set theory
Applications
Areas of study
Objections to set theory
Set theory in mathematical education


Combinatorial set theory
Descriptive set theory
Fuzzy set theory
Inner model theory
Large cardinals
Determinacy
Forcing
Cardinal invariants
Set-theoretic topology


Set theory
Areas of study
Combinatorial set theory


Set theory
Areas of study
Descriptive set theory


Set theory
Areas of study
Fuzzy set theory


Set theory
Areas of study
Inner model theory


Set theory
Areas of study
Large cardinals


Set theory
Areas of study
Cardinal invariants


Set theory
Areas of study
Set-theoretic topology


Set theory
Set theory in mathematical education


Topological and differential geometric notions


Plane (mathematics)
Euclidean plane
Embedding in three-dimensional space


Plane (mathematics)
Further generalizations


Details
Example
History
Overview
Extensions to higher numbers of sets
Related concepts


Venn diagram
Extensions to higher numbers of sets


Venn diagram
Extensions to higher numbers of sets
Edwards–Venn diagrams


Venn diagram
Extensions to higher numbers of sets
Other diagrams


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


Line segment
In real or complex vector spaces


Triangles
Quadrilaterals
Circles and ellipses


Line segment
In other geometric shapes
Triangles


Line segment
In other geometric shapes
Quadrilaterals


Line segment
In other geometric shapes
Circles and ellipses


Definition
Properties
⊂ and ⊃ symbols
Examples of subsets
Other properties of inclusion


Usage note
Independence results in set theory
Applications to physical theory


Independence (mathematical logic)
Usage note


Independence (mathematical logic)
Independence results in set theory


Independence (mathematical logic)
Applications to physical theory


Minimal model of set theory is countable


Example
Properties
Representing subsets as functions
Relation to binomial theorem
Recursive definition
Subsets of limited cardinality
Power object
Functors and quantifiers


Power set
Representing subsets as functions


Power set
Subsets of limited cardinality


Early life and education
Career
Death
Selected publications


Union of two sets
Algebraic properties
Finite unions
Arbitrary unions
Notation encoding


Union (set theory)
Arbitrary unions
Notations


Definition and notation
Examples
Partitions and equivalence relations
Refinement of partitions
Noncrossing partitions
Counting partitions


Partition of a set
Definition and notation


Partition of a set
Partitions and equivalence relations


Partition of a set
Refinement of partitions


Partition of a set
Noncrossing partitions


Characterizations
Properties
Examples
Without the axiom of choice


Uncountable set
Without the axiom of choice


In a specific context
In ordinary mathematics
In set theory
In predicate calculus
In category theory
In type theory


Universe (mathematics)
In a specific context


Universe (mathematics)
In ordinary mathematics


Universe (mathematics)
In predicate calculus


Universe (mathematics)
In category theory


The fundamental properties of set algebra

Some additional laws for unions and intersections


The fundamental properties of set algebra


Algebra of sets
The principle of duality


Some additional laws for unions and intersections


Algebra of sets
Some additional laws for complements


Algebra of sets
The algebra of inclusion


Algebra of sets
The algebra of relative complements


Definition
Examples
More mathematical examples
Inverses
Composition
Cardinality
Properties
Category theory
Generalization to partial functions
Gallery


Batting line-up of a baseball or cricket team


Bijection
Examples
Seats and students of a classroom


Bijection
Generalization to partial functions


Set-theoretic definition
Examples
Most common implementation (set theory)
Cartesian products of several sets
Other forms
Definitions outside set theory


Cartesian product
Set-theoretic definition


A deck of cards
A two-dimensional coordinate system


Cartesian product
Examples
A deck of cards


Cartesian product
Examples
A two-dimensional coordinate system


Cartesian product
Most common implementation (set theory)

Non-commutativity and non-associativity
Intersections, unions, and subsets
Cardinality


Cartesian product
Most common implementation (set theory)
Non-commutativity and non-associativity


Cartesian product
Most common implementation (set theory)
Intersections, unions, and subsets


Cartesian product
Most common implementation (set theory)
Cardinality


Cartesian product
Cartesian products of several sets

n-ary Cartesian product
n-ary Cartesian power
Infinite Cartesian products


Cartesian product
Cartesian products of several sets
n-ary Cartesian product


Cartesian product
Cartesian products of several sets
n-ary Cartesian power


Cartesian product
Cartesian products of several sets
Infinite Cartesian products


Abbreviated form
Cartesian product of functions
Cylinder


Cartesian product
Other forms
Abbreviated form


Cartesian product
Other forms
Cartesian product of functions


Cartesian product
Definitions outside set theory


Cartesian product
Definitions outside set theory
Category theory


Cartesian product
Definitions outside set theory
Graph theory


Generalizations
Intersections
Disjoint unions and partitions


Disjoint sets
Disjoint unions and partitions


Defining the ordered pair using set theory


Ordered pair
Informal and formal definitions


Defining the ordered pair using set theory

Wiener's definition
Hausdorff's definition
Kuratowski's definition
Quine–Rosser definition
Cantor–Frege definition
Morse definition
Axiomatic definition


Proving that definitions satisfy the characteristic property


Hausdorff distance vs. symmetric difference


Symmetric difference
n-ary symmetric difference


Symmetric difference
Symmetric difference on measure spaces


Arguments for and against the continuum hypothesis


Continuum hypothesis
Cardinality of infinite sets


Continuum hypothesis
Independence from ZFC


Continuum hypothesis
The generalized continuum hypothesis

Implications of GCH for cardinal exponentiation


Implications of GCH for cardinal exponentiation


Notation and terminology
Definition
Algebraic properties
Arbitrary intersections
Nullary intersection


Intersection (set theory)
Notation and terminology


Intersection (set theory)
Definition
Intersecting and disjoint sets


Intersection (set theory)
Algebraic properties


Intersection (set theory)
Arbitrary intersections


Intersection (set theory)
Nullary intersection


Formal notation
History
Informal proof
Formal proof
Generalising De Morgan duality
Extension to predicate and modal logic
In intuitionistic logic
In computer engineering


Substitution form
Set theory and Boolean algebra
Engineering
Text searching


De Morgan's laws
Formal notation
Substitution form


De Morgan's laws
Formal notation
Set theory and Boolean algebra

Unions and intersections of any number of sets


Unions and intersections of any number of sets


De Morgan's laws
Formal notation
Engineering


De Morgan's laws
Formal notation
Text searching


Negation of a disjunction
Negation of a conjunction


De Morgan's laws
Informal proof
Negation of a disjunction


De Morgan's laws
Informal proof
Negation of a conjunction


De Morgan's laws
Formal proof
Conclusion


De Morgan's laws
Generalising De Morgan duality


De Morgan's laws
Extension to predicate and modal logic


De Morgan's laws
In intuitionistic logic


De Morgan's laws
In computer engineering


Definition
Prototypical examples
Affine structure
Metric structure
Isometries
Topology
Axiomatic definitions
Usage
Other geometric spaces


History of the definition
Motivation of the modern definition
Technical definition


Euclidean space
Definition
History of the definition


Euclidean space
Definition
Motivation of the modern definition


Euclidean space
Definition
Technical definition


Subspaces
Lines and segments
Parallelism


Euclidean space
Affine structure
Subspaces


Euclidean space
Affine structure
Lines and segments


Euclidean space
Affine structure
Parallelism


Distance and length
Orthogonality
Angle
Cartesian coordinates
Other coordinates


Euclidean space
Metric structure
Distance and length


Euclidean space
Metric structure
Orthogonality


Euclidean space
Metric structure
Cartesian coordinates


Euclidean space
Metric structure
Other coordinates


Isometry with prototypical examples
Euclidean group


Euclidean space
Isometries
Isometry with prototypical examples


Euclidean space
Isometries
Euclidean group


Affine space
Projective space
Non-Euclidean geometries
Curved spaces
Pseudo-Euclidean space


Euclidean space
Other geometric spaces
Affine space


Euclidean space
Other geometric spaces
Projective space


Euclidean space
Other geometric spaces
Non-Euclidean geometries


Euclidean space
Other geometric spaces
Curved spaces


Euclidean space
Other geometric spaces
Pseudo-Euclidean space


Elementary consequences of the group axioms


Group (mathematics)
Definition and illustration

First example: the integers
Definition
Notation and terminology
Second example: a symmetry group


Group (mathematics)
Definition and illustration
First example: the integers


Group (mathematics)
Definition and illustration
Definition


Group (mathematics)
Definition and illustration
Notation and terminology


Group (mathematics)
Definition and illustration
Second example: a symmetry group


Elementary consequences of the group axioms

Equivalent definition with relaxed axioms


Equivalent definition with relaxed axioms


Group homomorphisms
Subgroups
Cosets
Quotient groups
Presentations


Group (mathematics)
Basic concepts
Group homomorphisms


Group (mathematics)
Basic concepts
Subgroups


Group (mathematics)
Basic concepts
Cosets


Group (mathematics)
Basic concepts
Quotient groups


Group (mathematics)
Basic concepts
Presentations


Group (mathematics)
Examples and applications

General linear group and representation theory


Group (mathematics)
Examples and applications
Numbers


Group (mathematics)
Examples and applications
Numbers
Integers


Group (mathematics)
Examples and applications
Numbers
Rationals


Group (mathematics)
Examples and applications
Modular arithmetic


Group (mathematics)
Examples and applications
Cyclic groups


Group (mathematics)
Examples and applications
Symmetry groups


General linear group and representation theory


Group (mathematics)
Examples and applications
Galois groups


Finite abelian groups
Simple groups
Classification of finite simple groups


Group (mathematics)
Finite groups
Finite abelian groups


Group (mathematics)
Finite groups
Simple groups


Group (mathematics)
Finite groups
Classification of finite simple groups


Group (mathematics)
Groups with additional structure


Group (mathematics)
Groups with additional structure
Topological groups


Group (mathematics)
Groups with additional structure
Lie groups


History
Basic concepts
Branches of abstract algebra
Applications


Elementary algebra
Early group theory
Early ring theory
Early field theory
Other major areas
Modern algebra


