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"


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


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


Aaron Swartz incident
Limitations
Increasing public access


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


Minimal model of set theory is countable


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


Sketch of a proof that e is transcendental


Transcendental number
Numbers proven to be transcendental


Transcendental number
Possible transcendental numbers


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


Irreducible fraction
Embedding of integers
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Addition
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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


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


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


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Characteristics
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Restrictions, extensions, and variations

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First-order logic
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Free and bound variables


First-order logic
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Example: ordered abelian groups


Validity, satisfiability, and logical consequence

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Validity, satisfiability, and logical consequence


First-order logic
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Algebraizations


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First-order logic
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First-order logic
Deductive systems
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First-order logic
Deductive systems
Tableaux method


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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


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First-order logic
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First-order logic
Limitations
Formalizing natural languages


Restrictions, extensions, and variations

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Many-sorted logic
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Infinitary logics
Non-classical and modal logics
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Higher-order logics


Automated theorem proving and formal methods


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Articles, reviews, and essays


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Cambridge University Press
Controversies
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Cambridge University Press
Controversies
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Cambridge University Press
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References
Sources


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External links


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


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


Etymology
Comparison with theorem
Well-known lemmas


Lemma (mathematics)
Comparison with theorem


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


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


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


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


Definition
Basic concepts
Systems of logic
Areas of research
History


Premises, conclusions, and truth
Arguments and inferences
Fallacies
Definitory and strategic rules
Formal systems


Logic
Basic concepts
Premises, conclusions, and truth

Premises and conclusions
Internal structure
Logical truth
Truth tables


Logic
Basic concepts
Premises, conclusions, and truth
Premises and conclusions


Logic
Basic concepts
Premises, conclusions, and truth
Internal structure


Logic
Basic concepts
Premises, conclusions, and truth
Logical truth


Logic
Basic concepts
Premises, conclusions, and truth
Truth tables


Logic
Basic concepts
Arguments and inferences


Logic
Basic concepts
Arguments and inferences
Deductive


Logic
Basic concepts
Arguments and inferences
Ampliative


Logic
Basic concepts
Definitory and strategic rules


Aristotelian Classical Extended Deviant Informal


Logic
Systems of logic
Classical
Propositional logic


Logic
Systems of logic
Classical
First-order logic


Logic
Systems of logic
Extended
Modal logic


Logic
Systems of logic
Extended
Higher order logic


Philosophy of logic and philosophical logic


Logic
Areas of research
Mathematical logic


Logic
Areas of research
Computational logic


Logic
Areas of research
Formal semantics of natural language


Logic
Areas of research
Epistemology of logic


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


Characteristics
History
Generalized semantics
References
Further reading


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


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


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


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


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


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


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


Square of opposition (or logical square) and modal logic

Set-theoretical interpretation of categorical statements


Square of opposition
The problem of existential import


Square of opposition
Modern squares of opposition


Square of opposition
Logical hexagons and other bi-simplexes


Square of opposition (or logical square) and modal logic


Set-theoretical interpretation of categorical statements


Generic description of a propositional calculus

Interpretation of a truth-functional propositional calculus


Propositional calculus
Basic concepts
Closure under operations


Propositional calculus
Basic concepts
Argument


Generic description of a propositional calculus


Propositional calculus
Example 1. Simple axiom system


Propositional calculus
Example 2. Natural deduction system


Propositional calculus
Basic and derived argument forms


Propositional calculus
Proofs in propositional calculus

Example of a proof in natural deduction system

Example of a proof in a classical propositional calculus system


Example of a proof in natural deduction system


Example of a proof in a classical propositional calculus system


Propositional calculus
Soundness and completeness of the rules

Another outline for a completeness proof


Propositional calculus
Soundness and completeness of the rules
Sketch of a soundness proof


Propositional calculus
Soundness and completeness of the rules
Sketch of completeness proof

Verifying completeness for the classical propositional calculus system


Propositional calculus
Soundness and completeness of the rules
Sketch of completeness proof
Example


Verifying completeness for the classical propositional calculus system


Another outline for a completeness proof


Interpretation of a truth-functional propositional calculus

Interpretation of a sentence of truth-functional propositional logic


Interpretation of a sentence of truth-functional propositional logic


Propositional calculus
Alternative calculus

Axioms
Inference rule
Meta-inference rule
Example of a proof


Propositional calculus
Alternative calculus
Axioms


Propositional calculus
Alternative calculus
Inference rule


Propositional calculus
Alternative calculus
Meta-inference rule


Propositional calculus
Alternative calculus
Example of a proof


Propositional calculus
Equivalence to equational logics


Propositional calculus
Graphical calculi


Propositional calculus
Other logical calculi


List of formal systems
Other formal systems


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


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


Axiom schema
Finitely axiomatized theories


Nullary
Unary
Binary
Ternary
n-ary
Varying arity


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


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


Alphabet (formal languages)
Applications


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


Ground expression
Formal definitions
Ground term


Ground expression
Formal definitions
Ground atom


Ground expression
Formal definitions
Ground formula


Examples
Model-theoretic conservative extension


Conservative extension
Model-theoretic conservative extension


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


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


Extension by new constant and function names


Atomic formula
Atomic formula in first-order logic


Formal language
Formal systems
Propositional and predicate logic


Formation rule
Propositional and predicate logic


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


Ground expression
Formal definitions
Ground term


Ground expression
Formal definitions
Ground atom


Ground expression
Formal definitions
Ground formula


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


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


Inclusive and exclusive disjunction
Notation
Classical disjunction
Applications in computer science
Set theory
Natural language


Logical disjunction
Inclusive and exclusive disjunction


Logical disjunction
Classical disjunction

Semantics
Defined by other operators
Properties


Logical disjunction
Classical disjunction
Semantics


Logical disjunction
Classical disjunction
Defined by other operators


Logical disjunction
Classical disjunction
Properties


Logical disjunction
Applications in computer science

Bitwise operation
Logical operation
Constructive disjunction


Logical disjunction
Applications in computer science
Bitwise operation


Logical disjunction
Applications in computer science
Logical operation


Logical disjunction
Applications in computer science
Constructive disjunction


Basics
Properties
Universal closure
As adjoint


Universal quantification
Basics
Notation


Negation
Other connectives
Rules of inference
The empty set


Universal quantification
Properties
Negation


Universal quantification
Properties
Other connectives


Universal quantification
Properties
Rules of inference


Universal quantification
Properties
The empty set


Universal quantification
Universal closure


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
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


