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


List of axioms
Axiom of choice
Equivalents of AC


List of axioms
Axiom of choice
Stronger than AC


List of axioms
Axiom of choice
Weaker than AC


List of axioms
Axiom of choice
Alternates incompatible with AC


List of axioms
Other axioms of mathematical logic


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


Löwenheim–Skolem theorem
Discussion
Concepts


Löwenheim–Skolem theorem
Discussion
Concepts
Signatures


Löwenheim–Skolem theorem
Discussion
Concepts
Structures / Models


Löwenheim–Skolem theorem
Discussion
Consequences


Löwenheim–Skolem theorem
Proof sketch
Downward part


Löwenheim–Skolem theorem
Proof sketch
Upward part


Löwenheim–Skolem theorem
In other logics


Löwenheim–Skolem theorem
Historical notes


Historical publications
Secondary sources


Löwenheim–Skolem theorem
Sources
Historical publications


Löwenheim–Skolem theorem
Sources
Secondary sources


Aristotle's system
Basics
Term
Proposition
Singular terms
Syllogism in the first figure
Syllogism in the second figure
Syllogism in the third figure
Table of syllogisms
Decline of term logic
Revival
Boole’s acceptance of Aristotle


Term logic
Singular terms
The three figures


Term logic
Singular terms
The fourth figure


Term logic
Syllogism in the first figure


Term logic
Syllogism in the second figure


Term logic
Syllogism in the third figure


Term logic
Boole’s acceptance of Aristotle


Background
History
Formalization
Computability theory
Generalization


Programming consequences
Common pitfalls


Halting problem
Background
Programming consequences


Halting problem
Background
Common pitfalls


Halting problem
History
Origin of the halting problem


Representation as a set
Proof concept
Sketch of rigorous proof


Halting problem
Formalization
Representation as a set


Halting problem
Formalization
Proof concept


Halting problem
Formalization
Sketch of rigorous proof


Approximations
Gödel's incompleteness theorems


Halting problem
Computability theory
Approximations


Halting problem
Computability theory
Gödel's incompleteness theorems


Halting on all inputs
Recognizing partial solutions
Lossy computation
Oracle machines


Halting problem
Generalization
Halting on all inputs


Halting problem
Generalization
Recognizing partial solutions


Halting problem
Generalization
Lossy computation


Halting problem
Generalization
Oracle machines


Characteristics
History
Generalized semantics
References
Further reading


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


Compactness theorem
Applications
Robinson's principle


Compactness theorem
Applications
Upward Löwenheim–Skolem theorem


Compactness theorem
Applications
Non-standard analysis


Logical truths and analytic truths
Truth values and tautologies
Logical truth and logical constants
Logical truth and rules of inference
Logical truth and logical positivism
Non-classical logics


Logical truth
Logical truths and analytic truths


Logical truth
Truth values and tautologies


Logical truth
Logical truth and logical constants


Logical truth
Logical truth and rules of inference


Logical truth
Logical truth and logical positivism


Uncountable set
Consequences
Version for Quine's New Foundations


Cantor's diagonal argument
Uncountable set


Cantor's diagonal argument
Uncountable set
Real numbers


Cantor's diagonal argument
Uncountable set
General sets


Ordering of cardinals
Open questions
Diagonalization in broader context


Cantor's diagonal argument
Consequences
Ordering of cardinals


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


Cantor's diagonal argument
Consequences
Open questions


Cantor's diagonal argument
Consequences
Diagonalization in broader context


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


Efficient verification and the Boolean satisfiability problem

Tautologies versus validities in first-order logic


Tautology (logic)
Definition and examples


Tautology (logic)
Tautological implication


Tautology (logic)
Semantic completeness and soundness


Efficient verification and the Boolean satisfiability problem


Tautologies versus validities in first-order logic


Proof
When A is countably infinite
Related paradoxes
History
Generalizations


Cantor's theorem
When A is countably infinite


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


By Aristotle
By the logical positivists
By Russell


Proposition
Historical usage
By Aristotle


Proposition
Historical usage
By the logical positivists


Aristotelian logic
Syntactic characterization
Semantic characterization


Proposition
Treatment in logic
Aristotelian logic


Proposition
Treatment in logic
Syntactic characterization


Proposition
Treatment in logic
Semantic characterization


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

Obtaining infinitely many balls from one

Von Neumann paradox in the Euclidean plane


Banach–Tarski paradox
Banach and Tarski publication


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


Banach–Tarski paradox
A sketch of the proof

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


Banach–Tarski paradox
A sketch of the proof
Step 1


Banach–Tarski paradox
A sketch of the proof
Step 2


Banach–Tarski paradox
A sketch of the proof
Step 3


Banach–Tarski paradox
A sketch of the proof
Step 4


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


Obtaining infinitely many balls from one


Von Neumann paradox in the Euclidean plane


Definition
Examples
Incorrect inference
Applications


Example for definition #1
Example for definition #2


Inference
Examples
Example for definition #1


Inference
Examples
Example for definition #2


Bayesian statistics and probability logic


Inference
Applications
Inference engines


Inference
Applications
Inference engines
Prolog engine


Inference
Applications
Non-monotonic logic


History
Statement
General form
Discussion


Tarski's undefinability theorem
Statement


Tarski's undefinability theorem
General form


Tarski's undefinability theorem
Discussion


Logical equivalences
Examples
Relation to material equivalence


Logical equivalence
Logical equivalences

Logical equivalences involving conditional statements

Logical equivalences involving biconditionals


Logical equivalence
Logical equivalences
General logical equivalences


Logical equivalences involving conditional statements


Logical equivalences involving biconditionals


Logical equivalence
Relation to material equivalence


Relationship to the incompleteness theorems


Gödel's completeness theorem
Preliminaries


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


Gödel's completeness theorem
Statement
Gödel's original formulation


Gödel's completeness theorem
Statement
More general form


Gödel's completeness theorem
Statement
Model existence theorem


Gödel's completeness theorem
Statement
As a theorem of arithmetic


Gödel's completeness theorem
Consequences


Gödel's completeness theorem
Relationship to the compactness theorem


Gödel's completeness theorem
Completeness in other logics


Consistency and completeness in arithmetic and set theory


Notation
Definition
Basic results
Henkin's theorem
Sketch of proof


Consistency
First-order logic
Definition


Consistency
First-order logic
Basic results


Consistency
First-order logic
Henkin's theorem


Consistency
First-order logic
Sketch of proof


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


General theories (as expressed in formal language)