Abstract algebra
History
Elementary algebra


Abstract algebra
History
Early group theory


Abstract algebra
History
Early ring theory


Abstract algebra
History
Early field theory


Abstract algebra
History
Other major areas


Abstract algebra
Branches of abstract algebra


Abstract algebra
Branches of abstract algebra
Group theory


Abstract algebra
Branches of abstract algebra
Ring theory


Notations
Examples
Relation to Boolean algebras
Properties of Boolean rings
Unification


Boolean ring
Relation to Boolean algebras


Boolean ring
Properties of Boolean rings


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


Classic definition
Alternative definition


Field (mathematics)
Definition
Classic definition


Field (mathematics)
Definition
Alternative definition


Rational numbers
Real and complex numbers
Constructible numbers
A field with four elements


Field (mathematics)
Examples
Rational numbers


Field (mathematics)
Examples
Real and complex numbers


Field (mathematics)
Examples
Constructible numbers


Field (mathematics)
Examples
A field with four elements


The additive and the multiplicative group of a field


Field (mathematics)
Elementary notions
Consequences of the definition


Field (mathematics)
Elementary notions
Characteristic


Field (mathematics)
Elementary notions
Subfields and prime fields


Constructing fields within a bigger field


Field (mathematics)
Constructing fields
Constructing fields from rings


Field (mathematics)
Constructing fields
Constructing fields from rings
Field of fractions


Field (mathematics)
Constructing fields
Constructing fields from rings
Residue fields


Field (mathematics)
Constructing fields
Field extensions

Algebraic extensions
Transcendence bases


Field (mathematics)
Constructing fields
Field extensions
Algebraic extensions


Field (mathematics)
Constructing fields
Field extensions
Transcendence bases


Field (mathematics)
Constructing fields
Closure operations


Field (mathematics)
Fields with additional structure

Ordered fields
Topological fields
Differential fields


Field (mathematics)
Fields with additional structure
Ordered fields


Field (mathematics)
Fields with additional structure
Topological fields


Field (mathematics)
Fields with additional structure
Topological fields
Local fields


Field (mathematics)
Fields with additional structure
Differential fields


Field (mathematics)
Invariants of fields

Model theory of fields
The absolute Galois group
K-theory


Field (mathematics)
Invariants of fields
Model theory of fields


Field (mathematics)
Invariants of fields
The absolute Galois group


Field (mathematics)
Invariants of fields
K-theory


Finite fields: cryptography and coding theory


Field (mathematics)
Applications
Linear algebra and commutative algebra


Field (mathematics)
Applications
Geometry: field of functions


Field (mathematics)
Applications
Number theory: global fields


Field (mathematics)
Related notions
Division rings


Introduction
Common axioms
Common algebraic structures
Hybrid structures
Universal algebra
Category theory
Different meanings of "structure"


Equational axioms
Existential axioms
Non-equational axioms


Algebraic structure
Common axioms
Equational axioms


Algebraic structure
Common axioms
Existential axioms


Algebraic structure
Common axioms
Non-equational axioms


Algebraic structure
Common algebraic structures

One set with operations
Two sets with operations


Algebraic structure
Common algebraic structures
One set with operations


Algebraic structure
Common algebraic structures
Two sets with operations


Algebraic structure
Different meanings of "structure"


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


Ring (mathematics)
Definition
Notes on the definition


Some properties
Example: Integers modulo 4
Example: 2-by-2 matrices


Ring (mathematics)
Illustration
Some properties


Ring (mathematics)
Illustration
Example: Integers modulo 4


Ring (mathematics)
Illustration
Example: 2-by-2 matrices


Multiplicative identity and the term "ring"


Ring (mathematics)
History
Fraenkel and Noether


Commutative rings
Noncommutative rings
Non-rings


Ring (mathematics)
Basic examples
Commutative rings


Ring (mathematics)
Basic examples
Noncommutative rings


Ring (mathematics)
Basic examples
Non-rings


Products and powers
Elements in a ring
Subring
Ideal
Homomorphism
Quotient ring


Ring (mathematics)
Basic concepts
Products and powers


Ring (mathematics)
Basic concepts
Elements in a ring


Ring (mathematics)
Basic concepts
Subring


Ring (mathematics)
Basic concepts
Homomorphism


Ring (mathematics)
Basic concepts
Quotient ring


Direct product
Polynomial ring
Matrix ring and endomorphism ring
Limits and colimits of rings
Localization
Completion
Rings with generators and relations


Ring (mathematics)
Constructions
Direct product


Ring (mathematics)
Constructions
Polynomial ring


Ring (mathematics)
Constructions
Matrix ring and endomorphism ring


Ring (mathematics)
Constructions
Limits and colimits of rings


Ring (mathematics)
Constructions
Localization


Ring (mathematics)
Constructions
Completion


Ring (mathematics)
Constructions
Rings with generators and relations


Ring (mathematics)
Special kinds of rings

Domains
Division ring
Semisimple rings
Central simple algebra and Brauer group
Valuation ring


Ring (mathematics)
Special kinds of rings
Domains


Ring (mathematics)
Special kinds of rings
Division ring


Ring (mathematics)
Special kinds of rings
Semisimple rings


Ring (mathematics)
Special kinds of rings
Semisimple rings
Examples


Ring (mathematics)
Special kinds of rings
Semisimple rings
Properties


Ring (mathematics)
Special kinds of rings
Central simple algebra and Brauer group


Ring (mathematics)
Special kinds of rings
Valuation ring


Ring (mathematics)
Rings with extra structure


Ring (mathematics)
Some examples of the ubiquity of rings

Function field of an irreducible algebraic variety


Ring (mathematics)
Some examples of the ubiquity of rings
Cohomology ring of a topological space


Ring (mathematics)
Some examples of the ubiquity of rings
Burnside ring of a group


Ring (mathematics)
Some examples of the ubiquity of rings
Representation ring of a group ring


Function field of an irreducible algebraic variety


Ring (mathematics)
Some examples of the ubiquity of rings
Face ring of a simplicial complex


Ring (mathematics)
Category-theoretic description


Ring (mathematics)
Generalization
Nonassociative ring


Ring (mathematics)
Generalization
Semiring


Ring (mathematics)
Other ring-like objects

Ring object in a category
Ring scheme
Ring spectrum


Ring (mathematics)
Other ring-like objects
Ring object in a category


Ring (mathematics)
Other ring-like objects
Ring scheme


Ring (mathematics)
Other ring-like objects
Ring spectrum


Definitions
In topology
Binary relations
Other examples
Closure operator
Notes
References


Closure (mathematics)
Definitions
In algebraic structures


Closure (mathematics)
Closure operator
Closure operator vs. closed sets


Formula
Examples
A special case
Formula generalization
In probability
Other formulas
Applications
Diluted inclusion–exclusion principle
Proof of main statement


Inclusion–exclusion principle
Examples
Counting integers


Inclusion–exclusion principle
Examples
Counting derangements


Inclusion–exclusion principle
A special case


Inclusion–exclusion principle
Formula generalization


Inclusion–exclusion principle
In probability


Inclusion–exclusion principle
In probability
Special case


Inclusion–exclusion principle
Other formulas


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


Inclusion–exclusion principle
Applications
Counting derangements


Inclusion–exclusion principle
Applications
Counting intersections


Inclusion–exclusion principle
Applications
Graph coloring


Inclusion–exclusion principle
Applications
Bipartite graph perfect matchings


Inclusion–exclusion principle
Applications
Number of onto functions


Inclusion–exclusion principle
Applications
Permutations with forbidden positions


Inclusion–exclusion principle
Applications
Stirling numbers of the second kind


Inclusion–exclusion principle
Applications
Rook polynomials


Inclusion–exclusion principle
Applications
Euler's phi function


Inclusion–exclusion principle
Diluted inclusion–exclusion principle


Inclusion–exclusion principle
Proof of main statement


Inclusion–exclusion principle
Proof of main statement
Algebraic proof


Unions, intersections, and relative complements


Paradoxes
Cantor's theory
Axiomatic theories
Consistency
Utility


Naive set theory
Method
Axiomatic theories


Naive set theory
Sets, membership and equality

Note on consistency
Membership
Equality
Empty set


Naive set theory
Sets, membership and equality
Note on consistency


Naive set theory
Sets, membership and equality
Membership


Naive set theory
Sets, membership and equality
Equality


Naive set theory
Sets, membership and equality
Empty set


Naive set theory
Universal sets and absolute complements


Naive set theory
Ordered pairs and Cartesian products


Naive set theory
Paradoxes in early set theory


History
Examples
Types
Decimal expansions
Irrational powers
Open questions
In constructive mathematics
Set of all irrationals