Logical equality
Alternative descriptions


Introducing new function symbols
Doing without functional predicates


Functional predicate
Introducing new function symbols


Functional predicate
Doing without functional predicates


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


Reduction to ordinary existential and universal quantification


Uniqueness quantification
Proving uniqueness


Basics
Properties
As adjoint
Encoding
Origin


Negation
Rules of inference
The empty set


Existential quantification
Properties
Negation


Existential quantification
Properties
Rules of inference


Existential quantification
Properties
The empty set


Programming language and ordinary language


Double negation
Distributivity
Linearity
Self dual
Negations of quantifiers


Negation
Properties
Negations of quantifiers


Translating sentences from a natural language into an artificial language


Assumptions
Atomic sentences
Atomic formulae
Compound sentences
Compound formulae


Atomic sentence
Examples
Atomic sentences


Atomic sentence
Examples
Atomic formulae


Atomic sentence
Examples
Compound sentences


Atomic sentence
Examples
Compound formulae


Atomic sentence
Philosophical significance


Equivalence to Turing machines
Other properties


Spectrum of a sentence
Properties
Equivalence to Turing machines


Spectrum of a sentence
Properties
Other properties


Definition
Other conventions
Use of signatures in logic and algebra
Many-sorted signatures


Signature (logic)
Use of signatures in logic and algebra


Signature (logic)
Many-sorted signatures


Signature (logic)
Many-sorted signatures
Symbol types


Signature (logic)
Many-sorted signatures
Signature


Overview
Can words be modeled as formal symbols?
References


Symbol (formal)
Can words be modeled as formal symbols?


Propositional logic
First-order logic
Algebra


Substitution (logic)
Propositional logic


Substitution (logic)
Propositional logic
Definition


Substitution (logic)
Propositional logic
Tautologies


Character string-oriented languages and utilities


String (computer science)
String datatypes

String length
Character encoding
Implementations
Representations
Security concerns


String (computer science)
String datatypes
String length


String (computer science)
String datatypes
Character encoding


String (computer science)
String datatypes
Implementations


String (computer science)
String datatypes
Representations

Null-terminated
Byte- and bit-terminated
Length-prefixed
Strings as records
Other representations


String (computer science)
String datatypes
Representations
Null-terminated


String (computer science)
String datatypes
Representations
Byte- and bit-terminated


String (computer science)
String datatypes
Representations
Length-prefixed


String (computer science)
String datatypes
Representations
Strings as records


String (computer science)
String datatypes
Representations
Other representations


String (computer science)
String datatypes
Security concerns


String (computer science)
Literal strings


String (computer science)
Non-text strings


String (computer science)
String processing algorithms


String (computer science)
Character string functions


Concatenation and substrings
Prefixes and suffixes
Reversal
Rotations
Lexicographical ordering
String operations
Topology


String (computer science)
Formal theory
Concatenation and substrings


String (computer science)
Formal theory
Prefixes and suffixes


String (computer science)
Formal theory
Reversal


String (computer science)
Formal theory
Rotations


String (computer science)
Formal theory
Lexicographical ordering


String (computer science)
Formal theory
String operations


String (computer science)
Formal theory
Topology


Definition
Arithmetic undefinability
Computability properties
Model-theoretic properties
True theory of second-order arithmetic
Notes
References
External links


True arithmetic
Arithmetic undefinability


True arithmetic
Computability properties


True arithmetic
Model-theoretic properties


True arithmetic
True theory of second-order arithmetic


Historical second-order formulation
Peano arithmetic as first-order theory


Peano axioms
Historical second-order formulation

Defining arithmetic operations and relations


Defining arithmetic operations and relations


Peano axioms
Historical second-order formulation
Models

Set-theoretic models
Interpretation in category theory


Peano axioms
Historical second-order formulation
Models
Set-theoretic models


Peano axioms
Historical second-order formulation
Models
Interpretation in category theory


Peano axioms
Historical second-order formulation
Consistency


Peano axioms
Peano arithmetic as first-order theory

Equivalent axiomatizations
Undecidability and incompleteness
Nonstandard models


Peano axioms
Peano arithmetic as first-order theory
Equivalent axiomatizations


Peano axioms
Peano arithmetic as first-order theory
Undecidability and incompleteness


Peano axioms
Peano arithmetic as first-order theory
Nonstandard models


Peano axioms
Peano arithmetic as first-order theory
Nonstandard models
Overspill


Signatures
Models
Informal semantics
Descriptive signs


History
Notation
Specific kinds of variables
Moduli spaces
Conventional variable names


Variable (mathematics)
Specific kinds of variables

Dependent and independent variables
Examples


Variable (mathematics)
Specific kinds of variables
Dependent and independent variables


Variable (mathematics)
Specific kinds of variables
Examples


Variable (mathematics)
Specific kinds of variables
Examples
Example: the ideal gas law


Variable (mathematics)
Conventional variable names


Expressive power
Idea of decidability
Complexity
Decidable extensions
Undecidable extensions


Skolem arithmetic
Undecidable extensions


Formal definition
Operations with terms
Related concepts


Term structure vs. representation
Structural equality
Ground and linear terms
Building formulas from terms


Term (logic)
Formal definition
Term structure vs. representation


Term (logic)
Formal definition
Structural equality


Term (logic)
Formal definition
Ground and linear terms


Term (logic)
Formal definition
Building formulas from terms


Term (logic)
Related concepts
Sorted terms


Term (logic)
Related concepts
Lambda terms


Term (logic)
Related concepts
Lambda terms
Motivation


Term (logic)
Related concepts
Lambda terms
Definition


Definition
Friedman's grand conjecture
Related systems


Elementary function arithmetic
Definition


Elementary function arithmetic
Friedman's grand conjecture


Elementary function arithmetic
Related systems


Robinson arithmetic
Axioms
Variant axiomatizations


Primitive recursive arithmetic
Language and axioms


Primitive recursive arithmetic
Logic-free calculus


Definition
Models
Subsystems
Projective determinacy
Coding mathematics


Second-order arithmetic
Definition
Syntax


Second-order arithmetic
Definition
Semantics


Second-order arithmetic
Definition
Axioms

Basic
Induction and comprehension schema


Second-order arithmetic
Definition
Axioms
Basic


Second-order arithmetic
Definition
Axioms
Induction and comprehension schema


Second-order arithmetic
Definition
The full system


Definable functions
More types of models


Second-order arithmetic
Models
Definable functions


Second-order arithmetic
Models
More types of models


Arithmetical comprehension
The arithmetical hierarchy for formulas
Recursive comprehension
Weaker systems
Stronger systems