General theories (as expressed in formal language)

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


Theory (mathematical logic)
First-order theories

Syntactic consequence in a first-order theory


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


Syntactic consequence in a first-order theory


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


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


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


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


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)


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


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


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


Information theory
Historical background


Information theory
Quantities of information

Kullback–Leibler divergence (information gain)


Information theory
Quantities of information
Entropy of an information source


Information theory
Quantities of information
Joint entropy


Information theory
Quantities of information
Conditional entropy (equivocation)


Information theory
Quantities of information
Mutual information (transinformation)


Kullback–Leibler divergence (information gain)


Information theory
Quantities of information
Directed Information


Information theory
Quantities of information
Other quantities


Information theory
Coding theory
Source theory


Information theory
Coding theory
Source theory
Rate


Information theory
Coding theory
Channel capacity

Channels with memory and directed information


Information theory
Coding theory
Channel capacity
Capacity of particular channel models


Channels with memory and directed information


Information theory
Applications to other fields

Intelligence uses and secrecy applications

Integrated process organization of neural information


Intelligence uses and secrecy applications


Information theory
Applications to other fields
Pseudorandom number generation


Information theory
Applications to other fields
Seismic exploration


Information theory
Applications to other fields
Semiotics


Integrated process organization of neural information


Information theory
Applications to other fields
Miscellaneous applications


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


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


Well-definedness under an equivalence relation

Fundamental theorem of equivalence relations

Equivalence relations and mathematical logic


Alternative definition using relational algebra


Simple example
Equivalence relations
Relations that are not equivalences


Equivalence relation
Examples
Simple example


Equivalence relation
Examples
Equivalence relations


Equivalence relation
Examples
Relations that are not equivalences


Equivalence relation
Connections to other relations


Well-definedness under an equivalence relation


Equivalence relation
Related important definitions

Equivalence class
Quotient set
Projection
Equivalence kernel
Partition


Equivalence relation
Related important definitions
Equivalence class


Equivalence relation
Related important definitions
Quotient set


Equivalence relation
Related important definitions
Projection


Equivalence relation
Related important definitions
Equivalence kernel


Equivalence relation
Related important definitions
Partition


Equivalence relation
Related important definitions
Partition
Counting partitions


Fundamental theorem of equivalence relations


Equivalence relation
Comparing equivalence relations


Equivalence relation
Generating equivalence relations


Equivalence relation
Algebraic structure

Group theory
Categories and groupoids
Lattices


Equivalence relation
Algebraic structure
Group theory


Equivalence relation
Algebraic structure
Categories and groupoids


Equivalence relation
Algebraic structure
Lattices


Equivalence relations and mathematical logic


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


Ordinal number
Ordinals extend the natural numbers


Definition of an ordinal as an equivalence class


Ordinal number
Definitions
Well-ordered sets


Ordinal number
Definitions
Von Neumann definition of ordinals


Ordinal number
Definitions
Other definitions


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


Ordinal number
Transfinite induction
Transfinite recursion


Ordinal number
Transfinite induction
Successor and limit ordinals


Ordinal number
Transfinite induction
Indexing classes of ordinals


Ordinal number
Transfinite induction
Closed unbounded sets and classes


Initial ordinal of a cardinal
Cofinality


Ordinal number
Ordinals and cardinals
Initial ordinal of a cardinal


Ordinal number
Ordinals and cardinals
Cofinality


Ordinal number
Some "large" countable ordinals


Quantification scope
Semantics
Properties


Necessary and sufficient conditions for finiteness


Finite set
Set-theoretic definitions of finiteness


Finite set
Set-theoretic definitions of finiteness
Other concepts of finiteness


Predicate (mathematical logic)
Predicates in different systems


Defining the ordered pair using set theory


Ordered pair
Informal and formal definitions


Defining the ordered pair using set theory

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


Proving that definitions satisfy the characteristic property


Examples
Properties
Transitive closure
Transitive models of set theory


Transitive set
Transitive models of set theory


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


Forcing (mathematics)
Intuition
The role of the model


Forcing (mathematics)
Forcing conditions and forcing posets


Forcing (mathematics)
Forcing conditions and forcing posets
Examples


Forcing (mathematics)
P-names and interpretations


Forcing (mathematics)
P-names and interpretations
Rigorous definitions


Forcing (mathematics)
Forcing
Internal definition


Forcing (mathematics)
The countable chain condition


Forcing (mathematics)
Boolean-valued models


Forcing (mathematics)
Meta-mathematical explanation


Forcing (mathematics)
Logical explanation


Theory
Urelements in set theory
Quine atoms
References
External links


Inhabited set
Comparison with nonempty sets


Formal definition
Examples and non-examples
Properties


Computable set
Examples and non-examples


Examples, properties, and sufficient conditions


Ultrafilter (set theory)
Generalization to ultra prefilters


Ultrafilter (set theory)
Characterizations

Grills and filter-grills
Free or principal


Ultrafilter (set theory)
Characterizations
Grills and filter-grills


Ultrafilter (set theory)
Characterizations
Free or principal


Ultrafilter (set theory)
The ultrafilter lemma

Relationships to other statements under ZF


Relationships to other statements under ZF

Statements that cannot be deduced
Equivalent statements
Weaker statements


Ultrafilter (set theory)
Ordering on ultrafilters


Ultrafilter (set theory)
Ultrafilters on ℘(ω)


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
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