Ancient Greece
India
Middle Ages
Modern period


Irrational number
History
Ancient Greece


Irrational number
Examples
General roots


Irrational number
Types
Transcendental/algebraic


Irrational number
In constructive mathematics


Irrational number
Set of all irrationals


Sketch of a proof that e is transcendental


Transcendental number
Numbers proven to be transcendental


Transcendental number
Possible transcendental numbers


Philosophy, religion, literature and Cantor's mathematics


Youth and studies
Teacher and researcher
Later years and death


Georg Cantor
Biography
Youth and studies


Georg Cantor
Biography
Teacher and researcher


Georg Cantor
Biography
Later years and death


Number theory, trigonometric series and ordinals


Georg Cantor
Mathematical work
Set theory

Absolute infinite, well-ordering theorem, and paradoxes


Georg Cantor
Mathematical work
Set theory
One-to-one correspondence


Georg Cantor
Mathematical work
Set theory
Continuum hypothesis


Absolute infinite, well-ordering theorem, and paradoxes


Role and characterizations in mathematics


Name
Definition
Irrationality and normality
Transcendence
Continued fractions
Approximate value and digits
Complex numbers and Euler's identity


Pi
Fundamentals
Irrationality and normality


Pi
Fundamentals
Approximate value and digits


Pi
Fundamentals
Complex numbers and Euler's identity


Antiquity
Polygon approximation era
Infinite series
Irrationality and transcendence
Adoption of the symbol π


Pi
History
Infinite series
Rate of convergence


Pi
History
Irrationality and transcendence


Computer era and iterative algorithms
Motives for computing π
Rapidly convergent series
Monte Carlo methods
Spigot algorithms


Pi
Modern quest for more digits
Computer era and iterative algorithms


Pi
Modern quest for more digits
Motives for computing π


Pi
Modern quest for more digits
Rapidly convergent series


Pi
Modern quest for more digits
Monte Carlo methods


Pi
Modern quest for more digits
Spigot algorithms


Role and characterizations in mathematics

Fourier transform and Heisenberg uncertainty principle

The gamma function and Stirling's approximation

Cauchy distribution and potential theory


Fourier transform and Heisenberg uncertainty principle


The gamma function and Stirling's approximation


Cauchy distribution and potential theory


Describing physical phenomena
Memorizing digits
In popular culture
In computer culture


Pi
Outside mathematics
Describing physical phenomena


Pi
Outside mathematics
Memorizing digits


Pi
Outside mathematics
In popular culture


Pi
Outside mathematics
In computer culture


Notation
Properties
In other areas of mathematics
Questioned existence


Empty set
Properties
Operations on the empty set


Extended real numbers
Topology
Category theory
Set theory


Empty set
In other areas of mathematics
Extended real numbers


Empty set
In other areas of mathematics
Topology


Empty set
In other areas of mathematics
Category theory


Empty set
In other areas of mathematics
Set theory


Axiomatic set theory
Philosophical issues


Empty set
Questioned existence
Axiomatic set theory


Empty set
Questioned existence
Philosophical issues


History
Axioms
Motivation via the cumulative hierarchy
Metamathematics
Criticisms


2. Axiom of regularity (also called the axiom of foundation)

3. Axiom schema of specification (or of separation, or of restricted comprehension)


Zermelo–Fraenkel set theory
Axioms
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)


Zermelo–Fraenkel set theory
Axioms
4. Axiom of pairing


Zermelo–Fraenkel set theory
Axioms
5. Axiom of union


Zermelo–Fraenkel set theory
Axioms
6. Axiom schema of replacement


Zermelo–Fraenkel set theory
Axioms
7. Axiom of infinity


Zermelo–Fraenkel set theory
Axioms
8. Axiom of power set


Zermelo–Fraenkel set theory
Axioms
9. Well-ordering axiom


Zermelo–Fraenkel set theory
Axioms
Zorn's lemma


Zermelo–Fraenkel set theory
Motivation via the cumulative hierarchy


Zermelo–Fraenkel set theory
Metamathematics

Virtual classes
Von Neumann–Bernays–Gödel set theory
Consistency
Independence
Proposed additions


Zermelo–Fraenkel set theory
Metamathematics
Virtual classes


Zermelo–Fraenkel set theory
Metamathematics
Von Neumann–Bernays–Gödel set theory


Zermelo–Fraenkel set theory
Metamathematics
Consistency


Zermelo–Fraenkel set theory
Metamathematics
Independence


Zermelo–Fraenkel set theory
Metamathematics
Proposed additions


Natural domain
Other uses
Set theoretical notions


Domain of a function
Natural domain
Examples


Domain of a function
Set theoretical notions


Absolute complement
Relative complement
Complementary relation
LaTeX notation
In programming languages


Complement (set theory)
Absolute complement


Complement (set theory)
Absolute complement
Definition


Complement (set theory)
Absolute complement
Examples


Complement (set theory)
Absolute complement
Properties


Complement (set theory)
Relative complement


Complement (set theory)
Relative complement
Definition


Complement (set theory)
Relative complement
Examples


Complement (set theory)
Relative complement
Properties


Complement (set theory)
Complementary relation


Complement (set theory)
In programming languages


Statements consistent with the negation of AC


Nomenclature ZF, AC, and ZFC
Variants
Restriction to finite sets


Axiom of choice
Statement
Nomenclature ZF, AC, and ZFC


Axiom of choice
Statement
Restriction to finite sets


Axiom of choice
Criticism and acceptance


Axiom of choice
In constructive mathematics


Axiom of choice
Equivalents
Category theory


Results requiring AC (or weaker forms) but weaker than it


Axiom of choice
Weaker forms
Possibly equivalent implications of AC


Axiom of choice
Stronger forms of the negation of AC


Axiom of choice
Axiom of choice in type theory


In the foundations of mathematics and set theory


Total, univalent relation
Set exponentiation


Function (mathematics)
Definition
Total, univalent relation


Function (mathematics)
Definition
Set exponentiation


Functional notation
Arrow notation
Index notation
Dot notation
Specialized notations


Function (mathematics)
Notation
Functional notation


Function (mathematics)
Notation
Arrow notation


Function (mathematics)
Notation
Index notation


Function (mathematics)
Notation
Dot notation


Function (mathematics)
Notation
Specialized notations


Function (mathematics)
Specifying a function

By listing function values
By a formula
Inverse and implicit functions
Using differential calculus
By recurrence


Function (mathematics)
Specifying a function
By listing function values


Function (mathematics)
Specifying a function
By a formula


Function (mathematics)
Specifying a function
Inverse and implicit functions


Function (mathematics)
Specifying a function
Using differential calculus


Function (mathematics)
Specifying a function
By recurrence


Function (mathematics)
Representing a function


Function (mathematics)
Representing a function
Graphs and plots


Function (mathematics)
Representing a function
Tables


Function (mathematics)
Representing a function
Bar chart


Function (mathematics)
General properties

Injective, surjective and bijective functions


Function (mathematics)
General properties
Standard functions


Function (mathematics)
General properties
Function composition


Function (mathematics)
General properties
Image and preimage


Injective, surjective and bijective functions


Function (mathematics)
General properties
Restriction and extension


Function (mathematics)
Multivariate function


Function (mathematics)
In calculus
Real function


Function (mathematics)
In calculus
Vector-valued function


Function (mathematics)
Multi-valued functions


In the foundations of mathematics and set theory


Function (mathematics)
In computer science


Examples and notation
Formal definition and basic properties
Limits and convergence
Series
Use in other fields of mathematics


Examples
Indexing
Defining a sequence by recursion


Sequence
Examples and notation
Defining a sequence by recursion


Sequence
Formal definition and basic properties

Definition
Finite and infinite
Increasing and decreasing
Bounded
Subsequences
Other types of sequences


Sequence
Formal definition and basic properties
Definition


Sequence
Formal definition and basic properties
Finite and infinite


Sequence
Formal definition and basic properties
Increasing and decreasing


Sequence
Formal definition and basic properties
Bounded


Sequence
Formal definition and basic properties
Subsequences


Sequence
Formal definition and basic properties
Other types of sequences


Formal definition of convergence
Applications and important results
Cauchy sequences
Infinite limits


Sequence
Limits and convergence
Formal definition of convergence


Sequence
Limits and convergence
Applications and important results


Sequence
Limits and convergence
Cauchy sequences


Sequence
Limits and convergence
Infinite limits


Sequence
Use in other fields of mathematics

Topology
Analysis
Linear algebra
Abstract algebra
Set theory
Computing
Streams


Sequence
Use in other fields of mathematics
Topology


Sequence
Use in other fields of mathematics
Topology
Product topology


Sequence
Use in other fields of mathematics
Analysis


Sequence
Use in other fields of mathematics
Analysis
Sequence spaces


Sequence
Use in other fields of mathematics
Linear algebra


Sequence
Use in other fields of mathematics
Abstract algebra

Free monoid
Exact sequences
Spectral sequences


Sequence
Use in other fields of mathematics
Abstract algebra
Free monoid


Sequence
Use in other fields of mathematics
Abstract algebra
Exact sequences


Sequence
Use in other fields of mathematics
Abstract algebra
Spectral sequences


Sequence
Use in other fields of mathematics
Set theory


Sequence
Use in other fields of mathematics
Computing


Sequence
Use in other fields of mathematics
Streams


Formal systems: completeness, consistency, and effective axiomatization


Formal systems: completeness, consistency, and effective axiomatization

Effective axiomatization
Completeness
Consistency
Systems which contain arithmetic
Conflicting goals