Second-order arithmetic
Subsystems
Arithmetical comprehension


Second-order arithmetic
Subsystems
The arithmetical hierarchy for formulas


Second-order arithmetic
Subsystems
Recursive comprehension


Second-order arithmetic
Subsystems
Weaker systems


Second-order arithmetic
Subsystems
Stronger systems


Second-order arithmetic
Projective determinacy


Second-order arithmetic
Coding mathematics


Euclid's method and style of presentation


Basis in earlier work
Transmission of the text


Euclid's Elements
History
Basis in earlier work


Euclid's Elements
History
Transmission of the text


Euclid's Elements
Influence
In modern mathematics


Translations
Currently in print
Free versions


Euclid's Elements
Editions
Currently in print


Euclid's Elements
Editions
Free versions


Axiomatic systems
Euclidean geometry
Non-Euclidean geometry
Other geometries


Foundations of geometry
Axiomatic systems


Foundations of geometry
Axiomatic systems
Properties of axiomatic systems


Foundations of geometry
Euclidean geometry

Pieri and the Italian school of geometers


Foundations of geometry
Euclidean geometry
Euclid's Elements


Foundations of geometry
Euclidean geometry
A critique of Euclid


Foundations of geometry
Euclidean geometry
Pasch and Peano


Pieri and the Italian school of geometers


Foundations of geometry
Euclidean geometry
Hilbert's axioms

Changes in Hilbert's axioms
Consistency and independence


Foundations of geometry
Euclidean geometry
Hilbert's axioms
Changes in Hilbert's axioms


Foundations of geometry
Euclidean geometry
Hilbert's axioms
Consistency and independence


Foundations of geometry
Euclidean geometry
Birkhoff's axioms


Foundations of geometry
Euclidean geometry
Birkhoff's axioms
Postulates


Foundations of geometry
Euclidean geometry
School geometry

SMSG (School Mathematics Study Group) axioms

UCSMP (University of Chicago School Mathematics Project) axioms


Foundations of geometry
Euclidean geometry
School geometry
Mac Lane's axioms


SMSG (School Mathematics Study Group) axioms


UCSMP (University of Chicago School Mathematics Project) axioms


Foundations of geometry
Euclidean geometry
Other systems


Foundations of geometry
Non-Euclidean geometry

Euclid's parallel postulate
Neutral (or absolute) geometry
Hyperbolic geometry
Elliptic geometry


Foundations of geometry
Non-Euclidean geometry
Euclid's parallel postulate


Foundations of geometry
Non-Euclidean geometry
Neutral (or absolute) geometry

Relation to other geometries
Incompleteness


Foundations of geometry
Non-Euclidean geometry
Neutral (or absolute) geometry
Relation to other geometries


Foundations of geometry
Non-Euclidean geometry
Neutral (or absolute) geometry
Incompleteness


Foundations of geometry
Non-Euclidean geometry
Hyperbolic geometry


Foundations of geometry
Non-Euclidean geometry
Elliptic geometry


Foundations of geometry
Non-Euclidean geometry
Elliptic geometry
Spherical geometry


Foundations of geometry
Other geometries

Projective geometry
Affine geometry
Ordered geometry
Finite geometry


Foundations of geometry
Other geometries
Projective geometry


Foundations of geometry
Other geometries
Affine geometry


Foundations of geometry
Other geometries
Ordered geometry


Foundations of geometry
Other geometries
Finite geometry


History
Definition
Examples
Homomorphisms and isomorphisms
Boolean rings
Ideals and filters
Representations
Axiomatics
Generalizations


Boolean algebra (structure)
Homomorphisms and isomorphisms


Boolean algebra (structure)
Boolean rings


Boolean algebra (structure)
Ideals and filters


Boolean algebra (structure)
Representations


Boolean algebra (structure)
Generalizations


Axiomatic definitions
Explicit constructions of models


Construction of the real numbers
Axiomatic definitions

Axioms
Tarski's axiomatization of the reals


Construction of the real numbers
Axiomatic definitions
Axioms

On the least upper bound property
On models


Construction of the real numbers
Axiomatic definitions
Axioms
On the least upper bound property


Construction of the real numbers
Axiomatic definitions
Axioms
On models


Construction of the real numbers
Axiomatic definitions
Tarski's axiomatization of the reals


Construction of the real numbers
Explicit constructions of models

Construction from integers (Eudoxus reals)


Construction of the real numbers
Explicit constructions of models
Construction from Cauchy sequences


Construction of the real numbers
Explicit constructions of models
Construction by Dedekind cuts


Construction of the real numbers
Explicit constructions of models
Construction using hyperreal numbers


Construction of the real numbers
Explicit constructions of models
Construction from surreal numbers


Construction from integers (Eudoxus reals)


Construction of the real numbers
Explicit constructions of models
Other constructions


Minimal axioms for Boolean algebra
References


Definition
Basis
Truth tables
Examples
Boolean algebras of Boolean operations
Axiomatizing Boolean algebras
Underlying lattice structure
Boolean homomorphisms
Infinitary extensions
Other definitions of Boolean algebra


Boolean algebras canonically defined
Definition


Boolean algebras canonically defined
Basis


Boolean algebras canonically defined
Truth tables


Boolean algebras canonically defined
Examples

Bit vectors
Power set algebra
Representation theorems
Other examples


Boolean algebras canonically defined
Examples
Bit vectors


Boolean algebras canonically defined
Examples
Power set algebra


Boolean algebras canonically defined
Examples
Representation theorems


Boolean algebras canonically defined
Examples
Other examples


Boolean algebras canonically defined
Boolean algebras of Boolean operations


Boolean algebras canonically defined
Axiomatizing Boolean algebras


Boolean algebras canonically defined
Underlying lattice structure


Boolean algebras canonically defined
Boolean homomorphisms


Boolean algebras canonically defined
Infinitary extensions


Boolean algebras canonically defined
Other definitions of Boolean algebra


Consistency, completeness, and normal forms

Different presentations of natural deduction

Comparison with other foundational approaches


Natural deduction
Judgments and propositions


Natural deduction
Introduction and elimination


Natural deduction
Hypothetical derivations


Consistency, completeness, and normal forms


Natural deduction
First and higher-order extensions


Different presentations of natural deduction

Tree-like presentations
Sequential presentations


Natural deduction
Proofs and type theory


Natural deduction
Proofs and type theory
Example: Dependent Type Theory


Natural deduction
Classical and modal logics


Comparison with other foundational approaches


Example: Hilbert systems for two propositional logics


Rule of inference
Admissibility and derivability


Overview
The axioms
Discussion
Comparison with Hilbert


Fundamental relations
Congruence axioms
Betweenness axioms
Congruence and betweenness