Operation (mathematics)
Types of operation


Ontology
Axioms
Discussion
History
Footnotes
References
External links


Metamathematics
Peano arithmetic
Infinite sets


General set theory
Discussion
Metamathematics


General set theory
Discussion
Peano arithmetic


General set theory
Discussion
Infinite sets


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


Binary operation
Properties and examples


Binary operation
Binary operations as ternary relations


Binary operation
External binary operations


Axioms
Related definitions
Theorems
Metalogic


Comparison with Zermelo-Fraenkel set theory


Kripke–Platek set theory
Axioms
Empty set


Kripke–Platek set theory
Related definitions


Admissible sets
Cartesian products exist


Kripke–Platek set theory
Theorems
Admissible sets


Kripke–Platek set theory
Theorems
Cartesian products exist


Introduction
Subtheories of ZF
Sorted theories


Constructive outlook
History and overview


Constructive set theory
Introduction
Constructive outlook

Preliminary on the use of intuitionistic logic


Preliminary on the use of intuitionistic logic


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


Constructive set theory
Introduction
Constructive outlook
Metalogic


Constructive set theory
Introduction
Constructive outlook
Anti-classical principles


Constructive set theory
Introduction
History and overview


Constructive set theory
Introduction
History and overview
Models


Constructive set theory
Subtheories of ZF

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


Constructive set theory
Subtheories of ZF
Notation


Constructive set theory
Subtheories of ZF
Notation
Language


Constructive set theory
Subtheories of ZF
Notation
Variables


Constructive set theory
Subtheories of ZF
Notation
Classes


Constructive set theory
Subtheories of ZF
Equality


Constructive set theory
Subtheories of ZF
Merging sets


Constructive set theory
Subtheories of ZF
Set existence


Constructive set theory
Subtheories of ZF
BCST

Separation
No universal set
Predicativity
Replacement


Constructive set theory
Subtheories of ZF
BCST
Separation


Constructive set theory
Subtheories of ZF
BCST
No universal set


Constructive set theory
Subtheories of ZF
BCST
Predicativity


Constructive set theory
Subtheories of ZF
BCST
Replacement


Constructive set theory
Subtheories of ZF
ECST

Weaker formulations of infinity
Number bounds


Constructive set theory
Subtheories of ZF
ECST
Weaker formulations of infinity


Constructive set theory
Subtheories of ZF
ECST
Number bounds


Constructive set theory
Subtheories of ZF
Functions

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


Constructive set theory
Subtheories of ZF
Functions
Finiteness


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


Constructive set theory
Subtheories of ZF
Functions
Total functional relations


Constructive set theory
Subtheories of ZF
Functions
Characteristic functions


Constructive set theory
Subtheories of ZF
Functions
Computable sets


Constructive set theory
Subtheories of ZF
Functions
Boundedness criteria


Constructive set theory
Subtheories of ZF
Choice functions

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


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


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


Constructive set theory
Subtheories of ZF
Choice functions
Regularity implies PEM


Constructive set theory
Subtheories of ZF
Arithmetic

Recursion
Moderate induction in ECST
Induction without infinite sets


Constructive set theory
Subtheories of ZF
Arithmetic
Recursion


Constructive set theory
Subtheories of ZF
Arithmetic
Moderate induction in ECST


Constructive set theory
Subtheories of ZF
Arithmetic
Induction without infinite sets


Constructive set theory
Subtheories of ZF
Exponentiation

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


Constructive set theory
Subtheories of ZF
Exponentiation
On countable sets


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


Constructive set theory
Subtheories of ZF
Exponentiation
Metalogic


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


Constructive set theory
Subtheories of ZF
Analysis

Cauchy sequences
Cauchy reals
Towards the Dedekind reals
Constructive schools


Constructive set theory
Subtheories of ZF
Analysis
Cauchy sequences


Constructive set theory
Subtheories of ZF
Analysis
Cauchy reals


Constructive set theory
Subtheories of ZF
Analysis
Towards the Dedekind reals


Constructive set theory
Subtheories of ZF
Analysis
Constructive schools


Constructive set theory
Subtheories of ZF
Non-constructive principles


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


Constructive set theory
Subtheories of ZF
Induction

Mathematical induction
Set Induction
Metalogic
Relation to ZF


Constructive set theory
Subtheories of ZF
Induction
Mathematical induction


Constructive set theory
Subtheories of ZF
Induction
Set Induction


Constructive set theory
Subtheories of ZF
Induction
Metalogic


Constructive set theory
Subtheories of ZF
Induction
Relation to ZF


Constructive set theory
Subtheories of ZF
Strong Collection


Constructive set theory
Subtheories of ZF
Strong Collection
Metalogic


Constructive set theory
Subtheories of ZF
Constructive Zermelo–Fraenkel

Binary refinement
Subset Collection
Fullness


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


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


Constructive set theory
Subtheories of ZF
Constructive Zermelo–Fraenkel
Fullness


Constructive set theory
Subtheories of ZF
Metalogic of CZF

Unprovable claims
Ordinal analysis
Models
Indecomposability
Breaking with ZF


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Unprovable claims


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Ordinal analysis


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Models


Constructive set theory
Subtheories of ZF
Metalogic of CZF
Indecomposability


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


Constructive set theory
Subtheories of ZF
Intuitionistic Zermelo–Fraenkel


Constructive set theory
Subtheories of ZF
Intuitionistic Zermelo–Fraenkel
Metalogic


Constructive set theory
Subtheories of ZF
Intuitionistic Z


Constructive set theory
Subtheories of ZF
Intuitionistic KP


Constructive set theory
Bishop style set theory
Category theories


Constructive set theory
Sorted theories
Constructive set theory


Constructive set theory
Sorted theories
Bishop style set theory


Constructive set theory
Sorted theories
Category theories


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


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

Axiom schema versus class existence theorem


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


Axiom schema versus class existence theorem


Von Neumann–Bernays–Gödel set theory
Axiomatization of NBG

Definitions and axioms of extensionality and pairing

Class existence axioms and axiom of regularity


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


Definitions and axioms of extensionality and pairing


Class existence axioms and axiom of regularity


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


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


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


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


Von Neumann–Bernays–Gödel set theory
History

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


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


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


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


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


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


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


Von Neumann–Bernays–Gödel set theory
Category theory


Von Neumann–Bernays–Gödel set theory
Notes


Von Neumann–Bernays–Gödel set theory
References


Von Neumann–Bernays–Gödel set theory
Bibliography


Von Neumann–Bernays–Gödel set theory
External links


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


Ackermann set theory
Relation to Zermelo–Fraenkel set theory


Ackermann set theory
AST and category theory


The consistency problem and related partial results


Natural numbers and the axiom of infinity


New Foundations
Quinean set theory
Axioms and stratification


New Foundations
Quinean set theory
Finite axiomatization


New Foundations
Quinean set theory
Ordered pairs


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


New Foundations
Consequences
Admissibility of useful large sets


New Foundations
Consequences
Resolution of set-theoretic paradoxes

Burali-Forti paradox and the T operation


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


Burali-Forti paradox and the T operation


New Foundations
Consequences
Cartesian closure


Self-sufficiency of mathematical foundations in NFU


New Foundations
Models of NFU
Facts about the automorphism j


New Foundations
Strong axioms of infinity


New Foundations
The system ML (Mathematical Logic)


