Double negation
Distributivity
Linearity
Self dual
Negations of quantifiers


Negation
Properties
Negations of quantifiers


Programming language and ordinary 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


Ground expression
Formal definitions
Ground term


Ground expression
Formal definitions
Ground atom


Ground expression
Formal definitions
Ground formula


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


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


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


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


Introducing new function symbols
Doing without functional predicates


Functional predicate
Introducing new function symbols


Functional predicate
Doing without functional predicates


Equivalence to Turing machines
Other properties


Spectrum of a sentence
Properties
Equivalence to Turing machines


Spectrum of a sentence
Properties
Other properties


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
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


Signatures
Models
Informal semantics
Descriptive signs


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


Overview
Can words be modeled as formal symbols?
References


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


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


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


Propositional logic
First-order logic
Algebra


Substitution (logic)
Propositional logic


Substitution (logic)
Propositional logic
Definition


Substitution (logic)
Propositional logic
Tautologies


Minimal axioms for Boolean algebra
References


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 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


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


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


Expressive power
Idea of decidability
Complexity
Decidable extensions
Undecidable extensions


Skolem arithmetic
Undecidable extensions


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


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


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


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


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


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


Definition
Friedman's grand conjecture
Related systems


Elementary function arithmetic
Definition


Elementary function arithmetic
Friedman's grand conjecture


Elementary function arithmetic
Related systems


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


Robinson arithmetic
Axioms
Variant axiomatizations


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


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


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


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


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


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


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


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


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


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


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


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


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


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


Primitive recursive arithmetic
Language and axioms


Primitive recursive arithmetic
Logic-free calculus


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


List of possible spectra of a countable theory


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


Definition
Example
As subobjects
Submodel


Substructure (mathematics)
As subobjects


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


Sufficient condition for completeness of model-complete theories


Model complete theory
Model companion and model completion


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


Propositional satisfiability for classical logic


Satisfiability
Reduction of validity to satisfiability


Satisfiability
Satisfiability in first-order logic


Satisfiability
Satisfiability in model theory


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


Truth value
Intuitionistic and constructive logic


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


The inductive definition
Natural languages


Formal definition
Examples of types
Stone spaces
Omitting types theorem
References


Type (model theory)
Omitting types theorem


Problems in NP not known to be in P or NP-complete


P versus NP problem
Reasons to believe P ≠ NP or P = NP


P versus NP problem
Consequences of solution


P versus NP problem
Consequences of solution
P = NP


P versus NP problem
Consequences of solution
P ≠ NP


P versus NP problem
Results about difficulty of proof


P versus NP problem
Logical characterizations


P versus NP problem
Polynomial-time algorithms


P versus NP problem
Formal definitions
P and NP


P versus NP problem
Formal definitions
P and NP
Example


P versus NP problem
Formal definitions
NP-completeness


Definition
Invariance theorem
History and context
Basic results
Compression
Chaitin's incompleteness theorem
Minimum message length
Kolmogorov randomness
Relation to entropy
Conditional versions


Kolmogorov complexity
Invariance theorem

Informal treatment
A more formal treatment


Kolmogorov complexity
Invariance theorem
Informal treatment


Kolmogorov complexity
Invariance theorem
A more formal treatment


Kolmogorov complexity
History and context


Uncomputability of Kolmogorov complexity


Uncomputability of Kolmogorov complexity

A naive attempt at a program to compute K


A naive attempt at a program to compute K


Kolmogorov complexity
Basic results
Chain rule for Kolmogorov complexity


Kolmogorov complexity
Chaitin's incompleteness theorem


Kolmogorov complexity
Minimum message length


Kolmogorov complexity
Kolmogorov randomness


Kolmogorov complexity
Relation to entropy


Kolmogorov complexity
Conditional versions


Formal definition
Equivalent formulations
Examples
Properties
Remarks


Computably enumerable set
Formal definition


Computably enumerable set
Equivalent formulations


Formal definition
Background
Properties
Why some NP problems are hard to solve
Equivalence of definitions
Relationship to other classes
Other characterizations
Examples


NP (complexity)
Background
Verifier-based definition


NP (complexity)
Background
Machine-definition


NP (complexity)
Why some NP problems are hard to solve


NP (complexity)
Equivalence of definitions


NP (complexity)
Relationship to other classes


P
Integer factorization
NP-complete problems


NP (complexity)
Examples
Integer factorization


NP (complexity)
Examples
NP-complete problems

Boolean satisfiability
Travelling salesman
Subgraph isomorphism


NP (complexity)
Examples
NP-complete problems
Boolean satisfiability


NP (complexity)
Examples
NP-complete problems
Travelling salesman


NP (complexity)
Examples
NP-complete problems
Subgraph isomorphism


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


Direct limits of ultrapowers (ultralimits)


Uncomputable functions and unsolvable problems


Computable function
Characteristics of computable functions


Computable function
Computable sets and relations


Computable function
Church–Turing thesis


Relation to recursively defined functions

Total functions that are not provably total


Relation to recursively defined functions


Total functions that are not provably total


Computable function
Extensions of computability

Relative computability
Higher recursion theory
Hyper-computation


Computable function
Extensions of computability
Relative computability


Computable function
Extensions of computability
Higher recursion theory


Computable function
Extensions of computability
Hyper-computation


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


Statement in Church's and Turing's words


Circa 1930–1952
Later developments
The thesis as a definition


Church–Turing thesis
History
Circa 1930–1952


Church–Turing thesis
History
Later developments


Church–Turing thesis
History
The thesis as a definition


Church–Turing thesis
Success of the thesis


Church–Turing thesis
Informal usage in proofs


Church–Turing thesis
Philosophical implications


Church–Turing thesis
Non-computable functions


Decidability of a logical system
Decidability of a theory
Some decidable theories
Some undecidable theories
Semidecidability
Relationship with completeness
Relationship to computability
In context of games