Tarski's axioms
The axioms
Fundamental relations


Tarski's axioms
The axioms
Congruence axioms


Tarski's axioms
The axioms
Congruence axioms
Commentary


Tarski's axioms
The axioms
Betweenness axioms


Tarski's axioms
The axioms
Congruence and betweenness


Axiomatic basis of non-Euclidean geometry


Background
Discovery of non-Euclidean geometry
Terminology


Non-Euclidean geometry
History
Background


Non-Euclidean geometry
History
Discovery of non-Euclidean geometry


Non-Euclidean geometry
History
Terminology


Non-Euclidean geometry
Models
Elliptic geometry


Non-Euclidean geometry
Models
Hyperbolic geometry


Non-Euclidean geometry
Uncommon properties


Non-Euclidean geometry
Kinematic geometries


History
Structural proof theory
Ordinal analysis
Provability logic
Reverse mathematics
Functional interpretations
Formal and informal proof
Proof-theoretic semantics


As a description of the structure of space


Axioms
Parallel postulate
Methods of proof


Euclidean geometry
The Elements
Parallel postulate


Euclidean geometry
The Elements
Methods of proof


Euclidean geometry
Notation and terminology

Naming of points and figures
Complementary and supplementary angles
Modern versions of Euclid's notation


Euclidean geometry
Notation and terminology
Naming of points and figures


Euclidean geometry
Notation and terminology
Complementary and supplementary angles


Euclidean geometry
Notation and terminology
Modern versions of Euclid's notation


Euclidean geometry
Some important or well known results

Pons asinorum
Congruence of triangles
Triangle angle sum
Pythagorean theorem
Thales' theorem
Scaling of area and volume


Euclidean geometry
Some important or well known results
Pons asinorum


Euclidean geometry
Some important or well known results
Congruence of triangles


Euclidean geometry
Some important or well known results
Triangle angle sum


Euclidean geometry
Some important or well known results
Pythagorean theorem


Euclidean geometry
Some important or well known results
Thales' theorem


Euclidean geometry
Some important or well known results
Scaling of area and volume


Euclidean geometry
System of measurement and arithmetic


Archimedes and Apollonius
17th century: Descartes
18th century
19th century
20th century and relativity


Euclidean geometry
Later history
Archimedes and Apollonius


Euclidean geometry
Later history
17th century: Descartes


Euclidean geometry
Later history
18th century


Euclidean geometry
Later history
19th century

Higher dimensions
Non-Euclidean geometry


Euclidean geometry
Later history
19th century
Higher dimensions


Euclidean geometry
Later history
19th century
Non-Euclidean geometry


Euclidean geometry
Later history
20th century and relativity


As a description of the structure of space


Euclidean geometry
Treatment of infinity


Euclidean geometry
Treatment of infinity
Infinite objects


Euclidean geometry
Treatment of infinity
Infinite processes


Classical logic
Modern standards of rigor
Axiomatic formulations


Euclidean geometry
Logical basis
Classical logic


Euclidean geometry
Logical basis
Modern standards of rigor


Euclidean geometry
Logical basis
Axiomatic formulations


Overview
Proving logical formulas
The system LK
Variants


Distinction between natural deduction and sequent calculus


Sequent calculus
Overview
Hilbert-style deduction systems


Sequent calculus
Overview
Natural deduction systems


Sequent calculus
Overview
Sequent calculus systems


Sequent calculus
Overview
Origin of word "sequent"


Sequent calculus
Proving logical formulas

Reduction trees
Relation to standard axiomatizations


Sequent calculus
Proving logical formulas
Reduction trees


Sequent calculus
Proving logical formulas
Relation to standard axiomatizations


Inference rules
An intuitive explanation
Example derivations
Relation to analytic tableaux
Structural rules
Properties of the system LK


Sequent calculus
The system LK
Inference rules


Sequent calculus
The system LK
An intuitive explanation


Sequent calculus
The system LK
Example derivations


Sequent calculus
The system LK
Relation to analytic tableaux


Sequent calculus
The system LK
Structural rules


Sequent calculus
The system LK
Properties of the system LK


Intuitionistic sequent calculus: System LJ


Sequent calculus
Variants
Minor structural alternatives


Sequent calculus
Variants
Substructural logics


Properties
Relative consistency
Models
Axiomatic method


Example: The Peano axiomatization of natural numbers


Axiomatic system
Axiomatic method
History


Axiomatic system
Axiomatic method
Issues


Axiomatic system
Axiomatic method
Axiomatization


The axioms
How these axioms imply a field


Tarski's axiomatization of the reals
The axioms


Tarski's axiomatization of the reals
How these axioms imply a field


Editions and translations of Grundlagen der Geometrie


I. Incidence
II. Order
III. Congruence
IV. Parallels
V. Continuity


Hilbert's axioms
The axioms
I. Incidence


Hilbert's axioms
The axioms
III. Congruence


Hilbert's axioms
The axioms
IV. Parallels


Hilbert's axioms
The axioms
V. Continuity


Hilbert's axioms
Hilbert's discarded axiom


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


Formal deductions
Conservative extensions
Metatheorems
Some useful theorems and their proofs
Alternative axiomatizations
Further connections


Hilbert system
Formal deductions
Logical axioms


Existential quantification
Conjunction and disjunction


Hilbert system
Conservative extensions
Existential quantification


Hilbert system
Conservative extensions
Conjunction and disjunction


Hilbert system
Some useful theorems and their proofs


Hilbert system
Alternative axiomatizations


Classical propositional calculus systems


Classical propositional calculus systems

Implication and negation
Implication and falsum
Negation and disjunction
Sheffer's stroke


List of Hilbert systems
Implicational propositional calculus


List of Hilbert systems
Intuitionistic and intermediate logics


List of Hilbert systems
Positive implicational calculus


List of Hilbert systems
Positive propositional calculus


List of Hilbert systems
Equivalential calculus


History
Definition
Upper bound
Examples
Table of ordinal analyses


Ordinal analysis
Definition
Ordinal notations


Theories with proof-theoretic ordinal ω2

Theories with proof-theoretic ordinal ω3

Theories with proof-theoretic ordinal ωn (for n = 2, 3, ... ω)

Theories with proof-theoretic ordinal ωω

Theories with proof-theoretic ordinal ε0

Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0

Theories with proof-theoretic ordinal the Bachmann–Howard ordinal

Theories with larger proof-theoretic ordinals


Ordinal analysis
Examples
Theories with proof-theoretic ordinal ω


Theories with proof-theoretic ordinal ω2


Theories with proof-theoretic ordinal ω3


Theories with proof-theoretic ordinal ωn (for n = 2, 3, ... ω)


Theories with proof-theoretic ordinal ωω


Theories with proof-theoretic ordinal ε0


Theories with proof-theoretic ordinal the Feferman–Schütte ordinal Γ0


Theories with proof-theoretic ordinal the Bachmann–Howard ordinal


Theories with larger proof-theoretic ordinals


Ordinal analysis
Table of ordinal analyses


Ordinal analysis
Table of ordinal analyses
Key


Fundamental notions of first-order model theory

Connections to related branches of mathematical logic


Fundamental notions of first-order model theory

Compactness and the Löwenheim-Skolem theorem


Compactness and the Löwenheim-Skolem theorem


Definable sets
Eliminating quantifiers
Minimality
Definable and interpretable structures