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


Morse–Kelley set theory
MK axioms and ontology


Morse–Kelley set theory
Discussion
Model theory


Morse–Kelley set theory
Discussion
History


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


Axioms
Implementation in the Mizar system
Implementation in Metamath


Tarski–Grothendieck set theory
Implementation in the Mizar system


Tarski–Grothendieck set theory
Implementation in Metamath


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


Formal language
Language-specification formalisms


Programming languages
Formal theories, systems, and proofs


Formal language
Applications
Programming languages


Formal language
Applications
Formal theories, systems, and proofs


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


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


Logical conjunction
Definition
Truth table


Logical conjunction
Definition
Defined by other operators


Logical conjunction
Introduction and elimination rules


Logical conjunction
Negation
Other proof strategies


Logical conjunction
Applications in computer engineering


Logical conjunction
Set-theoretic correspondence


Ground expression
Formal definitions
Ground term


Ground expression
Formal definitions
Ground atom


Ground expression
Formal definitions
Ground formula


Programming language and ordinary language


Double negation
Distributivity
Linearity
Self dual
Negations of quantifiers


Negation
Properties
Negations of quantifiers


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


Metalanguage
Types of metalanguage
Embedded


Metalanguage
Types of metalanguage
Ordered


Metalanguage
Types of metalanguage
Nested


Deductive systems
Metavariables
Metatheories and metatheorems
Interpretations


Metalanguage
Types of expressions
Deductive systems


Metalanguage
Types of expressions
Metavariables


Metalanguage
Types of expressions
Metatheories and metatheorems


Metalanguage
Types of expressions
Interpretations


Examples
Syntax versus semantics
Variables


Expression (mathematics)
Syntax versus semantics

Syntax
Semantics
Formal languages and lambda calculus


Expression (mathematics)
Syntax versus semantics
Syntax


Expression (mathematics)
Syntax versus semantics
Semantics


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


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


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


Notation
Definitions
Formal properties
Discrepancies with natural language


Semantics
Truth table
Deductive definition


Material conditional
Definitions
Semantics


Material conditional
Definitions
Truth table


Material conditional
Definitions
Deductive definition


Material conditional
Discrepancies with natural language


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


Nullary
Unary
Binary
Ternary
n-ary
Varying arity


Definition
Properties
Rules of inference
Colloquial usage


Logical biconditional
Definition
Truth table


Logical biconditional
Definition
Venn diagrams


Logical biconditional
Rules of inference

Biconditional introduction
Biconditional elimination


Logical biconditional
Rules of inference
Biconditional introduction


Logical biconditional
Rules of inference
Biconditional elimination


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


Automata theory
Automata
Informal description


Automata theory
Automata
Formal definition


Automata theory
Variant definitions of automata


Discrete, continuous, and hybrid automata


Automata theory
Hierarchy in terms of powers


Automata theory
Connection to category theory


List of formal systems
Other formal systems


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


Examples
Formal explanation
Natural language


Free variables and bound variables
Examples


Free variables and bound variables
Examples
Variable-binding operators


Free variables and bound variables
Formal explanation


Free variables and bound variables
Formal explanation
Function expressions


Free variables and bound variables
Natural language


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


Axiom schema
Finitely axiomatized theories


Reduction to ordinary existential and universal quantification


Uniqueness quantification
Proving uniqueness


Atomic formula
Atomic formula in first-order logic


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


Well-formed formula
Propositional calculus


Well-formed formula
Atomic and open formulas


Well-formed formula
Properties applicable to formulas


Well-formed formula
Usage of the terminology


History
Introductory example
Formal definition
The Chomsky hierarchy
Analytic grammars


Formal grammar
Introductory example
Example 1


Formal grammar
Introductory example
Examples 2 and 3


Some mathematical constructs regarding formal grammars


Formal grammar
Formal definition
The syntax of grammars


Formal grammar
Formal definition
Example


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


Formal grammar
The Chomsky hierarchy
Context-free grammars


Formal grammar
The Chomsky hierarchy
Regular grammars


Formal grammar
The Chomsky hierarchy
Other forms of generative grammars


Formal grammar
The Chomsky hierarchy
Recursive grammars


Ground expression
Formal definitions
Ground term


Ground expression
Formal definitions
Ground atom


Ground expression
Formal definitions
Ground formula


Overview
Can words be modeled as formal symbols?
References


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


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


Propositional logic
First-order logic
Algebra


Substitution (logic)
Propositional logic


Substitution (logic)
Propositional logic
Definition


Substitution (logic)
Propositional logic
Tautologies


Formal language
Formal systems
Propositional and predicate logic


Formation rule
Propositional and predicate logic


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


Examples
Model-theoretic conservative extension


Conservative extension
Model-theoretic conservative extension


Logical equality
Alternative descriptions


Introducing new function symbols
Doing without functional predicates


Functional predicate
Introducing new function symbols


Functional predicate
Doing without functional predicates


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


Alphabet (formal languages)
Applications


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


Extension by definitions
Definition of relation symbols


Extension by definitions
Definition of function symbols


Extension by definitions
Extensions by definitions


Signatures
Models
Informal semantics
Descriptive signs


Extension by new constant and function names


Equivalence to Turing machines
Other properties


Spectrum of a sentence
Properties
Equivalence to Turing machines


Spectrum of a sentence
Properties
Other properties


Symbols
Formal language
Formation rules
Propositions
Formal theories
Formal systems
Interpretations