Gödel's incompleteness theorems
First incompleteness theorem

Syntactic form of the Gödel sentence
Truth of the Gödel sentence
Relationship with the liar paradox
Extensions of Gödel's original result


Gödel's incompleteness theorems
First incompleteness theorem
Syntactic form of the Gödel sentence


Gödel's incompleteness theorems
First incompleteness theorem
Truth of the Gödel sentence


Gödel's incompleteness theorems
First incompleteness theorem
Relationship with the liar paradox


Gödel's incompleteness theorems
First incompleteness theorem
Extensions of Gödel's original result


Gödel's incompleteness theorems
Second incompleteness theorem

Expressing consistency
The Hilbert–Bernays conditions
Implications for consistency proofs


Gödel's incompleteness theorems
Second incompleteness theorem
Expressing consistency


Gödel's incompleteness theorems
Second incompleteness theorem
The Hilbert–Bernays conditions


Gödel's incompleteness theorems
Second incompleteness theorem
Implications for consistency proofs


Gödel's incompleteness theorems
Examples of undecidable statements

Undecidable statements provable in larger systems


Undecidable statements provable in larger systems


Gödel's incompleteness theorems
Relationship with computability


Gödel's incompleteness theorems
Proof sketch for the first theorem

Construction of a statement about "provability"


Gödel's incompleteness theorems
Proof sketch for the first theorem
Arithmetization of syntax


Construction of a statement about "provability"


Gödel's incompleteness theorems
Proof sketch for the first theorem
Diagonalization


Gödel's incompleteness theorems
Proof sketch for the first theorem
Proof via Berry's paradox


Gödel's incompleteness theorems
Proof sketch for the first theorem
Computer verified proofs


Gödel's incompleteness theorems
Proof sketch for the second theorem


Gödel's incompleteness theorems
Discussion and implications

Consequences for logicism and Hilbert's second problem

Appeals to the incompleteness theorems in other fields


Consequences for logicism and Hilbert's second problem


Gödel's incompleteness theorems
Discussion and implications
Minds and machines


Gödel's incompleteness theorems
Discussion and implications
Paraconsistent logic


Appeals to the incompleteness theorems in other fields


Announcement
Generalization and acceptance
Criticisms


Gödel's incompleteness theorems
History
Announcement


Gödel's incompleteness theorems
History
Generalization and acceptance


Gödel's incompleteness theorems
History
Criticisms


Gödel's incompleteness theorems
History
Criticisms
Finsler


Gödel's incompleteness theorems
History
Criticisms
Zermelo


Gödel's incompleteness theorems
History
Criticisms
Wittgenstein


Examples
Paradoxes
Classes in formal set theories
Notes
References
External links


Class (set theory)
Classes in formal set theories


Properties of the category of sets
Foundations for the category of sets


Category of sets
Properties of the category of sets


Category of sets
Foundations for the category of sets


Definition
Examples
Properties
Related notions


Dense set
Definition
Density in metric spaces


Internal sets in the ultrapower construction


Internal set
Internal subsets of the reals


Definition
Fuzzy logic
Fuzzy number and only number
Fuzzy categories
Fuzzy relation equation
Entropy
Extensions


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


Fuzzy set
Definition
Crisp sets related to a fuzzy set


Fuzzy set
Definition
Fuzzy set operations


Fuzzy set
Definition
Disjoint fuzzy sets


Fuzzy set
Definition
Distance and similarity


Fuzzy set
Definition
Neutrosophic fuzzy sets


Fuzzy set
Definition
Pythagorean fuzzy sets


Examples
Properties
Related concepts
Special types of set families


Family of sets
Related concepts
Covers and topologies


Family of sets
Special types of set families


Vopěnka's Alternative Set Theory
Other alternative set theories


Alternative set theory
Vopěnka's Alternative Set Theory


Alternative set theory
Other alternative set theories


History
Axioms and primitive notions
Various systems
Set theory
Mathematics
Natural language
Metaphysics
Important surveys


Mereological constitution
Mereological composition


Mereology
Metaphysics
Mereological constitution


Mereology
Metaphysics
Mereological composition

Fundamentality
Special composition question (SCQ)


Mereology
Metaphysics
Mereological composition
Fundamentality


Mereology
Metaphysics
Mereological composition
Special composition question (SCQ)


History
Examples
Definition
Basic properties and operations
Counting multisets
Applications
Generalizations


Multiset
Basic properties and operations


Generalization and connection to the negative binomial series


Multiset
Counting multisets
Recurrence relation


Multiset
Counting multisets
Generating series


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


Bertrand Russell
Biography
Early life and background


Bertrand Russell
Biography
Childhood and adolescence


Bertrand Russell
Biography
University and first marriage


Bertrand Russell
Biography
First World War


Bertrand Russell
Biography
G. H. Hardy on the Trinity controversy


Bertrand Russell
Biography
Between the wars


Bertrand Russell
Biography
Second World War


Bertrand Russell
Biography
Political causes


Bertrand Russell
Biography
Final years, death and legacy


Bertrand Russell
Biography
Final years, death and legacy
Marriages and issue


Bertrand Russell
Biography
Titles and honours from birth


Philosophy
Religion
Society
Freedom of opinion and expression
Education


Bertrand Russell
Views
Freedom of opinion and expression


History
Past editors
References
External links


Comparison with other identifier schemes


Digital object identifier
Nomenclature and syntax


Digital object identifier
Nomenclature and syntax
Display


Digital object identifier
Features and benefits


Digital object identifier
IDF organizational structure


Digital object identifier
Standardization


Website design
Search
Journal article parameters


PubMed
Characteristics
Website design
PubMed for handhelds/mobiles


PubMed
Characteristics
Search
Standard search


PubMed
Characteristics
Search
Comprehensive search


PubMed
Characteristics
Journal article parameters

Publication Type: Clinical queries/systematic reviews


Publication Type: Clinical queries/systematic reviews


PubMed
Characteristics
Journal article parameters
Secondary ID


Positions
Publications
Articles, reviews, and essays


Joseph Dauben
Articles, reviews, and essays


Related publishers, imprints, and series


Early life and education
Career
Accomplishments
Books by Halmos


Classic reprints
History
Publications
References
External links


Nobel prize winners published by Cambridge University Press

Organisational governance and operational structure

Cambridge University Press partnerships and acquisitions


Cambridge University Press
Print and typographic heritage


Cambridge University Press
Print and typographic heritage
People


Cambridge University Press
Print and typographic heritage
Publications


Cambridge University Press
Current publications


Nobel prize winners published by Cambridge University Press


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Relationship with the University of Cambridge


Relationship with the University of Cambridge


Cambridge University Press partnerships and acquisitions


Cambridge University Press
Digital developments


Cambridge University Press
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Tax exemption controversy
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Cambridge University Press
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Tax exemption controversy


Cambridge University Press
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Cambridge University Press
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Cambridge University Press
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Community


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Corporate social responsibility
Environment


Cambridge University Press
References
Citations


Cambridge University Press
References
Sources


Cambridge University Press
External links


Etymology
Historical development
Mathematical logic


Early Greeks
Modern development
Other sciences


Axiom
Historical development
Early Greeks


Axiom
Historical development
Modern development


Axiom
Historical development
Other sciences


Logical axioms
Non-logical axioms
Role in mathematical logic
Further discussion


Axiom
Mathematical logic
Logical axioms
Examples


Axiom
Mathematical logic
Non-logical axioms


Axiom
Mathematical logic
Non-logical axioms
Examples


Axiom
Mathematical logic
Role in mathematical logic


Axiom
Mathematical logic
Role in mathematical logic
Deductive systems and completeness


Axiom
Mathematical logic
Further discussion


ZF (the Zermelo–Fraenkel axioms without the axiom of choice)


Equivalents of AC
Stronger than AC
Weaker than AC
Alternates incompatible with AC


List of axioms
Axiom of choice
Equivalents of AC


List of axioms
Axiom of choice
Stronger than AC


List of axioms
Axiom of choice
Weaker than AC


List of axioms
Axiom of choice
Alternates incompatible with AC


List of axioms
Other axioms of mathematical logic


Formal language
Formal grammar
Formal systems
Interpretations


Overview
Historical background
Quantities of information
Coding theory
Applications to other fields


Information theory
Historical background


Information theory
Quantities of information

Kullback–Leibler divergence (information gain)


Information theory
Quantities of information
Entropy of an information source


Information theory
Quantities of information
Joint entropy


Information theory
Quantities of information
Conditional entropy (equivocation)


Information theory
Quantities of information
Mutual information (transinformation)


Kullback–Leibler divergence (information gain)


Information theory
Quantities of information
Directed Information


Information theory
Quantities of information
Other quantities


Information theory
Coding theory
Source theory


Information theory
Coding theory
Source theory
Rate


Information theory
Coding theory
Channel capacity

Channels with memory and directed information


Information theory
Coding theory
Channel capacity
Capacity of particular channel models


Channels with memory and directed information


Information theory
Applications to other fields

Intelligence uses and secrecy applications

Integrated process organization of neural information


Intelligence uses and secrecy applications


Information theory
Applications to other fields
Pseudorandom number generation


Information theory
Applications to other fields
Seismic exploration


Information theory
Applications to other fields
Semiotics


Integrated process organization of neural information


Information theory
Applications to other fields
Miscellaneous applications


Definitions
Historical overview
Branches of philosophy
Methods of philosophy
Applied and professional philosophy