Model theory
Definability
Definable sets


Model theory
Definability
Eliminating quantifiers


Model theory
Definability
Definable and interpretable structures


Basic notions
Structures and types
Stone spaces


Realising and omitting types
Ultraproducts


Model theory
Constructing models
Realising and omitting types


Model theory
Constructing models
Ultraproducts


Model theory
Categoricity
ω-categoricity


Model theory
Categoricity
Uncountable categoricity


The stability hierarchy
Geometric stability theory


Model theory
Stability theory
The stability hierarchy


Model theory
Stability theory
Geometric stability theory


Model theory
Non-elementary model theory


Connections to related branches of mathematical logic


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


General properties of truth-functional interpretations


Interpretation (logic)
Formal languages
Example


Interpretation (logic)
Formal languages
Logical constants


Interpretation (logic)
Interpretation of a theory


Interpretation (logic)
Interpretations for propositional logic


Interpretation (logic)
First-order logic

Interpretations of a first-order language


Interpretation (logic)
First-order logic
Formal languages for first-order logic


Interpretations of a first-order language


Interpretation (logic)
First-order logic
Example of a first-order interpretation


Interpretation (logic)
First-order logic
Non-empty domain requirement


Interpretation (logic)
First-order logic
Interpreting equality


Interpretation (logic)
First-order logic
Many-sorted first-order logic


Interpretation (logic)
Higher-order predicate logics


Interpretation (logic)
Non-classical interpretations


Interpretation (logic)
Intended interpretations


Interpretation (logic)
Intended interpretations
Example


Interpretation (logic)
Other concepts of interpretation


Axiomatic set theory
Set theory of the real line
Order theory
Abstract algebra
Number theory
Measure theory
Topology
Functional analysis
Model theory
Computability theory
References
External links


List of statements independent of ZFC
Axiomatic set theory


List of statements independent of ZFC
Set theory of the real line


List of statements independent of ZFC
Order theory


List of statements independent of ZFC
Abstract algebra


List of statements independent of ZFC
Number theory


List of statements independent of ZFC
Measure theory


List of statements independent of ZFC
Topology


List of statements independent of ZFC
Functional analysis


List of statements independent of ZFC
Model theory


List of statements independent of ZFC
Computability theory


List of statements independent of ZFC
References


List of statements independent of ZFC
External links


Definition Bi-interpretability Example References


Interpretation (model theory)
Definition


Interpretation (model theory)
Bi-interpretability


Interpretation (model theory)
References


Pythagorean proof of the existence of irrational numbers

Impossible constructions sought by the ancient Greeks

Incompleteness of axiomatic systems: Gödel's proof

Integer solutions of Diophantine equations: Hilbert's tenth problem


Proof of impossibility
Types of proof
By contradiction


Proof of impossibility
Types of proof
By contradiction
By descent


Proof of impossibility
Types of disproof


Pythagorean proof of the existence of irrational numbers


Impossible constructions sought by the ancient Greeks

Angle trisection and doubling the cube
Squaring the circle
Constructing an equilateral n-gon


Proof of impossibility
Euclid's parallel postulate


Proof of impossibility
Fermat's Last Theorem


Proof of impossibility
Richard's paradox


Incompleteness of axiomatic systems: Gödel's proof


Proof of impossibility
Halting problem: Turing's first proof


Proof of impossibility
String compressibility: Chaitin's proof


Integer solutions of Diophantine equations: Hilbert's tenth problem


Proof of impossibility
In social science


Proof of impossibility
In natural science


Axiomatisability
Zero-one laws
Descriptive complexity theory
Applications
History
Citations
References
Further reading
External links


Characterisation of a class of structures


Finite model theory
Axiomatisability
Characterisation of a single structure

Extension to a fixed number of structures


Finite model theory
Axiomatisability
Characterisation of a single structure
Problem


Finite model theory
Axiomatisability
Characterisation of a single structure
Approach


Extension to a fixed number of structures


Finite model theory
Axiomatisability
Characterisation of a single structure
Extension to an infinite structure


Finite model theory
Descriptive complexity theory


Finite model theory
Applications
Database theory


Finite model theory
Applications
Querying and search


Elementary substructures and elementary extensions


Elementary equivalence
Elementarily equivalent structures


Elementary equivalence
Tarski–Vaught test


Elementary equivalence
Elementary embeddings


Definition
Motivation
Examples
Relationship to prime models
Notes
References


Saturated model
Relationship to prime models


History and motivation
Examples
Properties


Categorical theory
History and motivation


Structure of countable non-standard models


Non-standard model of arithmetic
Existence

From the compactness theorem
From the incompleteness theorems
From an ultraproduct


Non-standard model of arithmetic
Existence
From the compactness theorem


Non-standard model of arithmetic
Existence
From the incompleteness theorems

Arithmetic unsoundness for models with ~G true


Arithmetic unsoundness for models with ~G true


Non-standard model of arithmetic
Existence
From an ultraproduct


Non-standard model of arithmetic
References


Non-standard model of arithmetic
References
Citations


Non-standard model of arithmetic
References
Sources


Sufficient condition for completeness of model-complete theories


Model complete theory
Model companion and model completion


Definition
Example
As subobjects
Submodel


Substructure (mathematics)
As subobjects


List of possible spectra of a countable theory


The inductive definition
Natural languages


Definition and etymology
Major theories
Formal theories
Folk beliefs
Notable views


Correspondence Coherence Pragmatic Constructivist Consensus


Truth
Major theories
Substantive
Correspondence


Truth
Major theories
Substantive
Coherence


Truth
Major theories
Substantive
Pragmatic


Truth
Major theories
Substantive
Constructivist


Truth
Major theories
Substantive
Consensus


Deflationary
Performative
Redundancy and related
Philosophical skepticism


Truth
Major theories
Minimalist
Deflationary


Truth
Major theories
Minimalist
Performative


Truth
Major theories
Minimalist
Redundancy and related


Truth
Major theories
Minimalist
Philosophical skepticism


Logic
Mathematics
Tarski's semantics
Kripke's semantics


Truth
Formal theories
Tarski's semantics


Truth
Formal theories
Kripke's semantics


Ancient Greek philosophy
Medieval philosophy
Modern philosophy
Theological views


Truth
Notable views
Ancient Greek philosophy


Changing concepts of truth in the Middle Ages


Truth
Notable views
Medieval philosophy
Avicenna (980–1037)


Truth
Notable views
Medieval philosophy
Aquinas (1225–1274)