Syntax (logic)
Syntactic entities
Symbols


Syntax (logic)
Syntactic entities
Formal language


Syntax (logic)
Syntactic entities
Formation rules


Syntax (logic)
Syntactic entities
Propositions


Syntax (logic)
Syntactic entities
Formal theories


Syntax (logic)
Syntactic entities
Formal systems

Syntactic consequence within a formal system

Syntactic completeness of a formal system


Syntactic consequence within a formal system


Syntactic completeness of a formal system


Syntax (logic)
Syntactic entities
Interpretations


Cartesian products ⨯ of finitely many sets

Sequences and collections of families of sets


List of set identities and relations
Notation


List of set identities and relations
Notation
Elementary set operations


List of set identities and relations
One subset involved


List of set identities and relations
Two sets involved

Formulas for binary set operations ⋂, ⋃, \, and ∆


Formulas for binary set operations ⋂, ⋃, \, and ∆


List of set identities and relations
Two sets involved
De Morgan's laws


List of set identities and relations
Two sets involved
Commutativity


List of set identities and relations
Two sets involved
Other identities involving two sets


List of set identities and relations
Two sets involved
Subsets ⊆ and supersets ⊇

Set equality
Meets, Joins, and lattice properties


List of set identities and relations
Two sets involved
Subsets ⊆ and supersets ⊇
Set equality


List of set identities and relations
Two sets involved
Subsets ⊆ and supersets ⊇
Meets, Joins, and lattice properties


List of set identities and relations
Three sets involved

Associativity
Distributivity
Two set subtractions
One set subtraction
Three operations on three sets
Other simplifications


List of set identities and relations
Three sets involved
Associativity


List of set identities and relations
Three sets involved
Distributivity

Distributivity and symmetric difference ∆


Distributivity and symmetric difference ∆


List of set identities and relations
Three sets involved
Distributivity
Distributivity and set subtraction \


List of set identities and relations
Three sets involved
Two set subtractions


List of set identities and relations
Three sets involved
Two set subtractions
(L\M)\R


List of set identities and relations
Three sets involved
Two set subtractions
L\(M\R)


List of set identities and relations
Three sets involved
One set subtraction


List of set identities and relations
Three sets involved
One set subtraction
(L\M) ⁎ R


List of set identities and relations
Three sets involved
One set subtraction
L\(M ⁎ R)


List of set identities and relations
Three sets involved
One set subtraction
(L ⁎ M)\R


List of set identities and relations
Three sets involved
One set subtraction
L ⁎ (M\R)


List of set identities and relations
Three sets involved
Three operations on three sets

(L • M) ⁎ (M • R)
(L • M) ⁎ (R\M)
(L\M) ⁎ (L\R)


List of set identities and relations
Three sets involved
Three operations on three sets
(L • M) ⁎ (M • R)


List of set identities and relations
Three sets involved
Three operations on three sets
(L • M) ⁎ (R\M)


List of set identities and relations
Three sets involved
Three operations on three sets
(L\M) ⁎ (L\R)


List of set identities and relations
Three sets involved
Other simplifications


Cartesian products ⨯ of finitely many sets

Binary ⋂ of finite ⨯
Binary ⋃ of finite ⨯
Difference \ of finite ⨯
Finite ⨯ of differences \
Symmetric difference ∆ and finite ⨯


List of set identities and relations
Arbitrary families of sets

Commutativity and associativity of ⋃ and ⋂

Cartesian products Π of arbitrarily many sets


List of set identities and relations
Arbitrary families of sets
Definitions


List of set identities and relations
Arbitrary families of sets
Distributing unions and intersections

Binary ⋂ of arbitrary ⋃'s
Binary ⋃ of arbitrary ⋂'s
Arbitrary ⋂'s and arbitrary ⋃'s


List of set identities and relations
Arbitrary families of sets
Distributing unions and intersections
Binary ⋂ of arbitrary ⋃'s


List of set identities and relations
Arbitrary families of sets
Distributing unions and intersections
Binary ⋃ of arbitrary ⋂'s


List of set identities and relations
Arbitrary families of sets
Distributing unions and intersections
Arbitrary ⋂'s and arbitrary ⋃'s


List of set identities and relations
Arbitrary families of sets
Distributing subtraction over ⋃ and ⋂


Commutativity and associativity of ⋃ and ⋂


Cartesian products Π of arbitrarily many sets

Intersections ⋂ of Π
Unions ⋃ of Π
Difference \ of Π
Symmetric difference ∆ of Π


List of set identities and relations
Functions and sets

(Pre)Images of arbitrary unions ⋃'s and intersections ⋂'s

(Pre)Images of set operations on (pre)images


List of set identities and relations
Functions and sets
Definitions


(Pre)Images of arbitrary unions ⋃'s and intersections ⋂'s


List of set identities and relations
Functions and sets
(Pre)Images of binary set operations

Counter-examples: images of operations not distributing

Conditions guaranteeing that images distribute over set operations

Exact formulas/equalities for images of set operations


Counter-examples: images of operations not distributing


Conditions guaranteeing that images distribute over set operations


Exact formulas/equalities for images of set operations


(Pre)Images of set operations on (pre)images


List of set identities and relations
Functions and sets
(Pre)Images and Cartesian products Π


List of set identities and relations
Functions and sets
(Pre)Image of a single set