Decidability (logic)
Decidability of a logical system


Decidability (logic)
Decidability of a theory


Decidability (logic)
Some decidable theories


Decidability (logic)
Some undecidable theories


Decidability (logic)
Relationship with completeness


Decidability (logic)
Relationship to computability


Decidability (logic)
In context of games


Example: the halting problem in computability theory

Relationship with Gödel's incompleteness theorem


Example: the halting problem in computability theory


Relationship with Gödel's incompleteness theorem


Undecidable problem
Examples of undecidable problems


Undecidable problem
Examples of undecidable statements


Definition
Examples
Use in first-order Peano arithmetic
Elimination of parameters
Relationship to recursive functions
Limitations
Variants
Finitism and consistency results
History


Converting predicates to numeric functions

Operations on integers and rational numbers

Some common primitive recursive functions


Primitive recursive function
Examples
Addition


Primitive recursive function
Examples
Doubling


Primitive recursive function
Examples
Multiplication


Primitive recursive function
Examples
Predecessor


Primitive recursive function
Examples
Truncated subtraction


Converting predicates to numeric functions


Primitive recursive function
Examples
Predicate "Is zero"


Primitive recursive function
Examples
Predicate "Less or equal"


Primitive recursive function
Examples
Predicate "Greater or equal"


Primitive recursive function
Examples
If-then-else


Primitive recursive function
Examples
Junctors


Primitive recursive function
Examples
Equality predicate


Primitive recursive function
Examples
Other operations on natural numbers


Operations on integers and rational numbers


Some common primitive recursive functions


Primitive recursive function
Use in first-order Peano arithmetic


Primitive recursive function
Elimination of parameters


Primitive recursive function
Relationship to recursive functions


Primitive recursive function
Limitations


Constant functions
Weak primitive recursion
Iterative functions
Additional primitive recursive forms
Computer language definition


Primitive recursive function
Variants
Constant functions


Primitive recursive function
Variants
Weak primitive recursion


Primitive recursive function
Variants
Iterative functions


Primitive recursive function
Variants
Additional primitive recursive forms


Primitive recursive function
Variants
Computer language definition


Primitive recursive function
Finitism and consistency results


Lambda calculus and programming languages


Lambda calculus
Explanation and applications


Lambda calculus
History
Origin of the λ symbol


Lambda calculus
Informal description
Motivation


Lambda calculus
Informal description
The lambda calculus

Lambda terms
Functions that operate on functions
Alpha equivalence
Free variables
Capture-avoiding substitutions
β-reduction


Lambda calculus
Informal description
The lambda calculus
Lambda terms


Lambda calculus
Informal description
The lambda calculus
Functions that operate on functions


Lambda calculus
Informal description
The lambda calculus
Alpha equivalence


Lambda calculus
Informal description
The lambda calculus
Free variables


Lambda calculus
Informal description
The lambda calculus
Capture-avoiding substitutions


Lambda calculus
Informal description
The lambda calculus
β-reduction


Definition
Notation
Free and bound variables


Lambda calculus
Formal definition
Definition


Lambda calculus
Formal definition
Notation


Lambda calculus
Formal definition
Free and bound variables


Lambda calculus
Reduction
α-conversion
Substitution


Lambda calculus
Normal forms and confluence


Arithmetic in lambda calculus
Logic and predicates
Pairs


Lambda calculus
Encoding datatypes
Arithmetic in lambda calculus


Lambda calculus
Encoding datatypes
Logic and predicates


Lambda calculus
Encoding datatypes
Pairs


Lambda calculus
Additional programming techniques

Named constants
Recursion and fixed points
Standard terms
Abstraction elimination


Lambda calculus
Additional programming techniques
Named constants


Lambda calculus
Additional programming techniques
Recursion and fixed points


Lambda calculus
Additional programming techniques
Standard terms


Lambda calculus
Additional programming techniques
Abstraction elimination


Lambda calculus and programming languages

Anonymous functions
Parallelism and concurrency


Lambda calculus
Variations and extensions


Formal definitions
Informal definition
In language
In mathematics
In computer science
In biology
In business
In art
In culture


Recursively defined sets
Finite subdivision rules
Functional recursion
Proofs involving recursive definitions
Recursive optimization
The recursion theorem


Recursion
In mathematics
Recursively defined sets

Example: the natural numbers
Example: Proof procedure


Recursion
In mathematics
Recursively defined sets
Example: the natural numbers


Recursion
In mathematics
Recursively defined sets
Example: Proof procedure


Recursion
In mathematics
Finite subdivision rules


Recursion
In mathematics
Functional recursion


Recursion
In mathematics
Proofs involving recursive definitions


Recursion
In mathematics
Recursive optimization


Recursion
In mathematics
The recursion theorem


Recursion
In mathematics
The recursion theorem
Proof of uniqueness


Additional details required to visualize or implement Turing machines


Turing machine
Overview
Physical description


Additional details required to visualize or implement Turing machines

Alternative definitions
The "state"
"State" diagrams


Turing machine
Choice c-machines, oracle o-machines


Turing machine
Universal Turing machines


Turing machine
Comparison with real machines


Turing machine
Comparison with real machines
Limitations

Computational complexity theory
Interaction


Turing machine
Comparison with real machines
Limitations
Computational complexity theory