Western philosophy
Middle Eastern philosophy
Eastern philosophy
African philosophy
Indigenous American philosophy


Philosophy
Historical overview
Western philosophy


Philosophy
Historical overview
Western philosophy
Ancient era


Philosophy
Historical overview
Western philosophy
Medieval era


Philosophy
Historical overview
Western philosophy
Modern era


Philosophy
Historical overview
Middle Eastern philosophy

Pre-Islamic philosophy
Islamic philosophy


Philosophy
Historical overview
Middle Eastern philosophy
Pre-Islamic philosophy


Philosophy
Historical overview
Middle Eastern philosophy
Islamic philosophy


Philosophy
Historical overview
Eastern philosophy

Indian philosophy
Buddhist philosophy
East Asian philosophy


Philosophy
Historical overview
Eastern philosophy
Indian philosophy


Philosophy
Historical overview
Eastern philosophy
Buddhist philosophy


Philosophy
Historical overview
Eastern philosophy
East Asian philosophy


Philosophy
Historical overview
African philosophy


Philosophy
Historical overview
Indigenous American philosophy


Aesthetics
Ethics
Epistemology
Metaphysics
Logic
Mind and language
Philosophy of science
Political philosophy
Philosophy of religion
Metaphilosophy
Other subdivisions


Philosophy
Branches of philosophy
Aesthetics


Philosophy
Branches of philosophy
Ethics


Philosophy
Branches of philosophy
Epistemology


Philosophy
Branches of philosophy
Metaphysics


Philosophy
Branches of philosophy
Mind and language


Philosophy
Branches of philosophy
Philosophy of science


Philosophy
Branches of philosophy
Political philosophy


Philosophy
Branches of philosophy
Philosophy of religion


Philosophy
Branches of philosophy
Metaphilosophy


Philosophy
Branches of philosophy
Other subdivisions


Philosophy
Methods of philosophy
Philosophical progress


Philosophy
Applied and professional philosophy


Philosophy
Applied and professional philosophy
Women in philosophy


Etymology
Comparison with theorem
Well-known lemmas


Lemma (mathematics)
Comparison with theorem


Proof theory and constructive mathematics


Mathematical logic
History
Early history


Mathematical logic
History
19th century
Foundational theories


Set theory and paradoxes
Symbolic logic
Beginnings of the other branches


Mathematical logic
History
20th century
Set theory and paradoxes


Mathematical logic
History
20th century
Symbolic logic


Mathematical logic
History
20th century
Beginnings of the other branches


Mathematical logic
Formal logical systems

First-order logic
Other classical logics
Nonclassical and modal logic
Algebraic logic


Mathematical logic
Formal logical systems
First-order logic


Mathematical logic
Formal logical systems
Other classical logics


Mathematical logic
Formal logical systems
Nonclassical and modal logic


Mathematical logic
Formal logical systems
Algebraic logic


Mathematical logic
Recursion theory
Algorithmically unsolvable problems


Mathematical logic
Connections with computer science


Mathematical logic
Foundations of mathematics


A priori property of logical consequence


Syntactic consequence
Semantic consequence


Logical consequence
Proofs and models
Syntactic consequence


Logical consequence
Proofs and models
Semantic consequence


Modal-formal accounts
Warrant-based accounts
Non-monotonic logical consequence


Logical consequence
Modal accounts
Modal-formal accounts


Logical consequence
Modal accounts
Warrant-based accounts


Logical consequence
Modal accounts
Non-monotonic logical consequence


Induced substructures and closed subsets


Structure (mathematical logic)
Definition

Domain
Signature
Interpretation function
Examples


Structure (mathematical logic)
Definition
Domain


Structure (mathematical logic)
Definition
Signature


Structure (mathematical logic)
Definition
Interpretation function


Structure (mathematical logic)
Definition
Examples


Structure (mathematical logic)
Homomorphisms and embeddings

Homomorphisms
Embeddings
Example
Homomorphism problem


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphisms


Structure (mathematical logic)
Homomorphisms and embeddings
Embeddings


Structure (mathematical logic)
Homomorphisms and embeddings
Example


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphism problem


Structure (mathematical logic)
Structures and first-order logic

Satisfaction relation
Definable relations


Structure (mathematical logic)
Structures and first-order logic
Satisfaction relation


Structure (mathematical logic)
Structures and first-order logic
Definable relations

Definability with parameters
Implicit definability


Structure (mathematical logic)
Structures and first-order logic
Definable relations
Definability with parameters


Structure (mathematical logic)
Structures and first-order logic
Definable relations
Implicit definability


Structure (mathematical logic)
Many-sorted structures


Structure (mathematical logic)
Other generalizations

Partial algebras
Structures for typed languages
Higher-order languages
Structures that are proper classes


Structure (mathematical logic)
Other generalizations
Partial algebras


Structure (mathematical logic)
Other generalizations
Structures for typed languages


Structure (mathematical logic)
Other generalizations
Higher-order languages


Structure (mathematical logic)
Other generalizations
Structures that are proper classes


Theoremhood and truth
Epistemological considerations
Informal account of theorems
Relation with scientific theories
Terminology
Layout
Lore
Theorems in logic


Theorem
Relation with scientific theories


Syntax and semantics
Interpretation of a formal theorem
Theorems and theories


Theorem
Theorems in logic
Syntax and semantics


Theorem
Theorems in logic
Interpretation of a formal theorem


Theorem
Theorems in logic
Theorems and theories


General theories (as expressed in formal language)


General theories (as expressed in formal language)

Subtheories and extensions
Deductive theories
Consistency and completeness
Interpretation of a theory
Theories associated with a structure


Theory (mathematical logic)
First-order theories

Syntactic consequence in a first-order theory


Theory (mathematical logic)
First-order theories
Derivation in a first-order theory


Syntactic consequence in a first-order theory


Theory (mathematical logic)
First-order theories
Interpretation of a first-order theory


Theory (mathematical logic)
First-order theories
First-order theories with identity


Theory (mathematical logic)
First-order theories
Topics related to first-order theories


History
Introduction
Differences from set theory
Technical details
Interpretations
List of type theories
Applications


Basics
Functions
Common "by rule" types and terms


Type theory
Introduction
Basics
Terms and types


Type theory
Introduction
Basics
Computation


Type theory
Introduction
Functions
Lambda terms


Type theory
Introduction
Functions
Dependent typing


Type theory
Introduction
Functions
Universes


Type theory
Introduction
Common "by rule" types and terms

Empty type
Unit type
Boolean type
Product type
Sum type
Natural numbers
Identity type
Inductive types


Type theory
Introduction
Common "by rule" types and terms
Empty type


Type theory
Introduction
Common "by rule" types and terms
Unit type


Type theory
Introduction
Common "by rule" types and terms
Boolean type


Type theory
Introduction
Common "by rule" types and terms
Product type


Type theory
Introduction
Common "by rule" types and terms
Sum type


Type theory
Introduction
Common "by rule" types and terms
Natural numbers


Type theory
Introduction
Common "by rule" types and terms
Identity type


Type theory
Introduction
Common "by rule" types and terms
Inductive types


Terms
Judgements
Rules
Properties of type theories
Axioms
Decision problems
Research areas


Type theory
Technical details
Judgements


Type theory
Technical details
Properties of type theories


Type theory
Technical details
Decision problems


Type theory
Technical details
Decision problems
Type inhabitation


Type theory
Technical details
Decision problems
Type inference


Type theory
Technical details
Research areas


Types are propositions; terms are proofs


Type theory
Interpretations
Curry-Howard correspondence


Type theory
Interpretations
Intuitionistic logic


Type theory
Interpretations
Constructive mathematics


Type theory
Interpretations
Category theory


Type theory
List of type theories
Active research


Mathematical foundations
Proof assistants
Programming languages
Linguistics
Social sciences


Type theory
Applications
Mathematical foundations


Type theory
Applications
Proof assistants


Type theory
Applications
Programming languages


Type theory
Applications
Social sciences


Basics
Paradoxes of the infinite sets
Paradoxes of the Supertask
Paradoxes of proof and definability


Cardinal numbers
Ordinal numbers
Power sets


Paradoxes of set theory
Basics
Cardinal numbers


Paradoxes of set theory
Basics
Ordinal numbers


Paradoxes of set theory
Basics
Power sets


Paradoxes of set theory
Paradoxes of the infinite sets

Paradoxes of enumeration
Je le vois, mais je ne crois pas
Paradoxes of well-ordering


Paradoxes of set theory
Paradoxes of the infinite sets
Paradoxes of enumeration


Paradoxes of set theory
Paradoxes of the infinite sets
Je le vois, mais je ne crois pas


Paradoxes of set theory
Paradoxes of the infinite sets
Paradoxes of well-ordering


Paradoxes of set theory
Paradoxes of the Supertask

The diary of Tristram Shandy
The Ross-Littlewood paradox


Paradoxes of set theory
Paradoxes of the Supertask
The diary of Tristram Shandy


Paradoxes of set theory
Paradoxes of the Supertask
The Ross-Littlewood paradox


Paradoxes of set theory
Paradoxes of proof and definability

Early paradoxes: the set of all sets
Paradoxes by change of language
Paradox of Löwenheim and Skolem