Containments ⊆ and intersections ⋂ of images and preimages


Containments ⊆ and intersections ⋂ of images and preimages


Sequences and collections of families of sets


Definitions of categories of families of sets


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


True arithmetic
Arithmetic undefinability


True arithmetic
Computability properties


True arithmetic
Model-theoretic properties


True arithmetic
True theory of second-order arithmetic


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


Peano axioms
Historical second-order formulation

Defining arithmetic operations and relations


Defining arithmetic operations and relations


Peano axioms
Historical second-order formulation
Models

Set-theoretic models
Interpretation in category theory


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


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


Peano axioms
Historical second-order formulation
Consistency


Peano axioms
Peano arithmetic as first-order theory

Equivalent axiomatizations
Undecidability and incompleteness
Nonstandard models


Peano axioms
Peano arithmetic as first-order theory
Equivalent axiomatizations


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


Peano axioms
Peano arithmetic as first-order theory
Nonstandard models


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


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


Primitive recursive arithmetic
Language and axioms


Primitive recursive arithmetic
Logic-free calculus


Definition
Models
Subsystems
Projective determinacy
Coding mathematics


Second-order arithmetic
Definition
Syntax


Second-order arithmetic
Definition
Semantics


Second-order arithmetic
Definition
Axioms

Basic
Induction and comprehension schema


Second-order arithmetic
Definition
Axioms
Basic


Second-order arithmetic
Definition
Axioms
Induction and comprehension schema


Second-order arithmetic
Definition
The full system


Definable functions
More types of models


Second-order arithmetic
Models
Definable functions


Second-order arithmetic
Models
More types of models


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


Second-order arithmetic
Subsystems
Arithmetical comprehension


Second-order arithmetic
Subsystems
The arithmetical hierarchy for formulas


Second-order arithmetic
Subsystems
Recursive comprehension


Second-order arithmetic
Subsystems
Weaker systems


Second-order arithmetic
Subsystems
Stronger systems


Second-order arithmetic
Projective determinacy


Second-order arithmetic
Coding mathematics


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


Minimal axioms for Boolean algebra
References


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


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


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


Definition
Friedman's grand conjecture
Related systems


Elementary function arithmetic
Definition


Elementary function arithmetic
Friedman's grand conjecture


Elementary function arithmetic
Related systems


Expressive power
Idea of decidability
Complexity
Decidable extensions
Undecidable extensions


Skolem arithmetic
Undecidable extensions


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


Robinson arithmetic
Axioms
Variant axiomatizations


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


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


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


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


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


Example: Hilbert systems for two propositional logics


Rule of inference
Admissibility and derivability


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


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


Formal system
Background
Inference and entailment

Formal language
Deductive system
Logical system


Formal system
Background
Inference and entailment
Formal language


Formal system
Background
Inference and entailment
Deductive system


Formal system
Background
Inference and entailment
Logical system


Formal system
History
Formalism
Hilbert's program


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


Hilbert system
Formal deductions
Logical axioms


Existential quantification
Conjunction and disjunction


Hilbert system
Conservative extensions
Existential quantification


Hilbert system
Conservative extensions
Conjunction and disjunction


Hilbert system
Some useful theorems and their proofs


Hilbert system
Alternative axiomatizations


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


Classical propositional calculus systems


Classical propositional calculus systems

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


List of Hilbert systems
Implicational propositional calculus


List of Hilbert systems
Intuitionistic and intermediate logics


List of Hilbert systems
Positive implicational calculus


List of Hilbert systems
Positive propositional calculus


List of Hilbert systems
Equivalential calculus


History
Definition
Upper bound
Examples
Table of ordinal analyses


Ordinal analysis
Definition
Ordinal notations


Theories with proof-theoretic ordinal ω2

Theories with proof-theoretic ordinal ω3

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

Theories with proof-theoretic ordinal ωω

Theories with proof-theoretic ordinal ε0

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

Theories with proof-theoretic ordinal the Bachmann–Howard ordinal

Theories with larger proof-theoretic ordinals


Ordinal analysis
Examples
Theories with proof-theoretic ordinal ω


Theories with proof-theoretic ordinal ω2


Theories with proof-theoretic ordinal ω3


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


Theories with proof-theoretic ordinal ωω


Theories with proof-theoretic ordinal ε0


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


Theories with proof-theoretic ordinal the Bachmann–Howard ordinal


Theories with larger proof-theoretic ordinals


Ordinal analysis
Table of ordinal analyses


Ordinal analysis
Table of ordinal analyses
Key


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


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


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


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


Propositional satisfiability for classical logic


Satisfiability
Reduction of validity to satisfiability


Satisfiability
Satisfiability in first-order logic


Satisfiability
Satisfiability in model theory


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


Formal definition
Equivalent formulations
Examples
Properties
Remarks


Computably enumerable set
Formal definition


Computably enumerable set
Equivalent formulations


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


Truth value
Intuitionistic and constructive logic


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


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


Definition Bi-interpretability Example References


Interpretation (model theory)
Definition


Interpretation (model theory)
Bi-interpretability


Interpretation (model theory)
References


Semantic theory of truth
Tarski's theory of truth


Semantic theory of truth
Kripke's theory of truth


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


Direct limits of ultrapowers (ultralimits)


The inductive definition
Natural languages


History and motivation
Examples
Properties


Categorical theory
History and motivation


Truth predicate
Languages which allow a truth predicate


Elementary substructures and elementary extensions


Elementary equivalence
Elementarily equivalent structures


Elementary equivalence
Tarski–Vaught test


Elementary equivalence
Elementary embeddings


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


Induced substructures and closed subsets


Structure (mathematical logic)
Definition

Domain
Signature
Interpretation function
Examples


Structure (mathematical logic)
Definition
Domain


Structure (mathematical logic)
Definition
Signature


Structure (mathematical logic)
Definition
Interpretation function


Structure (mathematical logic)
Definition
Examples


Structure (mathematical logic)
Homomorphisms and embeddings

Homomorphisms
Embeddings
Example
Homomorphism problem


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphisms


Structure (mathematical logic)
Homomorphisms and embeddings
Embeddings


Structure (mathematical logic)
Homomorphisms and embeddings
Example


Structure (mathematical logic)
Homomorphisms and embeddings
Homomorphism problem


Structure (mathematical logic)
Structures and first-order logic

Satisfaction relation
Definable relations


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


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

Definability with parameters
Implicit definability


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


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


Structure (mathematical logic)
Many-sorted structures


Structure (mathematical logic)
Other generalizations

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


Structure (mathematical logic)
Other generalizations
Partial algebras


Structure (mathematical logic)
Other generalizations
Structures for typed languages


Structure (mathematical logic)
Other generalizations
Higher-order languages


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


Formal definition
Examples and non-examples
Properties


Computable set
Examples and non-examples


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


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


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


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


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


Sufficient condition for completeness of model-complete theories


Model complete theory
Model companion and model completion


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


Formal definition
Examples of types
Stone spaces
Omitting types theorem
References