Turing machine
Comparison with real machines
Limitations
Interaction


Historical background: computational machinery

The Entscheidungsproblem (the "decision problem"): Hilbert's tenth question of 1900

1937–1970: The "digital computer", the birth of "computer science"


Historical background: computational machinery


The Entscheidungsproblem (the "decision problem"): Hilbert's tenth question of 1900


Turing machine
History
Alan Turing's a-machine


1937–1970: The "digital computer", the birth of "computer science"


Turing machine
History
1970–present: as a model of computation


Interpretability of arithmetic
References


Axiom of adjunction
Interpretability of arithmetic


Timeline of mathematical logic
19th century


Timeline of mathematical logic
20th century


Timeline of mathematical logic
20th century
1950-1999


Definition
Small and large categories
Examples
Construction of new categories
Types of morphisms
Types of categories


Category (mathematics)
Small and large categories


Category (mathematics)
Construction of new categories


Category (mathematics)
Construction of new categories
Dual category


Category (mathematics)
Construction of new categories
Product categories


Category (mathematics)
Types of morphisms


Category (mathematics)
Types of categories


Zeno
Thomson
Benacerraf
Modern literature


Supertask
History
Zeno
Achilles and the tortoise


Philosophy of mathematics
Physical possibility
Super Turing machines


Supertask
History
Modern literature
Philosophy of mathematics


Supertask
History
Modern literature
Physical possibility


Supertask
History
Modern literature
Super Turing machines


Ross–Littlewood paradox
Benardete's paradox
Grim Reaper paradox
Davies' super-machine


Supertask
Prominent supertasks
Ross–Littlewood paradox


Supertask
Prominent supertasks
Benardete's paradox


Supertask
Prominent supertasks
Grim Reaper paradox


Supertask
Prominent supertasks
Davies' super-machine


Logic in the East
Logic in the West
Medieval logic
Traditional logic
Rise of modern logic
Modern logic


Logic in India
Hindu logic
Origin
Before Gautama
Dattatreya
Medhatithi Gautama
Panini
Nyaya-Vaisheshika
Jain Logic
Buddhist logic
Nagarjuna
Dignaga
Syllogism and influence
Logic in China


History of logic
Logic in the East
Logic in India


History of logic
Logic in the East
Hindu logic


History of logic
Logic in the East
Origin


History of logic
Logic in the East
Before Gautama


History of logic
Logic in the East
Dattatreya


History of logic
Logic in the East
Medhatithi Gautama


History of logic
Logic in the East
Panini


History of logic
Logic in the East
Nyaya-Vaisheshika


History of logic
Logic in the East
Jain Logic


History of logic
Logic in the East
Buddhist logic


History of logic
Logic in the East
Nagarjuna


History of logic
Logic in the East
Dignaga


History of logic
Logic in the East
Syllogism and influence


History of logic
Logic in the East
Logic in China


Prehistory of logic
Ancient Greece before Aristotle
Aristotle
Stoics


History of logic
Logic in the West
Prehistory of logic


History of logic
Logic in the West
Ancient Greece before Aristotle

Thales
Pythagoras
Heraclitus and Parmenides
Plato


History of logic
Logic in the West
Ancient Greece before Aristotle
Thales


History of logic
Logic in the West
Ancient Greece before Aristotle
Pythagoras


History of logic
Logic in the West
Ancient Greece before Aristotle
Heraclitus and Parmenides


History of logic
Logic in the West
Ancient Greece before Aristotle
Plato


History of logic
Logic in the West
Aristotle


History of logic
Logic in the West
Aristotle
The Organon


History of logic
Logic in the West
Stoics


Logic in the Middle East
Logic in medieval Europe


History of logic
Medieval logic
Logic in the Middle East


History of logic
Medieval logic
Logic in medieval Europe


The textbook tradition
Logic in Hegel's philosophy
Logic and psychology


History of logic
Traditional logic
The textbook tradition


History of logic
Traditional logic
Logic in Hegel's philosophy


History of logic
Traditional logic
Logic and psychology


Embryonic period
Algebraic period
Logicist period
Metamathematical period
Logic after WWII


History of logic
Modern logic
Embryonic period


History of logic
Modern logic
Algebraic period


History of logic
Modern logic
Logicist period


History of logic
Modern logic
Metamathematical period


History of logic
Modern logic
Logic after WWII


History
Major themes
Contemporary schools of thought
Arguments
Aesthetics
Journals


Philosophy of mathematics
History
Contemporary philosophy


Mathematical realism
Mathematical anti-realism


Philosophy of mathematics
Major themes
Mathematical realism


Philosophy of mathematics
Major themes
Mathematical anti-realism


Philosophy of mathematics
Contemporary schools of thought

Artistic
Platonism
Mathematicism
Logicism
Formalism
Conventionalism
Intuitionism
Structuralism
Embodied mind theories
Fictionalism
Social constructivism
Beyond the traditional schools


Philosophy of mathematics
Contemporary schools of thought
Artistic


Philosophy of mathematics
Contemporary schools of thought
Platonism


Philosophy of mathematics
Contemporary schools of thought
Mathematicism


Philosophy of mathematics
Contemporary schools of thought
Logicism


Philosophy of mathematics
Contemporary schools of thought
Formalism


Philosophy of mathematics
Contemporary schools of thought
Conventionalism


Philosophy of mathematics
Contemporary schools of thought
Intuitionism


Philosophy of mathematics
Contemporary schools of thought
Intuitionism
Constructivism