Paradoxes of set theory
Paradoxes of proof and definability
Early paradoxes: the set of all sets


Paradoxes of set theory
Paradoxes of proof and definability
Paradoxes by change of language


Paradoxes of set theory
Paradoxes of proof and definability
Paradoxes by change of language
König's paradox


Paradoxes of set theory
Paradoxes of proof and definability
Paradoxes by change of language
Richard's paradox


Paradoxes of set theory
Paradoxes of proof and definability
Paradox of Löwenheim and Skolem


Relationship to the incompleteness theorems


Gödel's completeness theorem
Preliminaries


Gödel's original formulation
More general form
Model existence theorem
As a theorem of arithmetic


Gödel's completeness theorem
Statement
Gödel's original formulation


Gödel's completeness theorem
Statement
More general form


Gödel's completeness theorem
Statement
Model existence theorem


Gödel's completeness theorem
Statement
As a theorem of arithmetic


Gödel's completeness theorem
Consequences


Gödel's completeness theorem
Relationship to the compactness theorem


Gödel's completeness theorem
Completeness in other logics


History
Statement
General form
Discussion


Tarski's undefinability theorem
Statement


Tarski's undefinability theorem
General form


Tarski's undefinability theorem
Discussion


Aaron Swartz incident
Limitations
Increasing public access


Connection with earlier work and the role of the axiom of choice

Obtaining infinitely many balls from one

Von Neumann paradox in the Euclidean plane


Banach–Tarski paradox
Banach and Tarski publication


Connection with earlier work and the role of the axiom of choice


Banach–Tarski paradox
A sketch of the proof

Step 1
Step 2
Step 3
Step 4
Some details, fleshed out


Banach–Tarski paradox
A sketch of the proof
Step 1


Banach–Tarski paradox
A sketch of the proof
Step 2


Banach–Tarski paradox
A sketch of the proof
Step 3


Banach–Tarski paradox
A sketch of the proof
Step 4


Banach–Tarski paradox
A sketch of the proof
Some details, fleshed out


Obtaining infinitely many balls from one


Von Neumann paradox in the Euclidean plane


Proof
When A is countably infinite
Related paradoxes
History
Generalizations


Cantor's theorem
When A is countably infinite


Uncountable set
Consequences
Version for Quine's New Foundations


Cantor's diagonal argument
Uncountable set


Cantor's diagonal argument
Uncountable set
Real numbers


Cantor's diagonal argument
Uncountable set
General sets


Ordering of cardinals
Open questions
Diagonalization in broader context


Cantor's diagonal argument
Consequences
Ordering of cardinals


Cantor's diagonal argument
Consequences
Ordering of cardinals
In the absence of excluded middle


Cantor's diagonal argument
Consequences
Open questions


Cantor's diagonal argument
Consequences
Diagonalization in broader context


Cantor's diagonal argument
Version for Quine's New Foundations


Robinson's principle
Upward Löwenheim–Skolem theorem
Non-standard analysis


Compactness theorem
Applications
Robinson's principle


Compactness theorem
Applications
Upward Löwenheim–Skolem theorem


Compactness theorem
Applications
Non-standard analysis


Background
History
Formalization
Computability theory
Generalization


Programming consequences
Common pitfalls


Halting problem
Background
Programming consequences


Halting problem
Background
Common pitfalls


Halting problem
History
Origin of the halting problem


Representation as a set
Proof concept
Sketch of rigorous proof


Halting problem
Formalization
Representation as a set


Halting problem
Formalization
Proof concept


Halting problem
Formalization
Sketch of rigorous proof


Approximations
Gödel's incompleteness theorems


Halting problem
Computability theory
Approximations


Halting problem
Computability theory
Gödel's incompleteness theorems


Halting on all inputs
Recognizing partial solutions
Lossy computation
Oracle machines


Halting problem
Generalization
Halting on all inputs


Halting problem
Generalization
Recognizing partial solutions


Halting problem
Generalization
Lossy computation


Halting problem
Generalization
Oracle machines


Contents
Early reviews
Second edition
Later reviews


The Principles of Mathematics
Early reviews


The Principles of Mathematics
Second edition


The Principles of Mathematics
Later reviews


EAN format used in barcodes, and upgrading


Issuing process
Registration group element
Registrant element


ISBN
Overview
Registration group element


ISBN
Overview
Registrant element
English language pattern


ISBN-10 check digits
ISBN-10 check digit calculation
ISBN-13 check digit calculation
ISBN-10 to ISBN-13 conversion
Errors in usage
eISBN


ISBN
Check digits
ISBN-10 check digit calculation


ISBN
Check digits
ISBN-13 check digit calculation


ISBN
Check digits
ISBN-10 to ISBN-13 conversion


Ramified types and the axiom of reducibility


Principia Mathematica
Scope of foundations laid


Contemporary construction of a formal theory


Principia Mathematica
Theoretical basis
Construction


Principia Mathematica
Theoretical basis
Primitive ideas


Principia Mathematica
Theoretical basis
Primitive propositions


An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)

An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)

Introduction to the notation of the theory of classes and relations


An introduction to the notation of "Section A Mathematical Logic" (formulas ✸1–✸5.71)


An introduction to the notation of "Section B Theory of Apparent Variables" (formulas ✸8–✸14.34)


Introduction to the notation of the theory of classes and relations


Principia Mathematica
Consistency and criticisms

Gödel 1930, 1931
Wittgenstein 1919, 1939
Gödel 1944


Principia Mathematica
Consistency and criticisms
Gödel 1930, 1931


Principia Mathematica
Consistency and criticisms
Wittgenstein 1919, 1939


Principia Mathematica
Consistency and criticisms
Gödel 1944


Part I Mathematical logic. Volume I ✸1 to ✸43

Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97

Part III Cardinal arithmetic. Volume II ✸100 to ✸126

Part IV Relation-arithmetic. Volume II ✸150 to ✸186

Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276

Part VI Quantity. Volume III ✸300 to ✸375


Part I Mathematical logic. Volume I ✸1 to ✸43


Part II Prolegomena to cardinal arithmetic. Volume I ✸50 to ✸97


Part III Cardinal arithmetic. Volume II ✸100 to ✸126


Part IV Relation-arithmetic. Volume II ✸150 to ✸186


Part V Series. Volume II ✸200 to ✸234 and volume III ✸250 to ✸276


Part VI Quantity. Volume III ✸300 to ✸375


Principia Mathematica
Comparison with set theory


Principia Mathematica
Differences between editions


Theorem
Discussion
Proof sketch
In other logics
Historical notes
References
Sources
External links


Löwenheim–Skolem theorem
Discussion
Concepts


Löwenheim–Skolem theorem
Discussion
Concepts
Signatures


Löwenheim–Skolem theorem
Discussion
Concepts
Structures / Models


Löwenheim–Skolem theorem
Discussion
Consequences


Löwenheim–Skolem theorem
Proof sketch
Downward part


Löwenheim–Skolem theorem
Proof sketch
Upward part


Löwenheim–Skolem theorem
In other logics


Löwenheim–Skolem theorem
Historical notes


Historical publications
Secondary sources


Löwenheim–Skolem theorem
Sources
Historical publications


Löwenheim–Skolem theorem
Sources
Secondary sources


Characteristics
History
Generalized semantics
References
Further reading


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’s acceptance of Aristotle


Term logic
Singular terms
The three figures


Term logic
Singular terms
The fourth figure


Term logic
Syllogism in the first figure


Term logic
Syllogism in the second figure


Term logic
Syllogism in the third figure


Term logic
Boole’s acceptance of Aristotle


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


Logical truth
Logical truths and analytic truths


Logical truth
Truth values and tautologies


Logical truth
Logical truth and logical constants


Logical truth
Logical truth and rules of inference


Logical truth
Logical truth and logical positivism


Logical equivalences
Examples
Relation to material equivalence


Logical equivalence
Logical equivalences

Logical equivalences involving conditional statements

Logical equivalences involving biconditionals


Logical equivalence
Logical equivalences
General logical equivalences


Logical equivalences involving conditional statements


Logical equivalences involving biconditionals


Logical equivalence
Relation to material equivalence


Definition
Examples
Incorrect inference
Applications


Example for definition #1
Example for definition #2


Inference
Examples
Example for definition #1


Inference
Examples
Example for definition #2


Bayesian statistics and probability logic


Inference
Applications
Inference engines


Inference
Applications
Inference engines
Prolog engine


Inference
Applications
Non-monotonic logic


Consistency and completeness in arithmetic and set theory


Notation
Definition
Basic results
Henkin's theorem
Sketch of proof


Consistency
First-order logic
Definition


Consistency
First-order logic
Basic results


Consistency
First-order logic
Henkin's theorem


Consistency
First-order logic
Sketch of proof


Historical usage
Relation to the mind
Treatment in logic
Objections to propositions