Type (model theory)
Omitting types theorem


List of possible spectra of a countable theory


Definition
Motivation
Examples
Relationship to prime models
Notes
References


Saturated model
Relationship to prime models


Definition
Example
As subobjects
Submodel


Substructure (mathematics)
As subobjects


Blackboard bold
History
Use in textbooks


"Definition" as anticipation of definition


Well-defined expression
Independence of representative


Well-defined expression
Independence of representative
Functions with one argument


Well-defined expression
Independence of representative
Operations


Well-defined expression
Well-defined notation


Well-defined expression
Other uses of the term


Etymology
Historical development
Mathematical logic


Early Greeks
Modern development
Other sciences


Axiom
Historical development
Early Greeks


Axiom
Historical development
Modern development


Axiom
Historical development
Other sciences


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


Axiom
Mathematical logic
Logical axioms
Examples


Axiom
Mathematical logic
Non-logical axioms


Axiom
Mathematical logic
Non-logical axioms
Examples


Axiom
Mathematical logic
Role in mathematical logic


Axiom
Mathematical logic
Role in mathematical logic
Deductive systems and completeness


Axiom
Mathematical logic
Further discussion


Definition
Examples
Injections can be undone
Injections may be made invertible
Other properties
Proving that functions are injective
Gallery


Injective function
Injections can be undone


Injective function
Injections may be made invertible


Injective function
Proving that functions are injective


Informal presentation
Formal presentation
Philosophical implications
Set-theoretic responses
History
Applied versions
Applications and related topics
Related paradoxes


Russell's paradox
Philosophical implications


Russell's paradox
Set-theoretic responses


Russell's paradox
Applications and related topics


Russell's paradox
Applications and related topics
Russell-like paradoxes


History
Applications
In calculus
Properties
Representations
Computing the digits
In computer culture
References
Further reading
External links


Compound interest
Bernoulli trials
Standard normal distribution
Derangements
Optimal planning problems
Asymptotics


e (mathematical constant)
Applications
Compound interest


e (mathematical constant)
Applications
Bernoulli trials


e (mathematical constant)
Applications
Standard normal distribution


e (mathematical constant)
Applications
Derangements


e (mathematical constant)
Applications
Optimal planning problems


e (mathematical constant)
Applications
Asymptotics


e (mathematical constant)
In calculus
Alternative characterizations


Calculus
Inequalities
Exponential-like functions
Number theory
Complex numbers


e (mathematical constant)
Properties
Calculus


e (mathematical constant)
Properties
Inequalities


e (mathematical constant)
Properties
Exponential-like functions


e (mathematical constant)
Properties
Number theory


e (mathematical constant)
Properties
Complex numbers


e (mathematical constant)
Representations


e (mathematical constant)
Representations
Stochastic representations


e (mathematical constant)
Representations
Known digits


e (mathematical constant)
Computing the digits


e (mathematical constant)
In computer culture


e (mathematical constant)
Further reading


e (mathematical constant)
External links


Etymology
Properties
Definitions
n-tuples of m-sets
Type theory


Tuple
Etymology
Names for tuples of specific lengths


Tuples as functions
Tuples as nested ordered pairs
Tuples as nested sets


Tuple
Definitions
Tuples as nested ordered pairs


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


Theorem
Relation with scientific theories


Syntax and semantics
Interpretation of a formal theorem
Theorems and theories


Theorem
Theorems in logic
Syntax and semantics


Theorem
Theorems in logic
Interpretation of a formal theorem


Theorem
Theorems in logic
Theorems and theories


History
Examples
Usage
Oblique type compared to italics
More complex usage


Italic type
Oblique type compared to italics


Italics within italics
Left-leaning italics
Upright italics
Parentheses
Substitutes
OpenType
Web pages
Unicode


Italic type
More complex usage
Italics within italics


Italic type
More complex usage
Left-leaning italics


Italic type
More complex usage
Left-leaning italics
Iranic font style


Italic type
More complex usage
Upright italics


Italic type
More complex usage
Parentheses


Italic type
More complex usage
Substitutes


Italic type
More complex usage
Web pages


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


Auxiliary operations
Auxiliary order relations
Integers and fractions as real numbers


Real number
Arithmetic
Auxiliary operations


Real number
Arithmetic
Auxiliary order relations


Real number
Arithmetic
Integers and fractions as real numbers


Real number
Topological completeness
"The complete ordered field"


Axiomatic approach
Construction from the rational numbers


Real number
Formal definitions
Axiomatic approach


Real number
Formal definitions
Construction from the rational numbers


Real number
Applications and connections


Real number
Applications and connections
Physics


Real number
Applications and connections
Logic


Real number
Applications and connections
Computation


Real number
Applications and connections
Set theory


Real number
Generalizations and extensions


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


Batting line-up of a baseball or cricket team


Bijection
Examples
Seats and students of a classroom


Bijection
Generalization to partial functions


Family
Career
Mathematical work
Philosophical work
Philosophical legacy
Writings


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)

Introduction to Wissenschaftslehre
Metaphysics
Satz an Sich (proposition in itself)
Ideas and objects
Sensation and simple ideas
Logic
Truth
Judgments and cognitions


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Introduction to Wissenschaftslehre


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Metaphysics


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Satz an Sich (proposition in itself)


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Ideas and objects


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Sensation and simple ideas


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Logic


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Truth


Bernard Bolzano
Philosophical work
Wissenschaftslehre (Theory of Science)
Judgments and cognitions


Bernard Bolzano
Writings
Translations and compilations


Historical context
Foundational crisis
Toward resolution of the crisis


Foundations of mathematics
Historical context

Platonism as a philosophy of mathematics


Foundations of mathematics
Historical context
Ancient Greek mathematics


Platonism as a philosophy of mathematics


Foundations of mathematics
Historical context
Aristotelian realism


Foundations of mathematics
Historical context
Middle Ages and Renaissance


Foundations of mathematics
Historical context
19th century

Real analysis
Group theory
Non-Euclidean geometries
Projective geometry
Boolean algebra and logic
Peano arithmetic