Philosophy of mathematics
Contemporary schools of thought
Intuitionism
Finitism


Philosophy of mathematics
Contemporary schools of thought
Structuralism


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories

Aristotelian realism
Psychologism
Empiricism


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories
Aristotelian realism


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories
Psychologism


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories
Empiricism


Philosophy of mathematics
Contemporary schools of thought
Fictionalism


Philosophy of mathematics
Contemporary schools of thought
Social constructivism


Philosophy of mathematics
Contemporary schools of thought
Beyond the traditional schools

Popper's two senses of number statements


Philosophy of mathematics
Contemporary schools of thought
Beyond the traditional schools
Unreasonable effectiveness


Popper's two senses of number statements


Philosophy of mathematics
Contemporary schools of thought
Beyond the traditional schools
Philosophy of language


Indispensability argument for realism
Epistemic argument against realism


Philosophy of mathematics
Arguments
Indispensability argument for realism


Philosophy of mathematics
Arguments
Epistemic argument against realism


Implicit structure of concrete categories


Categories, objects, and morphisms
Functors
Natural transformations
Other concepts
Historical notes


Category theory
Categories, objects, and morphisms


Category theory
Categories, objects, and morphisms
Categories


Category theory
Categories, objects, and morphisms
Morphisms


Universal constructions, limits, and colimits


Category theory
Other concepts
Equivalent categories


Category theory
Other concepts
Further concepts and results


Category theory
Other concepts
Higher-dimensional categories


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


An example of a logicist construction of the natural numbers: Russell's construction in the Principia

The unit class, impredicativity, and the vicious circle principle


Logicism
Intent, or goal, of logicism
History


Logicism
Epistemology, ontology and logicism


An example of a logicist construction of the natural numbers: Russell's construction in the Principia

Preliminaries
The definition of the natural numbers
Criticism


The unit class, impredicativity, and the vicious circle principle

A solution to impredicativity: a hierarchy of types


A solution to impredicativity: a hierarchy of types


Definition
Examples
Decidability
Complete problems
Function problems
Optimization problems


Pure existence proofs of polynomial-time algorithms


P (complexity)
Relationships to other classes


P (complexity)
Alternative characterizations


Formal statement
Use
Equivalent statements
Relation with other axioms
Notes
References


Axiom of dependent choice
Formal statement


Axiom of dependent choice
Equivalent statements


Axiom of dependent choice
Relation with other axioms


Formal definition
Examples and non-examples
Properties


Computable set
Examples and non-examples


Extracting the natural numbers from the infinite set


Axiom of infinity
Interpretation and consequences


Axiom of infinity
An apparently weaker version


Formal statement
Interpretation
In predicate logic without equality
In set theory with ur-elements


Axiom of extensionality
Formal statement


Axiom of extensionality
In predicate logic without equality


Axiom of extensionality
In set theory with ur-elements


Formal statement
Consequences
Alternatives
References


Axiom of pairing
Alternatives
Non-independence


Formal statement
Relation to Pairing
Relation to Replacement
Relation to Separation
Relation to Intersection
References
Further reading


Zermelo's models and the axiom of limitation of size


Axiom of limitation of size
Formal statement


Axiom of limitation of size
Implications of the axiom


Zermelo's models and the axiom of limitation of size

The models Vκ where κ is a strongly inaccessible cardinal


The models Vκ where κ is a strongly inaccessible cardinal


Axiom of limitation of size
Limitation of size doctrine


Axiom of limitation of size
Bibliography


The axiom of dependent choice and no infinite descending sequence of sets implies regularity

Regularity, the cumulative hierarchy, and types

Regularity in the presence of urelements


Axiom of regularity
Elementary implications of regularity

No infinite descending sequence of sets exists

Simpler set-theoretic definition of the ordered pair

For every two sets, only one can be an element of the other


Axiom of regularity
Elementary implications of regularity
No set is an element of itself


No infinite descending sequence of sets exists


Simpler set-theoretic definition of the ordered pair


Axiom of regularity
Elementary implications of regularity
Every set has an ordinal rank


For every two sets, only one can be an element of the other


The axiom of dependent choice and no infinite descending sequence of sets implies regularity


Axiom of regularity
Regularity and the rest of ZF(C) axioms


Axiom of regularity
Regularity and Russell's paradox


Regularity, the cumulative hierarchy, and types


Regularity in the presence of urelements


Incompatibility of the axiom of determinacy with the axiom of choice

Infinitary logic and the axiom of determinacy

Large cardinals and the axiom of determinacy


Axiom of determinacy
Types of game that are determined


Incompatibility of the axiom of determinacy with the axiom of choice


Infinitary logic and the axiom of determinacy


Large cardinals and the axiom of determinacy


Axiom of determinacy
Projective ordinals


Implications
In arithmetic
Significance
Statements true in L
References
External links


Axiom of constructibility
Statements true in L


Axiom of constructibility
External links


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


Logical equality
Alternative descriptions


Preliminaries, notation, and basic notions

Set theoretic properties and constructions


Preliminaries, notation, and basic notions


Filter (set theory)
Filters and prefilters

Finer/coarser, subordination, and meshing


Filter (set theory)
Filters and prefilters
Basic examples


Filter (set theory)
Filters and prefilters
Ultrafilters


Filter (set theory)
Filters and prefilters
Kernels

Classifying families by their kernels
Characterizing fixed ultra prefilters


Filter (set theory)
Filters and prefilters
Kernels
Classifying families by their kernels