By Aristotle
By the logical positivists
By Russell


Proposition
Historical usage
By Aristotle


Proposition
Historical usage
By the logical positivists


Aristotelian logic
Syntactic characterization
Semantic characterization


Proposition
Treatment in logic
Aristotelian logic


Proposition
Treatment in logic
Syntactic characterization


Proposition
Treatment in logic
Semantic characterization


Efficient verification and the Boolean satisfiability problem

Tautologies versus validities in first-order logic


Tautology (logic)
Definition and examples


Tautology (logic)
Tautological implication


Tautology (logic)
Semantic completeness and soundness


Efficient verification and the Boolean satisfiability problem


Tautologies versus validities in first-order logic


Examples
Applications
Category theoretic view
Relation with equality


Logarithm and exponential
Integers modulo 6
Relation-preserving isomorphism


Isomorphism
Examples
Logarithm and exponential


Isomorphism
Examples
Relation-preserving isomorphism


Isomorphism
Category theoretic view
Isomorphism vs. bijective morphism


Aleph-nought
Aleph-one
Continuum hypothesis
Aleph-omega
Aleph-α for general α
Fixed points of omega
Role of axiom of choice


Combinatorics
Set theory
Computability and complexity theory


Listing
Countable vs. uncountable
Ordinals
Comparison of cardinalities


Enumeration
Set theory
Countable vs. uncountable


Enumeration
Set theory
Countable vs. uncountable
Examples


Enumeration
Set theory
Countable vs. uncountable
Properties


Enumeration
Set theory
Comparison of cardinalities


Enumeration
Computability and complexity theory


Partial definition
Hierarchy of consistency strength
Motivations and epistemic status


Large cardinal
Hierarchy of consistency strength


Large cardinal
Motivations and epistemic status


Operation (mathematics)
Types of operation


Definition
Examples
Properties
Space of surjections
Gallery


Surjections as right invertible functions

Cardinality of the domain of a surjection

Induced surjection and induced bijection


Surjections as right invertible functions


Surjective function
Properties
Surjections as epimorphisms


Surjective function
Properties
Surjections as binary relations


Cardinality of the domain of a surjection


Surjective function
Properties
Composition and decomposition


Induced surjection and induced bijection


Surjective function
Space of surjections


Existence of a proper class of inaccessibles

α-inaccessible cardinals and hyper-inaccessible cardinals

Two model-theoretic characterisations of inaccessibility


Inaccessible cardinal
Models and consistency


Existence of a proper class of inaccessibles


α-inaccessible cardinals and hyper-inaccessible cardinals


Two model-theoretic characterisations of inaccessibility


Schröder–Bernstein theorem
Proof
Another proof


Schröder–Bernstein theorem
Proof
Examples


Simplified overview
Gödel's encoding
Lack of uniqueness
Application to formal arithmetic
Generalizations
Gödel  sets


Gödel numbering
Gödel's encoding
Example


Gödel numbering
Application to formal arithmetic

Expressing statements and proofs by numbers


Gödel numbering
Application to formal arithmetic
Recursion


Expressing statements and proofs by numbers


Terminology
Properties and examples
Notation
Binary operations as ternary relations
External binary operations


Binary operation
Properties and examples


Binary operation
Binary operations as ternary relations


Binary operation
External binary operations


Ontology
Axioms
Discussion
History
Footnotes
References
External links


Metamathematics
Peano arithmetic
Infinite sets


General set theory
Discussion
Metamathematics


General set theory
Discussion
Peano arithmetic


General set theory
Discussion
Infinite sets


MK axioms and ontology
Discussion
The axioms in Kelley's General Topology
Notes
References
External links


Morse–Kelley set theory
MK axioms and ontology


Morse–Kelley set theory
Discussion
Model theory


Morse–Kelley set theory
Discussion
History


Morse–Kelley set theory
The axioms in Kelley's General Topology


Introduction
Subtheories of ZF
Sorted theories


Constructive outlook
History and overview


Constructive set theory
Introduction
Constructive outlook

Preliminary on the use of intuitionistic logic


Preliminary on the use of intuitionistic logic


Constructive set theory
Introduction
Constructive outlook
Imposed restrictions on a set theory


Constructive set theory
Introduction
Constructive outlook
Metalogic


Constructive set theory
Introduction
Constructive outlook
Anti-classical principles


Constructive set theory
Introduction
History and overview


Constructive set theory
Introduction
History and overview
Models


Constructive set theory
Subtheories of ZF

Notation
Equality
Merging sets
Set existence
BCST
ECST
Functions
Choice functions
Arithmetic
Exponentiation
Analysis
Non-constructive principles
Induction
Strong Collection
Constructive Zermelo–Fraenkel
Metalogic of CZF
Intuitionistic Zermelo–Fraenkel
Intuitionistic Z
Intuitionistic KP


Constructive set theory
Subtheories of ZF
Notation


Constructive set theory
Subtheories of ZF
Notation
Language


Constructive set theory
Subtheories of ZF
Notation
Variables


Constructive set theory
Subtheories of ZF
Notation
Classes


Constructive set theory
Subtheories of ZF
Equality


Constructive set theory
Subtheories of ZF
Merging sets


Constructive set theory
Subtheories of ZF
Set existence


Constructive set theory
Subtheories of ZF
BCST

Separation
No universal set
Predicativity
Replacement


Constructive set theory
Subtheories of ZF
BCST
Separation


Constructive set theory
Subtheories of ZF
BCST
No universal set


Constructive set theory
Subtheories of ZF
BCST
Predicativity


Constructive set theory
Subtheories of ZF
BCST
Replacement


Constructive set theory
Subtheories of ZF
ECST

Weaker formulations of infinity
Number bounds


Constructive set theory
Subtheories of ZF
ECST
Weaker formulations of infinity


Constructive set theory
Subtheories of ZF
ECST
Number bounds


Constructive set theory
Subtheories of ZF
Functions

Finiteness
General note on programs and functions
Total functional relations
Characteristic functions
Computable sets
Boundedness criteria


Constructive set theory
Subtheories of ZF
Functions
Finiteness


Constructive set theory
Subtheories of ZF
Functions
General note on programs and functions


Constructive set theory
Subtheories of ZF
Functions
Total functional relations


Constructive set theory
Subtheories of ZF
Functions
Characteristic functions


Constructive set theory
Subtheories of ZF
Functions
Computable sets


Constructive set theory
Subtheories of ZF
Functions
Boundedness criteria


Constructive set theory
Subtheories of ZF
Choice functions

Diaconescu's theorem
Analysis of Diaconescu's theorem
Regularity implies PEM


Constructive set theory
Subtheories of ZF
Choice functions
Diaconescu's theorem


Constructive set theory
Subtheories of ZF
Choice functions
Analysis of Diaconescu's theorem


Constructive set theory
Subtheories of ZF
Choice functions
Regularity implies PEM


Constructive set theory
Subtheories of ZF
Arithmetic

Recursion
Moderate induction in ECST
Induction without infinite sets


Constructive set theory
Subtheories of ZF
Arithmetic
Recursion


Constructive set theory
Subtheories of ZF
Arithmetic
Moderate induction in ECST


Constructive set theory
Subtheories of ZF
Arithmetic
Induction without infinite sets


Constructive set theory
Subtheories of ZF
Exponentiation

On countable sets
The class of subsets of a set
Metalogic
Category and type theoretic notions


Constructive set theory
Subtheories of ZF
Exponentiation
On countable sets


Constructive set theory
Subtheories of ZF
Exponentiation
The class of subsets of a set


Constructive set theory
Subtheories of ZF
Exponentiation
Metalogic


Constructive set theory
Subtheories of ZF
Exponentiation
Category and type theoretic notions


Constructive set theory
Subtheories of ZF
Analysis

Cauchy sequences
Cauchy reals
Towards the Dedekind reals
Constructive schools


Constructive set theory
Subtheories of ZF
Analysis
Cauchy sequences


Constructive set theory
Subtheories of ZF
Analysis
Cauchy reals


Constructive set theory
Subtheories of ZF
Analysis
Towards the Dedekind reals


Constructive set theory
Subtheories of ZF
Analysis
Constructive schools


Constructive set theory
Subtheories of ZF
Non-constructive principles


Constructive set theory
Subtheories of ZF
Non-constructive principles
Infinite trees


Constructive set theory
Subtheories of ZF
Induction

Mathematical induction
Set Induction
Metalogic
Relation to ZF


Constructive set theory
Subtheories of ZF
Induction
Mathematical induction


Constructive set theory
Subtheories of ZF
Induction
Set Induction


Constructive set theory
Subtheories of ZF
Induction
Metalogic


Constructive set theory
Subtheories of ZF
Induction
Relation to ZF


Constructive set theory
Subtheories of ZF
Strong Collection


Constructive set theory
Subtheories of ZF
Strong Collection
Metalogic


Constructive set theory
Subtheories of ZF
Constructive Zermelo–Fraenkel

Binary refinement
Subset Collection
Fullness


Constructive set theory
Subtheories of ZF
Constructive Zermelo–Fraenkel
Binary refinement


Constructive set theory
Subtheories of ZF
Constructive Zermelo–Fraenkel
Subset Collection


Constructive set theory
Subtheories of ZF
Constructive Zermelo–Fraenkel
Fullness


Constructive set theory
Subtheories of ZF
Metalogic of CZF

Unprovable claims
Ordinal analysis
Models
Indecomposability
Breaking with ZF


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Unprovable claims


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Ordinal analysis


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Models


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Indecomposability


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Breaking with ZF


Constructive set theory
Subtheories of ZF
Intuitionistic Zermelo–Fraenkel


Constructive set theory
Subtheories of ZF
Intuitionistic Zermelo–Fraenkel
Metalogic