Kant (1724–1804)
Hegel (1770–1831)
Schopenhauer (1788–1860)
Kierkegaard (1813–1855)
Nietzsche (1844–1900)
Heidegger (1889–1976)
Sartre (1905–1980)
Camus (1913–1960)
Whitehead (1861–1947)
Peirce (1839–1914)
Nishida (1870–1945)
Fromm (1900–1980)
Foucault (1926–1984)
Baudrillard (1929–2007)
Other contemporary positions


Truth
Notable views
Modern philosophy
Kant (1724–1804)


Truth
Notable views
Modern philosophy
Hegel (1770–1831)


Truth
Notable views
Modern philosophy
Schopenhauer (1788–1860)


Truth
Notable views
Modern philosophy
Kierkegaard (1813–1855)


Truth
Notable views
Modern philosophy
Nietzsche (1844–1900)


Truth
Notable views
Modern philosophy
Heidegger (1889–1976)


Truth
Notable views
Modern philosophy
Sartre (1905–1980)


Truth
Notable views
Modern philosophy
Camus (1913–1960)


Truth
Notable views
Modern philosophy
Whitehead (1861–1947)


Truth
Notable views
Modern philosophy
Peirce (1839–1914)


Truth
Notable views
Modern philosophy
Nishida (1870–1945)


Truth
Notable views
Modern philosophy
Fromm (1900–1980)


Truth
Notable views
Modern philosophy
Foucault (1926–1984)


Truth
Notable views
Modern philosophy
Baudrillard (1929–2007)


Truth
Notable views
Modern philosophy
Other contemporary positions


Truth
Notable views
Theological views
Hinduism


Truth
Notable views
Theological views
Buddhism


Truth
Notable views
Theological views
Christianity


Origin
Tarski's theory of truth
Kripke's theory of truth


Semantic theory of truth
Tarski's theory of truth


Semantic theory of truth
Kripke's theory of truth


The big five subsystems of second-order arithmetic


Use of second-order arithmetic
Use of higher-order arithmetic


Reverse mathematics
General principles
Use of second-order arithmetic


Reverse mathematics
General principles
Use of higher-order arithmetic


The big five subsystems of second-order arithmetic

The base system RCA0
Weak Kőnig's lemma WKL0
Arithmetical comprehension ACA0
Arithmetical transfinite recursion ATR0
Π11 comprehension Π11-CA0


Reverse mathematics
Additional systems
Stronger Systems


Reverse mathematics
ω-models and β-models


Propositional satisfiability for classical logic


Satisfiability
Reduction of validity to satisfiability


Satisfiability
Satisfiability in first-order logic


Satisfiability
Satisfiability in model theory


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


Generalizations of the concept of number


Transfer principle
Transfer principle for the hyperreals


Transfer principle
Transfer principle for the hyperreals
Example


Transfer principle
Differences between R and *R


Transfer principle
Constructions of the hyperreals


Truth predicate
Languages which allow a truth predicate


Formal definition
Examples of types
Stone spaces
Omitting types theorem
References


Type (model theory)
Omitting types theorem


Computing
Classical logic
Intuitionistic and constructive logic
Multi-valued logic
Algebraic semantics
In other theories


Truth value
Intuitionistic and constructive logic


Direct limits of ultrapowers (ultralimits)


Relationships between definability, proof and computability


Computability theory
Turing computability


Relative computability and the Turing degrees

Rice's theorem and the arithmetical hierarchy

The lattice of computably enumerable sets


Relative computability and the Turing degrees


Computability theory
Areas of research
Other reducibilities


Rice's theorem and the arithmetical hierarchy


Computability theory
Areas of research
Reverse mathematics


Computability theory
Areas of research
Numberings


Computability theory
Areas of research
The priority method


The lattice of computably enumerable sets


Computability theory
Areas of research
Automorphism problems


Computability theory
Areas of research
Kolmogorov complexity


Computability theory
Areas of research
Frequency computation


Computability theory
Areas of research
Inductive inference


Computability theory
Areas of research
Generalizations of Turing computability


Computability theory
Areas of research
Continuous computability theory


Computability theory
Professional organizations


Use
Church numerals
Church Booleans
Predicates
Church pairs
List encodings


Calculation with Church numerals
Table of functions on Church numerals
Derivation of predecessor function
Division
Signed numbers
Plus and minus
Multiply and  divide
Rational and real numbers
Translation with other representations


Church encoding
Church numerals
Calculation with Church numerals


Church encoding
Church numerals
Table of functions on Church numerals


Church encoding
Church numerals
Derivation of predecessor function


Church encoding
Church numerals
Derivation of predecessor function
Another way of defining pred


Church encoding
Church numerals
Division


Church encoding
Church numerals
Signed numbers


Church encoding
Church numerals
Plus and minus


Church encoding
Church numerals
Multiply and  divide


Church encoding
Church numerals
Rational and real numbers


Church encoding
Church numerals
Translation with other representations


Two pairs as a list node
One pair as a list node
Represent the list using right fold
Represent the list using Scott encoding


Church encoding
List encodings
Two pairs as a list node


Church encoding
List encodings
One pair as a list node


Church encoding
List encodings
Represent the list using right fold


Church encoding
List encodings
Represent the list using Scott encoding


Overview
Natural language
Properties
Order of precedence
Computer science
Table and Hasse diagram


List of common logical connectives
History of notations
Redundancy


Logical connective
Overview
List of common logical connectives


Logical connective
Overview
History of notations


Logical connective
Table and Hasse diagram


Early history
Modern history
Basic structure
Types
Terms in syllogism
Existential import
Syllogistic fallacies
Other types of syllogism


Syllogism
Early history
Medieval syllogism


Syllogism
Early history
Medieval syllogism
Boethius


Syllogism
Early history
Medieval syllogism
Peter Abelard


Syllogism
Early history
Medieval syllogism
Jean Buridan


Syllogism
Modern history
Boole's acceptance of Aristotle


Barbara (AAA-1)
Celarent (EAE-1)
Darii (AII-1)
Ferio (EIO-1)
Baroco (AOO-2)
Bocardo (OAO-3)
Barbari (AAI-1)
Celaront (EAO-1)
Camestros (AEO-2)
Felapton (EAO-3)
Darapti (AAI-3)


Syllogism
Types
Examples
Barbara (AAA-1)


Syllogism
Types
Examples
Celarent (EAE-1)


Syllogism
Types
Examples
Bocardo (OAO-3)


Syllogism
Types
Examples
Barbari (AAI-1)


Syllogism
Types
Examples
Celaront (EAO-1)


Syllogism
Types
Examples
Camestros (AEO-2)


Syllogism
Types
Examples
Felapton (EAO-3)