Filter (set theory)
Filters and prefilters
Kernels
Characterizing fixed ultra prefilters


Finer/coarser, subordination, and meshing


Set theoretic properties and constructions

Trace and meshing
Images and preimages under functions
Products of prefilters
Set subtraction and some examples


Subordination is preserved by images and preimages


Nets to prefilters
Prefilters to nets
Subordinate filters and subnets


Filter (set theory)
Filters and nets
Nets to prefilters


Filter (set theory)
Filters and nets
Prefilters to nets


Filter (set theory)
Filters and nets
Prefilters to nets
Partially ordered net


Filter (set theory)
Filters and nets
Subordinate filters and subnets

Non–equivalence of subnets and subordinate filters


Non–equivalence of subnets and subordinate filters


Formal definition
Discussion
Axiomatizations
Graph models


Hereditarily finite set
Formal definition


Hereditarily finite set
Discussion
Ackermann coding


Hereditarily finite set
Discussion
Representation


Hereditarily finite set
Axiomatizations
Theories of finite sets


Hereditarily finite set
Axiomatizations
ZF


Existence
Additional properties
Variations
References


Example
Set theory definition
Category theory point of view


Disjoint union
Category theory point of view


Induction by cases
Transfinite recursion
Relationship to the axiom of choice


Transfinite induction
Induction by cases


Transfinite induction
Transfinite recursion


Transfinite induction
Relationship to the axiom of choice


History
Motivation
Formal definition
Cardinal arithmetic
The continuum hypothesis


Successor cardinal
Cardinal addition
Cardinal multiplication
Cardinal exponentiation


Cardinal number
Cardinal arithmetic
Successor cardinal


Cardinal number
Cardinal arithmetic
Cardinal addition


Cardinal number
Cardinal arithmetic
Cardinal addition
Subtraction


Cardinal number
Cardinal arithmetic
Cardinal multiplication


Cardinal number
Cardinal arithmetic
Cardinal multiplication
Division


Cardinal number
Cardinal arithmetic
Cardinal exponentiation


Cardinal number
Cardinal arithmetic
Cardinal exponentiation
Roots


Cardinal number
Cardinal arithmetic
Cardinal exponentiation
Logarithms


Cardinal number
The continuum hypothesis


Statement
Applications
Relation to other axiom schemas
History
References


Axiom schema of replacement
Applications


Axiom schema of replacement
Relation to other axiom schemas


Axiom schema of replacement
Relation to other axiom schemas
Collection


Axiom schema of replacement
Relation to other axiom schemas
Separation


Statement
Equivalent forms of MA(𝛋)
Consequences
Further development
References
Further reading


Martin's axiom
Equivalent forms of MA(𝛋)


Relation to the axiom schema of replacement


Axiom schema of specification
Unrestricted comprehension


Axiom schema of specification
In NBG class theory


Axiom schema of specification
In higher-order settings


Axiom schema of specification
In Quine's New Foundations


Axiom schema of specification
References


Axiom schema of specification
References
Citations


Preliminaries, notation, and basic notions

Set theoretic properties and constructions


Preliminaries, notation, and basic notions


Filter (set theory)
Filters and prefilters

Finer/coarser, subordination, and meshing


Filter (set theory)
Filters and prefilters
Basic examples


Filter (set theory)
Filters and prefilters
Ultrafilters


Filter (set theory)
Filters and prefilters
Kernels

Classifying families by their kernels
Characterizing fixed ultra prefilters


Filter (set theory)
Filters and prefilters
Kernels
Classifying families by their kernels


Filter (set theory)
Filters and prefilters
Kernels
Characterizing fixed ultra prefilters


Finer/coarser, subordination, and meshing


Set theoretic properties and constructions

Trace and meshing
Images and preimages under functions
Products of prefilters
Set subtraction and some examples


Subordination is preserved by images and preimages


Nets to prefilters
Prefilters to nets
Subordinate filters and subnets


Filter (set theory)
Filters and nets
Nets to prefilters


Filter (set theory)
Filters and nets
Prefilters to nets


Filter (set theory)
Filters and nets
Prefilters to nets
Partially ordered net


Filter (set theory)
Filters and nets
Subordinate filters and subnets

Non–equivalence of subnets and subordinate filters


Non–equivalence of subnets and subordinate filters


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


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


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


Arguments
Valid formula
Statements
Soundness
Satisfiability
Preservation


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


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)


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


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


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


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


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


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


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


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


Quantification scope
Semantics
Properties


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


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


Examples
In formulations of set theory
Assumptions


Hereditary set
In formulations of set theory


Expressiveness
Relationship with term logic
Variants
Footnotes


Monadic predicate calculus
Expressiveness


Monadic predicate calculus
Relationship with term logic


Definition
Legal matters
Usage
Open content


Free content
Legal matters
Public domain


Media
Software
Engineering and technology
Academia
Legislation


Free content
Usage
Engineering and technology


History
"Open content" definition
Open access
Open content and education
Textbooks
Licenses


Free content
Open content
"Open content" definition


Free content
Open content
Open content and education


The axioms of Zermelo set theory
Connection with standard set theory
Mac Lane set theory
The aim of Zermelo's paper
The axiom of separation
Cantor's theorem


Zermelo set theory
The axioms of Zermelo set theory


Zermelo set theory
Connection with standard set theory


Zermelo set theory
The aim of Zermelo's paper


Zermelo set theory
The axiom of separation


History
Openness
Policies and laws
Governance
Community
Language editions
Reception
Operation
Access to content
Cultural influence
Related projects