Foundations of mathematics
Historical context
19th century
Real analysis


Foundations of mathematics
Historical context
19th century
Group theory


Foundations of mathematics
Historical context
19th century
Non-Euclidean geometries


Foundations of mathematics
Historical context
19th century
Projective geometry


Foundations of mathematics
Historical context
19th century
Boolean algebra and logic


Foundations of mathematics
Historical context
19th century
Peano arithmetic


Foundations of mathematics
Foundational crisis


Foundations of mathematics
Foundational crisis
Philosophical views

Philosophical consequences of Gödel's completeness theorem


Foundations of mathematics
Foundational crisis
Philosophical views
Formalism


Foundations of mathematics
Foundational crisis
Philosophical views
Intuitionism


Foundations of mathematics
Foundational crisis
Philosophical views
Logicism


Foundations of mathematics
Foundational crisis
Philosophical views
Set-theoretic Platonism


Foundations of mathematics
Foundational crisis
Philosophical views
Indispensability argument for realism


Foundations of mathematics
Foundational crisis
Philosophical views
Rough-and-ready realism


Philosophical consequences of Gödel's completeness theorem


Foundations of mathematics
Foundational crisis
More paradoxes


Foundations of mathematics
Toward resolution of the crisis


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


Cartesian complex plane
Polar complex plane
Complex graphs


Complex number
Visualization
Cartesian complex plane


Complex number
Visualization
Polar complex plane


Complex number
Visualization
Polar complex plane
Modulus and argument


Complex number
Visualization
Complex graphs


Multiplication and division in polar form


Complex number
Relations and operations
Equality


Complex number
Relations and operations
Ordering


Complex number
Relations and operations
Conjugate


Complex number
Relations and operations
Addition and subtraction


Complex number
Relations and operations
Multiplication and square


Complex number
Relations and operations
Reciprocal and division


Complex number
Relations and operations
Square root


Complex number
Relations and operations
Exponential function


Complex number
Relations and operations
Exponential function
Functional equation


Complex number
Relations and operations
Exponential function
Euler's formula


Complex number
Relations and operations
Complex logarithm


Complex number
Relations and operations
Exponentiation


Complex number
Relations and operations
Exponentiation
Integer and fractional exponents


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


Complex number
Properties
Field structure


Complex number
Properties
Solutions of polynomial equations


Complex number
Properties
Algebraic characterization


Complex number
Properties
Characterization as a topological field


Matrix representation of complex numbers


Complex number
Formal construction
Construction as ordered pairs


Complex number
Formal construction
Construction as a quotient field


Complex exponential and related functions


Complex number
Complex analysis
Holomorphic functions


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


Complex number
Applications
Geometry
Shapes


Complex number
Applications
Geometry
Fractal geometry


Complex number
Applications
Geometry
Triangles


Complex number
Applications
Algebraic number theory


Complex number
Applications
Analytic number theory


Complex number
Applications
Improper integrals


Complex number
Applications
Dynamic equations


Complex number
Applications
Linear algebra


Complex number
Applications
In applied mathematics


Complex number
Applications
In applied mathematics
Control theory


Complex number
Applications
In applied mathematics
Signal analysis


Electromagnetism and electrical engineering


Complex number
Applications
In physics
Fluid dynamics


Complex number
Applications
In physics
Quantum mechanics


Complex number
Applications
In physics
Relativity


Complex number
Generalizations and related notions


Informal presentation
Formal presentation
Philosophical implications
Set-theoretic responses
History
Applied versions
Applications and related topics
Related paradoxes


Russell's paradox
Philosophical implications


Russell's paradox
Set-theoretic responses


Russell's paradox
Applications and related topics


Russell's paradox
Applications and related topics
Russell-like paradoxes


Sets
Notation and terminology
Cardinality of sets
Examples
Formal relation


Element (mathematics)
Notation and terminology


Element (mathematics)
Cardinality of sets


History
How sets are defined and set notation
Membership
The empty set
Singleton sets
Subsets
Euler and Venn diagrams
Special sets of numbers in mathematics
Functions
Cardinality
Power sets
Partitions
Basic operations
Applications
Principle of inclusion and exclusion


Set (mathematics)
History
Naive set theory


Set (mathematics)
History
Axiomatic set theory


Set (mathematics)
How sets are defined and set notation

Roster notation
Semantic definition
Set-builder notation
Classifying methods of definition


Set (mathematics)
How sets are defined and set notation
Roster notation


Set (mathematics)
How sets are defined and set notation
Roster notation
Infinite sets in roster notation


Set (mathematics)
How sets are defined and set notation
Semantic definition


Set (mathematics)
How sets are defined and set notation
Set-builder notation


Set (mathematics)
How sets are defined and set notation
Classifying methods of definition


Set (mathematics)
Euler and Venn diagrams


Set (mathematics)
Special sets of numbers in mathematics


Infinite sets and infinite cardinality
The continuum hypothesis


Set (mathematics)
Cardinality
Infinite sets and infinite cardinality


Set (mathematics)
Cardinality
The continuum hypothesis


Set (mathematics)
Principle of inclusion and exclusion


Examples
Properties
Field
Related fields
Algebraic integers
Special classes
Notes
References


Numbers defined by radicals
Closed-form number


Algebraic number
Related fields
Numbers defined by radicals


Algebraic number
Related fields
Closed-form number


Mathematical object
List of mathematical objects by branch


Relation between Euler and Venn diagrams


Relation between Euler and Venn diagrams

Example: Euler- to Venn-diagram and Karnaugh map


Example: Euler- to Venn-diagram and Karnaugh map


Properties
In category theory
Definition by indicator functions
Definition in Principia Mathematica


Singleton (mathematics)
In category theory


Singleton (mathematics)
Definition by indicator functions


Singleton (mathematics)
Definition in Principia Mathematica


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


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


Set theory
Areas of study
Combinatorial set theory


Set theory
Areas of study
Descriptive set theory


Set theory
Areas of study
Fuzzy set theory


Set theory
Areas of study
Inner model theory


Set theory
Areas of study
Large cardinals


Set theory
Areas of study
Cardinal invariants


Set theory
Areas of study
Set-theoretic topology