Syllogism
Types
Examples
Darapti (AAI-3)


Functional completeness of many-valued logics


Kleene (strong) K3 and Priest logic P3
Bochvar's internal three-valued logic
Belnap logic (B4)
Gödel logics Gk and G∞
Łukasiewicz logics Lv and L∞
Product logic Π
Post logics Pm
Rose logics


Many-valued logic
Examples
Kleene (strong) K3 and Priest logic P3


Many-valued logic
Examples
Bochvar's internal three-valued logic


Many-valued logic
Examples
Belnap logic (B4)


Many-valued logic
Examples
Gödel logics Gk and G∞


Many-valued logic
Examples
Łukasiewicz logics Lv and L∞


Many-valued logic
Examples
Product logic Π


Many-valued logic
Examples
Post logics Pm


Many-valued logic
Relation to classical logic


Many-valued logic
Relation to classical logic
Suszko's thesis


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


Arguments
Valid formula
Statements
Soundness
Satisfiability
Preservation


Pre-discovery
Representation of values
Logics
Applications


Three-valued logic
Representation of values


Kleene and Priest logics
Łukasiewicz logic
HT logic
Bochvar logic
Ternary Post logic
Modular algebras


Three-valued logic
Logics
Kleene and Priest logics


Three-valued logic
Logics
Łukasiewicz logic


Three-valued logic
Logics
Ternary Post logic


Three-valued logic
Logics
Modular algebras


Standard logical account of argument types

Defeasible arguments and argumentation schemes


Standard logical account of argument types


Defeasible arguments and argumentation schemes


Argument
Elliptical or ethymematic arguments


Examples
Representation
Analysis
Real polynomial form
Applications


Properties
Derived functions
Cryptographic analysis


Boolean function
Analysis
Derived functions


Boolean function
Analysis
Cryptographic analysis


Boolean function
Analysis
Cryptographic analysis
Linear approximation table


On the unit hypercube
On the symmetric hypercube


Boolean function
Real polynomial form
On the unit hypercube


Boolean function
Real polynomial form
On the symmetric hypercube


Logical systems
Relation to completeness


Soundness
Use in mathematical logic
Logical systems

Soundness
Strong soundness
Arithmetic soundness


Soundness
Use in mathematical logic
Logical systems
Soundness


Soundness
Use in mathematical logic
Logical systems
Strong soundness


Soundness
Use in mathematical logic
Logical systems
Arithmetic soundness


Soundness
Use in mathematical logic
Relation to completeness


History
Values
Operations
Laws
Diagrammatic representations
Boolean algebras
Axiomatizing Boolean algebra
Propositional logic
Applications


Boolean algebra
Operations
Basic operations


Boolean algebra
Operations
Secondary operations


Monotone laws
Nonmonotone laws
Completeness
Duality principle


Boolean algebra
Diagrammatic representations


Boolean algebra
Diagrammatic representations
Venn diagrams


Boolean algebra
Diagrammatic representations
Digital logic gates


Concrete Boolean algebras
Subsets as bit vectors
The prototypical Boolean algebra
Boolean algebras: the definition
Representable Boolean algebras


Boolean algebra
Boolean algebras
Concrete Boolean algebras


Boolean algebra
Boolean algebras
Subsets as bit vectors


Boolean algebra
Boolean algebras
The prototypical Boolean algebra


Boolean algebra
Boolean algebras
Boolean algebras: the definition


Boolean algebra
Boolean algebras
Representable Boolean algebras


Boolean algebra
Axiomatizing Boolean algebra


Deductive systems for propositional logic


Boolean algebra
Propositional logic
Applications


Computers
Two-valued logic
Boolean operations


Boolean algebra
Applications
Two-valued logic


Boolean algebra
Applications
Boolean operations

Natural language
Digital logic
Naive set theory
Video cards
Modeling and CAD
Boolean searches


Boolean algebra
Applications
Boolean operations
Natural language


Boolean algebra
Applications
Boolean operations
Digital logic


Boolean algebra
Applications
Boolean operations
Naive set theory


Boolean algebra
Applications
Boolean operations
Video cards


Boolean algebra
Applications
Boolean operations
Modeling and CAD


Boolean algebra
Applications
Boolean operations
Boolean searches


Unary operations
Binary operations
Size of truth tables
Applications
History


Logical true
Logical false
Logical identity
Logical negation


Truth table
Unary operations
Logical true


Truth table
Unary operations
Logical false


Truth table
Unary operations
Logical identity


Truth table
Unary operations
Logical negation


Truth table for all binary logical operators


Truth table
Binary operations
Logical conjunction (AND)


Truth table
Binary operations
Logical disjunction (OR)


Truth table
Binary operations
Logical implication


Truth table
Binary operations
Logical equality


Truth table
Binary operations
Exclusive disjunction


Truth table
Binary operations
Logical NAND


Truth table
Binary operations
Logical NOR


Truth table for most commonly used logical operators

Condensed truth tables for binary operators

Applications of truth tables in digital electronics


Truth table for most commonly used logical operators


Condensed truth tables for binary operators


Truth table
Applications
Truth tables in digital logic


Applications of truth tables in digital electronics


Reasons for nonexistence
Theories of universality


Regularity
Russell's paradox
Cantor's theorem


Universal set
Reasons for nonexistence
Regularity


Universal set
Reasons for nonexistence
Russell's paradox


Universal set
Reasons for nonexistence
Cantor's theorem


Restricted comprehension
Universal objects that are not sets


Universal set
Theories of universality
Restricted comprehension


Universal set
Theories of universality
Universal objects that are not sets


Predicate (mathematical logic)
Predicates in different systems


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


Preliminaries
Pure identity theories
Unary relations
Equivalence relations
Orders
Lattices
Graphs
Boolean algebras
Groups
Rings and fields
Geometry
Differential algebra
Addition
Arithmetic
Second order arithmetic
Set theories


List of first-order theories
Preliminaries


List of first-order theories
Pure identity theories


List of first-order theories
Unary relations


List of first-order theories
Equivalence relations


List of first-order theories
Boolean algebras


List of first-order theories
Rings and fields


List of first-order theories
Differential algebra


List of first-order theories
Second order arithmetic


List of first-order theories
Set theories


Examples
Syntax and fragments
Semantics
Expressive power
Deductive systems
Non-reducibility to first-order logic
Metalogical results
History and disputed value
Relation to computational complexity


Second-order logic
Non-reducibility to first-order logic


Second-order logic
History and disputed value


Second-order logic
Relation to computational complexity


Quantification scope
Semantics
Properties


Intuition
Forcing conditions and forcing posets
Generic filters
P-names and interpretations
Forcing
Consistency
Cohen forcing
The countable chain condition
Easton forcing
Random reals
Boolean-valued models
Meta-mathematical explanation
Logical explanation