Restrictions
Review of changes
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Wikipedia
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Wikipedia
Governance
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Wikipedia
Reception
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Wikipedia
Reception
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Coverage of topics and bias
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Wikipedia
Reception
Coverage of topics and systemic bias
Coverage of topics and bias


Wikipedia
Reception
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Wikipedia
Access to content
Content licensing


Wikipedia
Access to content
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Wikipedia
Access to content
Methods of access
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Wikipedia
Access to content
Chinese access


Trusted source to combat fake news
Readership
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Sister projects – Wikimedia
Publishing
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Wikipedia
Cultural influence
Trusted source to combat fake news


Wikipedia
Cultural influence
Readership
COVID-19 pandemic


Wikipedia
Cultural influence
Cultural significance


Wikipedia
Cultural influence
Cultural significance
Awards


Wikipedia
Cultural influence
Cultural significance
Satire


Wikipedia
Cultural influence
Sister projects – Wikimedia


Wikipedia
Cultural influence
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History Geography Economy Infrastructure Education Tourism Crime Demographics Municipalities


Agriculture
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Killings
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John von Neumann
Life and education
Family background


John von Neumann
Life and education
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John von Neumann
Life and education
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John von Neumann
Career and private life


John von Neumann
Career and private life
Illness and death


Set theory
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Lattice theory
Mathematical statistics
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John von Neumann
Mathematics
Set theory
Von Neumann paradox


John von Neumann
Mathematics
Proof theory


John von Neumann
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John von Neumann
Mathematics
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John von Neumann
Mathematics
Topological groups


John von Neumann
Mathematics
Functional analysis


John von Neumann
Mathematics
Operator algebras


John von Neumann
Mathematics
Lattice theory


John von Neumann
Mathematics
Mathematical statistics


Quantum mechanics
Fluid dynamics
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John von Neumann
Physics
Quantum mechanics

Von Neumann entropy
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Density matrix
Von Neumann measurement scheme
Quantum logic


John von Neumann
Physics
Quantum mechanics
Von Neumann entropy


John von Neumann
Physics
Quantum mechanics
Quantum mutual information


John von Neumann
Physics
Quantum mechanics
Density matrix


John von Neumann
Physics
Quantum mechanics
Von Neumann measurement scheme


John von Neumann
Physics
Quantum mechanics
Quantum logic


Game theory
Mathematical economics
Linear programming


John von Neumann
Economics
Mathematical economics


John von Neumann
Economics
Linear programming


Cellular automata, DNA and the universal constructor

Scientific computing and numerical analysis


Cellular automata, DNA and the universal constructor


Scientific computing and numerical analysis


John von Neumann
Computer science
Weather systems and global warming


John von Neumann
Computer science
Technological singularity hypothesis


Manhattan Project
Post war
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Consultancies


John von Neumann
Defense work
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John von Neumann
Defense work
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John von Neumann
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John von Neumann
Defense work
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John von Neumann
Personality
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John von Neumann
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John von Neumann
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John von Neumann
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John von Neumann
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John von Neumann
Selected works
Books authored / coauthored


John von Neumann
Selected works
Scholarly articles


John von Neumann
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John von Neumann
Selected works
Collected works


Definition
Properties
⊂ and ⊃ symbols
Examples of subsets
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Biography
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Triggering of mudflows
Mudflows and landslides
Mudflow geography
Largest recorded mudflow
Areas at risk


Classification
History
Geographical distribution
Phonology
Grammar
Vocabulary
Writing system
Dialects, accents and varieties
Sample text


Proto-Germanic to Old English
Middle English
Early Modern English
Spread of Modern English


English language
History
Proto-Germanic to Old English


English language
History
Early Modern English


English language
History
Spread of Modern English


English language
Geographical distribution

Three circles of English-speaking countries


Three circles of English-speaking countries


English language
Geographical distribution
Pluricentric English


English language
Geographical distribution
English as a global language


Consonants
Vowels
Phonotactics
Stress, rhythm and intonation
Regional variation


English language
Phonology
Stress, rhythm and intonation


English language
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English language
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Nouns and noun phrases

Adjectives
Determiners
Pronouns, case, and person
Prepositions


English language
Grammar
Nouns and noun phrases
Adjectives


English language
Grammar
Nouns and noun phrases
Determiners


English language
Grammar
Nouns and noun phrases
Pronouns, case, and person


English language
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Nouns and noun phrases
Prepositions


English language
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Tense, aspect and mood
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English language
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Tense, aspect and mood


English language
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Verbs and verb phrases
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English language
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Verbs and verb phrases
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Questions
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English language
Grammar
Syntax
Basic constituent order


English language
Grammar
Syntax
Clause syntax


English language
Grammar
Syntax
Auxiliary verb constructions


English language
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Syntax
Questions


English language
Grammar
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Discourse level syntax


English loanwords and calques in other languages


English language
Vocabulary
Word formation processes


English language
Vocabulary
Word origins


English language
Dialects, accents and varieties

Britain and Ireland
North America
Australia and New Zealand
Southeast Asia
Africa, the Caribbean, and South Asia
Non-native varieties


English language
Dialects, accents and varieties
Britain and Ireland


English language
Dialects, accents and varieties
North America


English language
Dialects, accents and varieties
Australia and New Zealand


English language
Dialects, accents and varieties
Southeast Asia


English language
Dialects, accents and varieties
Africa, the Caribbean, and South Asia