Constructive set theory
Subtheories of ZF
Intuitionistic Z


Constructive set theory
Subtheories of ZF
Intuitionistic KP


Constructive set theory
Bishop style set theory
Category theories


Constructive set theory
Sorted theories
Constructive set theory


Constructive set theory
Sorted theories
Bishop style set theory


Constructive set theory
Sorted theories
Category theories


Axioms
Related definitions
Theorems
Metalogic


Comparison with Zermelo-Fraenkel set theory


Kripke–Platek set theory
Axioms
Empty set


Kripke–Platek set theory
Related definitions


Admissible sets
Cartesian products exist


Kripke–Platek set theory
Theorems
Admissible sets


Kripke–Platek set theory
Theorems
Cartesian products exist


Classes in set theory
Axiomatization of NBG
History
NBG, ZFC, and MK
Category theory
Notes
References
Bibliography
External links


Von Neumann–Bernays–Gödel set theory
Classes in set theory

Axiom schema versus class existence theorem


Von Neumann–Bernays–Gödel set theory
Classes in set theory
The uses of classes


Axiom schema versus class existence theorem


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG

Definitions and axioms of extensionality and pairing

Class existence axioms and axiom of regularity


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG
Classes and sets


Definitions and axioms of extensionality and pairing


Class existence axioms and axiom of regularity


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG
Class existence theorem


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG
Extending the class existence theorem


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG
Set axioms


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG
Axiom of global choice


Von Neumann–Bernays–Gödel set theory
History

Von Neumann's 1925 axiom system
Von Neumann's 1929 axiom system
Bernays' axiom system
Gödel's axiom system (NBG)


Von Neumann–Bernays–Gödel set theory
History
Von Neumann's 1925 axiom system


Von Neumann–Bernays–Gödel set theory
History
Von Neumann's 1929 axiom system


Von Neumann–Bernays–Gödel set theory
History
Bernays' axiom system


Von Neumann–Bernays–Gödel set theory
History
Gödel's axiom system (NBG)


Von Neumann–Bernays–Gödel set theory
NBG, ZFC, and MK


Von Neumann–Bernays–Gödel set theory
NBG, ZFC, and MK
Models


Von Neumann–Bernays–Gödel set theory
Category theory


Von Neumann–Bernays–Gödel set theory
Notes


Von Neumann–Bernays–Gödel set theory
References


Von Neumann–Bernays–Gödel set theory
Bibliography


Von Neumann–Bernays–Gödel set theory
External links


Axioms
Implementation in the Mizar system
Implementation in Metamath


Tarski–Grothendieck set theory
Implementation in the Mizar system


Tarski–Grothendieck set theory
Implementation in Metamath


Formal system
Background
Inference and entailment

Formal language
Deductive system
Logical system


Formal system
Background
Inference and entailment
Formal language


Formal system
Background
Inference and entailment
Deductive system


Formal system
Background
Inference and entailment
Logical system


Formal system
History
Formalism
Hilbert's program


Alphabet (formal languages)
Applications


List of formal systems
Other formal systems


Examples
Syntax versus semantics
Variables


Expression (mathematics)
Syntax versus semantics

Syntax
Semantics
Formal languages and lambda calculus


Expression (mathematics)
Syntax versus semantics
Syntax


Expression (mathematics)
Syntax versus semantics
Semantics


Expression (mathematics)
Syntax versus semantics
Formal languages and lambda calculus


The language
The axioms
Relation to Zermelo–Fraenkel set theory
AST and category theory


Ackermann set theory
Relation to Zermelo–Fraenkel set theory


Ackermann set theory
AST and category theory


Nullary
Unary
Binary
Ternary
n-ary
Varying arity


History
Words over an alphabet
Definition
Examples
Language-specification formalisms
Operations on languages
Applications


Formal language
Language-specification formalisms


Programming languages
Formal theories, systems, and proofs


Formal language
Applications
Programming languages


Formal language
Applications
Formal theories, systems, and proofs


Formal language
Applications
Formal theories, systems, and proofs
Interpretations and models


Symbols
Formal language
Formation rules
Propositions
Formal theories
Formal systems
Interpretations


Syntax (logic)
Syntactic entities
Symbols


Syntax (logic)
Syntactic entities
Formal language


Syntax (logic)
Syntactic entities
Formation rules


Syntax (logic)
Syntactic entities
Propositions


Syntax (logic)
Syntactic entities
Formal theories


Syntax (logic)
Syntactic entities
Formal systems

Syntactic consequence within a formal system

Syntactic completeness of a formal system


Syntactic consequence within a formal system


Syntactic completeness of a formal system


Syntax (logic)
Syntactic entities
Interpretations


Formal definition
Finite axiomatization
Examples
Finitely axiomatized theories
In higher-order logic


Axiom schema
Finitely axiomatized theories


The consistency problem and related partial results


Natural numbers and the axiom of infinity


New Foundations
Quinean set theory
Axioms and stratification


New Foundations
Quinean set theory
Finite axiomatization


New Foundations
Quinean set theory
Ordered pairs


Admissibility of useful large sets
Resolution of set-theoretic paradoxes
Cartesian closure


New Foundations
Consequences
Admissibility of useful large sets


New Foundations
Consequences
Resolution of set-theoretic paradoxes

Burali-Forti paradox and the T operation


New Foundations
Consequences
Resolution of set-theoretic paradoxes
Cantor's paradox and Cantorian sets


Burali-Forti paradox and the T operation


New Foundations
Consequences
Cartesian closure


Self-sufficiency of mathematical foundations in NFU


New Foundations
Models of NFU
Facts about the automorphism j


New Foundations
Strong axioms of infinity


New Foundations
The system ML (Mathematical Logic)


History
Automata
Variant definitions of automata
Types of automata
Hierarchy in terms of powers
Applications
Automata simulators
Connection to category theory


Automata theory
Automata
Informal description


Automata theory
Automata
Formal definition


Automata theory
Variant definitions of automata


Discrete, continuous, and hybrid automata


Automata theory
Hierarchy in terms of powers


Automata theory
Connection to category theory


Ground expression
Formal definitions
Ground term


Ground expression
Formal definitions
Ground atom


Ground expression
Formal definitions
Ground formula


Examples
Formal explanation
Natural language


Free variables and bound variables
Examples


Free variables and bound variables
Examples
Variable-binding operators


Free variables and bound variables
Formal explanation


Free variables and bound variables
Formal explanation
Function expressions


Free variables and bound variables
Natural language


Extension by new constant and function names


Definition of relation symbols
Definition of function symbols
Extensions by definitions
Examples


Extension by definitions
Definition of relation symbols


Extension by definitions
Definition of function symbols


Extension by definitions
Extensions by definitions


Introduction
Propositional calculus
Predicate logic
Atomic and open formulas
Closed formulas
Properties applicable to formulas
Usage of the terminology


Well-formed formula
Propositional calculus


Well-formed formula
Atomic and open formulas


Well-formed formula
Properties applicable to formulas


Well-formed formula
Usage of the terminology


Definition
Properties
Rules of inference
Colloquial usage


Logical biconditional
Definition
Truth table


Logical biconditional
Definition
Venn diagrams


Logical biconditional
Rules of inference

Biconditional introduction
Biconditional elimination


Logical biconditional
Rules of inference
Biconditional introduction


Logical biconditional
Rules of inference
Biconditional elimination


Notation
Definition
Introduction and elimination rules
Negation
Properties
Applications in computer engineering
Set-theoretic correspondence
Natural language


Logical conjunction
Definition
Truth table


Logical conjunction
Definition
Defined by other operators


Logical conjunction
Introduction and elimination rules


Logical conjunction
Negation
Other proof strategies


Logical conjunction
Applications in computer engineering


Logical conjunction
Set-theoretic correspondence


History
Introductory example
Formal definition
The Chomsky hierarchy
Analytic grammars


Formal grammar
Introductory example
Example 1


Formal grammar
Introductory example
Examples 2 and 3


Some mathematical constructs regarding formal grammars


Formal grammar
Formal definition
The syntax of grammars


Formal grammar
Formal definition
Example


Context-free grammars
Regular grammars
Other forms of generative grammars
Recursive grammars


Formal grammar
The Chomsky hierarchy
Context-free grammars


Formal grammar
The Chomsky hierarchy
Regular grammars


Formal grammar
The Chomsky hierarchy
Other forms of generative grammars


Formal grammar
The Chomsky hierarchy
Recursive grammars


Atomic formula
Atomic formula in first-order logic


Examples
Model-theoretic conservative extension


Conservative extension
Model-theoretic conservative extension


Types of metalanguage
In natural language
Types of expressions
Role in metaphor
Metaprogramming


Metalanguage
Types of metalanguage
Embedded


Metalanguage
Types of metalanguage
Ordered


Metalanguage
Types of metalanguage
Nested


Deductive systems
Metavariables
Metatheories and metatheorems
Interpretations


Metalanguage
Types of expressions
Deductive systems


Metalanguage
Types of expressions
Metavariables


Metalanguage
Types of expressions
Metatheories and metatheorems


Metalanguage
Types of expressions
Interpretations


Notation
Definitions
Formal properties
Discrepancies with natural language


Semantics
Truth table
Deductive definition


Material conditional
Definitions
Semantics


Material conditional
Definitions
Truth table


Material conditional
Definitions
Deductive definition


Material conditional
Discrepancies with natural language


Formal language
Formal systems
Propositional and predicate logic


Formation rule
Propositional and predicate logic


Programming language and ordinary language