Forcing (mathematics)
Intuition
The role of the model


Forcing (mathematics)
Forcing conditions and forcing posets


Forcing (mathematics)
Forcing conditions and forcing posets
Examples


Forcing (mathematics)
P-names and interpretations


Forcing (mathematics)
P-names and interpretations
Rigorous definitions


Forcing (mathematics)
Forcing
Internal definition


Forcing (mathematics)
The countable chain condition


Forcing (mathematics)
Boolean-valued models


Forcing (mathematics)
Meta-mathematical explanation


Forcing (mathematics)
Logical explanation


An algebra of propositions, the propositional calculus


Relationship between propositional and predicate formulas


Propositional formula
Propositions
Identity


An algebra of propositions, the propositional calculus

Truth-value assignments, formula evaluations


Truth-value assignments, formula evaluations


Propositional formula
Propositional connectives

Connectives of rhetoric, philosophy and mathematics

CASE connective: IF ... THEN ... ELSE ...


Connectives of rhetoric, philosophy and mathematics


Propositional formula
Propositional connectives
Engineering connectives


CASE connective: IF ... THEN ... ELSE ...


Propositional formula
Propositional connectives
IDENTITY and evaluation


Propositional formula
More complex formulas

Definitions
Axiom and definition schemas
Substitution versus replacement


Propositional formula
More complex formulas
Definitions


Propositional formula
More complex formulas
Axiom and definition schemas


Propositional formula
More complex formulas
Substitution versus replacement


Propositional formula
Inductive definition


Laws of evaluation: Identity, nullity, and complement


Propositional formula
Parsing formulas
Connective seniority (symbol rank)


Propositional formula
Parsing formulas
Commutative and associative laws


Propositional formula
Parsing formulas
Distributive laws


Propositional formula
Parsing formulas
De Morgan's laws


Propositional formula
Parsing formulas
Laws of absorption


Propositional formula
Parsing formulas
Double negative (involution)


Propositional formula
Well-formed formulas (wffs)

Well-formed formulas versus valid formulas in inferences


Well-formed formulas versus valid formulas in inferences


Propositional formula
Reduced sets of connectives


Propositional formula
Reduced sets of connectives
The stroke (NAND)


Propositional formula
Reduced sets of connectives
IF ... THEN ... ELSE


Reduction by use of the map method (Veitch, Karnaugh)


Propositional formula
Normal forms
Reduction to normal form


Propositional formula
Normal forms
Reduction to normal form
Literal, term and alterm


Propositional formula
Normal forms
Reduction to normal form
Minterms


Reduction by use of the map method (Veitch, Karnaugh)

Produce the formula's truth table
Create the formula's Karnaugh map
Reduce minterms
Verify reduction with a truth table


Propositional formula
Impredicative propositions


Propositional formula
Propositional formula with "feedback"


Propositional formula
Propositional formula with "feedback"
Oscillation


Propositional formula
Propositional formula with "feedback"
Memory

Once-flip memory
Flip-flop memory
Clocked flip-flop memory


Propositional formula
Propositional formula with "feedback"
Memory
Once-flip memory


Propositional formula
Propositional formula with "feedback"
Memory
Flip-flop memory


Propositional formula
Propositional formula with "feedback"
Memory
Clocked flip-flop memory


Propositional formula
Historical development


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


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


Relations to logical conjunction and disjunction

Paucal, multal and other degree quantifiers


Relations to logical conjunction and disjunction


Quantifier (logic)
Algebraic approaches to quantification


Quantifier (logic)
Order of quantifiers (nesting)


Quantifier (logic)
Equivalent expressions


Quantifier (logic)
Equivalent expressions
Quantifier elimination


Quantifier (logic)
Range of quantification


Paucal, multal and other degree quantifiers


Ordinals extend the natural numbers
Definitions
Transfinite sequence
Transfinite induction
Arithmetic of ordinals
Ordinals and cardinals
Some "large" countable ordinals
Topology and ordinals
History


Ordinal number
Ordinals extend the natural numbers


Definition of an ordinal as an equivalence class


Ordinal number
Definitions
Well-ordered sets


Ordinal number
Definitions
Von Neumann definition of ordinals


Ordinal number
Definitions
Other definitions


Transfinite recursion
Successor and limit ordinals
Indexing classes of ordinals
Closed unbounded sets and classes


Ordinal number
Transfinite induction
Transfinite recursion


Ordinal number
Transfinite induction
Successor and limit ordinals


Ordinal number
Transfinite induction
Indexing classes of ordinals


Ordinal number
Transfinite induction
Closed unbounded sets and classes


Initial ordinal of a cardinal
Cofinality


Ordinal number
Ordinals and cardinals
Initial ordinal of a cardinal


Ordinal number
Ordinals and cardinals
Cofinality


Ordinal number
Some "large" countable ordinals


Examples
In formulations of set theory
Assumptions


Hereditary set
In formulations of set theory


Decidability and complexity of satisfiability


Monadic second-order logic
Computational complexity of evaluation


Decidability and complexity of satisfiability

Use of satisfiability of MSO on trees in verification


Use of satisfiability of MSO on trees in verification


Expressiveness
Relationship with term logic
Variants
Footnotes


Monadic predicate calculus
Expressiveness


Monadic predicate calculus
Relationship with term logic


Well-definedness under an equivalence relation

Fundamental theorem of equivalence relations

Equivalence relations and mathematical logic


Alternative definition using relational algebra


Simple example
Equivalence relations
Relations that are not equivalences


Equivalence relation
Examples
Simple example


Equivalence relation
Examples
Equivalence relations


Equivalence relation
Examples
Relations that are not equivalences


Equivalence relation
Connections to other relations


Well-definedness under an equivalence relation


Equivalence relation
Related important definitions

Equivalence class
Quotient set
Projection
Equivalence kernel
Partition


Equivalence relation
Related important definitions
Equivalence class


Equivalence relation
Related important definitions
Quotient set


Equivalence relation
Related important definitions
Projection


Equivalence relation
Related important definitions
Equivalence kernel


Equivalence relation
Related important definitions
Partition


Equivalence relation
Related important definitions
Partition
Counting partitions


Fundamental theorem of equivalence relations


Equivalence relation
Comparing equivalence relations


Equivalence relation
Generating equivalence relations


Equivalence relation
Algebraic structure

Group theory
Categories and groupoids
Lattices


Equivalence relation
Algebraic structure
Group theory


Equivalence relation
Algebraic structure
Categories and groupoids


Equivalence relation
Algebraic structure
Lattices


Equivalence relations and mathematical logic


Theory
Urelements in set theory
Quine atoms
References
External links


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)


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


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