English language
Dialects, accents and varieties
Non-native varieties


Biography
Mathematician
Awards
Published works


Relationship with astrology and esotericism


Number theory
Geometry
Algebra
Calculus and analysis
Discrete mathematics
Mathematical logic and set theory
Statistics and other decision sciences
Computational mathematics


Mathematics
Areas of mathematics
Number theory


Mathematics
Areas of mathematics
Geometry


Mathematics
Areas of mathematics
Algebra


Mathematics
Areas of mathematics
Calculus and analysis


Mathematics
Areas of mathematics
Discrete mathematics


Mathematics
Areas of mathematics
Mathematical logic and set theory


Mathematics
Areas of mathematics
Statistics and other decision sciences


Mathematics
Areas of mathematics
Computational mathematics


Mathematics
Symbolic notation and terminology


Pure and applied mathematics
Unreasonable effectiveness
Specific sciences


Mathematics
Relationship with sciences
Pure and applied mathematics


Mathematics
Relationship with sciences
Unreasonable effectiveness


Mathematics
Relationship with sciences
Specific sciences

Physics
Computing
Biology and chemistry
Earth sciences
Social sciences


Mathematics
Relationship with sciences
Specific sciences
Physics


Mathematics
Relationship with sciences
Specific sciences
Computing


Mathematics
Relationship with sciences
Specific sciences
Biology and chemistry


Mathematics
Relationship with sciences
Specific sciences
Earth sciences


Mathematics
Relationship with sciences
Specific sciences
Social sciences


Relationship with astrology and esotericism


Mathematics
Philosophy
Proposed definitions


Psychology (aesthetic, creativity and intuition)


Mathematics
Training and practice
Education


Mathematics
Cultural impact
Artistic expression


Mathematics
Cultural impact
Popularization


Background
Atlantic, Eastern and Central Pacific
Western Pacific
North Indian Ocean
South-West Indian Ocean
Australia and Fiji
Alternative scales


Tropical cyclone scales
Atlantic, Eastern and Central Pacific


Tropical cyclone scales
North Indian Ocean


Tropical cyclone scales
South-West Indian Ocean


Tropical cyclone scales
Australia and Fiji


Tropical cyclone scales
Alternative scales


Characteristics
Atlantic
East Pacific
Screaming eagle waves


Etymology
Characteristics
Pre-modern encyclopedias
Printed encyclopedias
Digital encyclopedias


Differences between encyclopedias and dictionaries


Encyclopedia
Characteristics
Four major elements


Encyclopedia
Characteristics
Encyclopedic dictionaries


Encyclopédie
Encyclopædia Britannica
Brockhaus Enzyklopädie
Encyclopedias in the US


Encyclopedia
Printed encyclopedias
Encyclopédie


Encyclopedia
Printed encyclopedias
Encyclopædia Britannica


Encyclopedia
Printed encyclopedias
Brockhaus Enzyklopädie


Encyclopedia
Printed encyclopedias
Encyclopedias in the US


Encyclopedia
Digital encyclopedias
Physical media


Encyclopedia
Digital encyclopedias
Online


Encyclopedia
Digital encyclopedias
Online
Free encyclopedias


Early life and education
Career
Awards and honours
Later life and death
Religious views
Legacy
Bibliography


Kurt Gödel
Early life and education
Childhood


Kurt Gödel
Early life and education
Studies in Vienna


Incompleteness theorems
Mid-1930s: further work and U.S. visits
Princeton, Einstein, U.S. citizenship


Kurt Gödel
Career
Incompleteness theorems


Kurt Gödel
Career
Mid-1930s: further work and U.S. visits


Kurt Gödel
Career
Princeton, Einstein, U.S. citizenship


Kurt Gödel
Bibliography
Important publications


Seasonal forecasts
Seasonal summary
Systems
Storm names
Season effects


2018 Pacific hurricane season
Seasonal forecasts


2018 Pacific hurricane season
Seasonal summary


Tropical Depression One-E
Hurricane Aletta
Hurricane Bud
Tropical Storm Carlotta
Tropical Storm Daniel
Tropical Storm Emilia
Hurricane Fabio
Tropical Storm Gilma
Tropical Depression Nine-E
Hurricane Hector
Tropical Storm Ileana
Hurricane John
Tropical Storm Kristy
Hurricane Lane
Hurricane Miriam
Hurricane Norman
Hurricane Olivia
Tropical Storm Paul
Tropical Depression Nineteen-E
Hurricane Rosa
Hurricane Sergio
Hurricane Walaka
Tropical Storm Tara
Tropical Storm Vicente
Hurricane Willa
Tropical Storm Xavier
Other systems


2018 Pacific hurricane season
Systems
Tropical Depression One-E


2018 Pacific hurricane season
Systems
Hurricane Aletta


2018 Pacific hurricane season
Systems
Hurricane Bud


2018 Pacific hurricane season
Systems
Tropical Storm Carlotta


2018 Pacific hurricane season
Systems
Tropical Storm Daniel


2018 Pacific hurricane season
Systems
Tropical Storm Emilia


2018 Pacific hurricane season
Systems
Hurricane Fabio


2018 Pacific hurricane season
Systems
Tropical Storm Gilma


2018 Pacific hurricane season
Systems
Tropical Depression Nine-E


2018 Pacific hurricane season
Systems
Hurricane Hector


2018 Pacific hurricane season
Systems
Tropical Storm Ileana


2018 Pacific hurricane season
Systems
Hurricane John


2018 Pacific hurricane season
Systems
Tropical Storm Kristy


2018 Pacific hurricane season
Systems
Hurricane Lane


2018 Pacific hurricane season
Systems
Hurricane Miriam


2018 Pacific hurricane season
Systems
Hurricane Norman


2018 Pacific hurricane season
Systems
Hurricane Olivia


2018 Pacific hurricane season
Systems
Tropical Storm Paul


2018 Pacific hurricane season
Systems
Tropical Depression Nineteen-E


2018 Pacific hurricane season
Systems
Hurricane Rosa


2018 Pacific hurricane season
Systems
Hurricane Sergio


2018 Pacific hurricane season
Systems
Hurricane Walaka


2018 Pacific hurricane season
Systems
Tropical Storm Tara


2018 Pacific hurricane season
Systems
Tropical Storm Vicente


2018 Pacific hurricane season
Systems
Hurricane Willa


2018 Pacific hurricane season
Systems
Tropical Storm Xavier


2018 Pacific hurricane season
Systems
Other systems


2018 Pacific hurricane season
Storm names


2018 Pacific hurricane season
Season effects


Background
Intensity
Classification and naming
Structure
Movement
Formation regions and warning centers
Preparations
Effects
Response
Climatology
Observation and forecasting
Related cyclone types


Factors that influence intensity
Formation
Rapid intensification
Dissipation
Methods for assessing intensity
Intensity metrics


Tropical cyclone
Intensity
Factors that influence intensity


Tropical cyclone
Intensity
Rapid intensification


Tropical cyclone
Intensity
Methods for assessing intensity


Tropical cyclone
Intensity
Intensity metrics


Tropical cyclone
Classification and naming


Tropical cyclone
Classification and naming
Classification


Tropical cyclone
Classification and naming
Naming


Tropical cyclone
Structure
Eye and center


Interaction with the mid-latitude westerlies


Tropical cyclone
Movement
Environmental steering


Tropical cyclone
Movement
Multiple storm interaction


Tropical cyclone
Formation regions and warning centers


Natural phenomena caused or worsened by tropical cyclones


Tropical cyclone
Effects
Effect on property and human life


Tropical cyclone
Effects
Environmental effects


Tropical cyclone
Climatology
Influence of climate change


Tropical cyclone
Observation and forecasting

Observation
Forecasting
Geopotential height


Tropical cyclone
Observation and forecasting
Observation


Tropical cyclone
Observation and forecasting
Forecasting


Tropical cyclone
Observation and forecasting
Geopotential height


Role for publishers in scholarly communication


Academic publishing
Publishers and business aspects

Crisis
Academic journal publishing reform


Academic publishing
Publishers and business aspects
Crisis


Academic publishing
Publishers and business aspects
Academic journal publishing reform


Academic publishing
Scholarly paper
Categories of papers


Academic publishing
Peer review
Rejection rate


Academic publishing
Publishing by discipline

Natural sciences
Social sciences
Humanities


Academic publishing
Publishing by discipline
Natural sciences


Academic publishing
Publishing by discipline
Social sciences


Academic publishing
Publishing by discipline
Humanities


Academic publishing
Open access journals


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


Galileo's paradox
Galileo on infinite sets


Stated in terms of von Neumann ordinals
Stated more generally
Resolutions of the paradox


Burali-Forti paradox
Stated in terms of von Neumann ordinals


Burali-Forti paradox
Stated more generally


Burali-Forti paradox
Resolutions of the paradox


Comparison with the usual definition of infinite set

Proof of equivalence to infinity, assuming axiom of countable choice


Comparison with the usual definition of infinite set


Dedekind-infinite set
Dedekind-infinite sets in ZF


Dedekind-infinite set
Relation to the axiom of choice


Proof of equivalence to infinity, assuming axiom of countable choice


Life
Work
Bibliography
References
External links


Life
Research in set theory
Zermelo's navigation problem
Publications


Ernst Zermelo
Zermelo's navigation problem


Tropical Storm Vicente (2018)
Meteorological history


Tropical Storm Vicente (2018)
Impact
Michoacán


Tropical Storm Vicente (2018)
Impact
Oaxaca


Tropical Storm Vicente (2018)
Impact
Veracruz


Tropical Storm Vicente (2018)
Impact
Elsewhere


Biography
Work
In popular culture
Bibliography
Filmography


Willard Van Orman Quine
Biography
Education


Willard Van Orman Quine
Biography
World War II


Willard Van Orman Quine
Biography
Personal


Willard Van Orman Quine
Biography
Harvard


Logic
Set theory
Metaphysics
Epistemology


Willard Van Orman Quine
Work
Metaphysics

Rejection of the analytic–synthetic distinction

Confirmation holism and ontological relativity

Indispensability argument for mathematical realism


Rejection of the analytic–synthetic distinction


Confirmation holism and ontological relativity


Willard Van Orman Quine
Work
Metaphysics
Existence and its contrary


Willard Van Orman Quine
Work
Metaphysics
Ontological commitment


Indispensability argument for mathematical realism


Willard Van Orman Quine
Work
Epistemology


Willard Van Orman Quine
In popular culture


Willard Van Orman Quine
Bibliography
Selected books


Willard Van Orman Quine
Bibliography
Important articles


Preliminaries, notation, and basic notions

Set theoretic properties and constructions


Preliminaries, notation, and basic notions


Filter (set theory)
Filters and prefilters

Finer/coarser, subordination, and meshing


Filter (set theory)
Filters and prefilters
Basic examples