α-inaccessible cardinals and hyper-inaccessible cardinals


Two model-theoretic characterisations of inaccessibility


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


Gödel numbering
Gödel's encoding
Example


Gödel numbering
Application to formal arithmetic

Expressing statements and proofs by numbers


Gödel numbering
Application to formal arithmetic
Recursion


Expressing statements and proofs by numbers


Necessary and sufficient conditions for finiteness


Finite set
Set-theoretic definitions of finiteness


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


The generalized continuum hypothesis holds in L

Constructible sets are definable from the ordinals


Constructible universe
Additional facts about the sets Lα


Constructible universe
L is a standard inner model of ZFC


Constructible universe
L is absolute and minimal


Constructible universe
L is absolute and minimal
L and large cardinals


Constructible universe
L can be well-ordered


Constructible universe
L has a reflection principle


The generalized continuum hypothesis holds in L


Constructible sets are definable from the ordinals


Constructible universe
Relative constructibility


Definition
Applications and interpretations
History
Philosophical perspectives


Finite and low cardinality stages of the hierarchy


Von Neumann universe
Applications and interpretations

Applications of V as models for set theories

Interpretation of V as the "set of all sets"


Applications of V as models for set theories


Interpretation of V as the "set of all sets"


Von Neumann universe
Applications and interpretations
V and the axiom of regularity


Von Neumann universe
Applications and interpretations
The existential status of V


Von Neumann universe
Philosophical perspectives


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


Inhabited set
Comparison with nonempty sets


Formal definition
Examples and non-examples
Properties


Computable set
Examples and non-examples


Combinatorics
Set theory
Computability and complexity theory


Listing
Countable vs. uncountable
Ordinals
Comparison of cardinalities


Enumeration
Set theory
Countable vs. uncountable


Enumeration
Set theory
Countable vs. uncountable
Examples


Enumeration
Set theory
Countable vs. uncountable
Properties


Enumeration
Set theory
Comparison of cardinalities


Enumeration
Computability and complexity theory


Schröder–Bernstein theorem
Proof
Another proof


Schröder–Bernstein theorem
Proof
Examples


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


Partial definition
Hierarchy of consistency strength
Motivations and epistemic status


Large cardinal
Hierarchy of consistency strength


Large cardinal
Motivations and epistemic status


Grothendieck universes and inaccessible cardinals


Examples
Applications
Category theoretic view
Relation with equality


Logarithm and exponential
Integers modulo 6
Relation-preserving isomorphism


Isomorphism
Examples
Logarithm and exponential


Isomorphism
Examples
Relation-preserving isomorphism


Isomorphism
Category theoretic view
Isomorphism vs. bijective morphism


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


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)


Definition
Inverse image
Notation for image and inverse image
Examples
Properties


Image of an element
Image of a subset
Image of a function
Generalization to binary relations


Image (mathematics)
Definition
Image of an element


Image (mathematics)
Definition
Image of a subset


Image (mathematics)
Definition
Image of a function


Image (mathematics)
Definition
Generalization to binary relations


Image (mathematics)
Notation for image and inverse image

Arrow notation
Star notation
Other terminology


Image (mathematics)
Notation for image and inverse image
Arrow notation


Image (mathematics)
Notation for image and inverse image
Star notation


Image (mathematics)
Notation for image and inverse image
Other terminology


General
Multiple functions
Multiple subsets of domain or codomain


Image (mathematics)
Properties
Multiple functions


Image (mathematics)
Properties
Multiple subsets of domain or codomain


Examples
Properties
Transitive closure
Transitive models of set theory


Transitive set
Transitive models of set theory


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


Ontology
Axioms
Discussion
History
Footnotes
References
External links


Metamathematics
Peano arithmetic
Infinite sets


General set theory
Discussion
Metamathematics


General set theory
Discussion
Peano arithmetic


General set theory
Discussion
Infinite sets


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


Morse–Kelley set theory
MK axioms and ontology


Morse–Kelley set theory
Discussion
Model theory


Morse–Kelley set theory
Discussion
History


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


Operation (mathematics)
Types of operation


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


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


NP (complexity)
Background
Verifier-based definition


NP (complexity)
Background
Machine-definition


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


NP (complexity)
Equivalence of definitions


NP (complexity)
Relationship to other classes


P
Integer factorization
NP-complete problems


NP (complexity)
Examples
Integer factorization


NP (complexity)
Examples
NP-complete problems

Boolean satisfiability
Travelling salesman
Subgraph isomorphism


NP (complexity)
Examples
NP-complete problems
Boolean satisfiability


NP (complexity)
Examples
NP-complete problems
Travelling salesman


NP (complexity)
Examples
NP-complete problems
Subgraph isomorphism


Pure existence proofs of polynomial-time algorithms


P (complexity)
Relationships to other classes


P (complexity)
Alternative characterizations


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


Formal definition
Examples and non-examples
Properties


Computable set
Examples and non-examples


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


Example: the halting problem in computability theory

Relationship with Gödel's incompleteness theorem


Example: the halting problem in computability theory


Relationship with Gödel's incompleteness theorem


Undecidable problem
Examples of undecidable problems


Undecidable problem
Examples of undecidable statements


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


Formal definition
Equivalent formulations
Examples
Properties
Remarks


Computably enumerable set
Formal definition


Computably enumerable set
Equivalent formulations


Definition
Examples
Decidability
Complete problems
Function problems
Optimization problems


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


Decidability (logic)
Decidability of a logical system


Decidability (logic)
Decidability of a theory


Decidability (logic)
Some decidable theories


Decidability (logic)
Some undecidable theories


Decidability (logic)
Relationship with completeness


Decidability (logic)
Relationship to computability


Decidability (logic)
In context of games


Implicit structure of concrete categories


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


Converting predicates to numeric functions

Operations on integers and rational numbers

Some common primitive recursive functions


Primitive recursive function
Examples
Addition


Primitive recursive function
Examples
Doubling


Primitive recursive function
Examples
Multiplication


Primitive recursive function
Examples
Predecessor


Primitive recursive function
Examples
Truncated subtraction


Converting predicates to numeric functions


Primitive recursive function
Examples
Predicate "Is zero"


Primitive recursive function
Examples
Predicate "Less or equal"


Primitive recursive function
Examples
Predicate "Greater or equal"


Primitive recursive function
Examples
If-then-else


Primitive recursive function
Examples
Junctors


Primitive recursive function
Examples
Equality predicate


Primitive recursive function
Examples
Other operations on natural numbers


Operations on integers and rational numbers


Some common primitive recursive functions


Primitive recursive function
Use in first-order Peano arithmetic


Primitive recursive function
Elimination of parameters


Primitive recursive function
Relationship to recursive functions


Primitive recursive function
Limitations


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


Primitive recursive function
Variants
Constant functions


Primitive recursive function
Variants
Weak primitive recursion


Primitive recursive function
Variants
Iterative functions


Primitive recursive function
Variants
Additional primitive recursive forms


Primitive recursive function
Variants
Computer language definition


Primitive recursive function
Finitism and consistency results


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


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


Recursion
In mathematics
Recursively defined sets

Example: the natural numbers
Example: Proof procedure


Recursion
In mathematics
Recursively defined sets
Example: the natural numbers


Recursion
In mathematics
Recursively defined sets
Example: Proof procedure


Recursion
In mathematics
Finite subdivision rules


Recursion
In mathematics
Functional recursion


Recursion
In mathematics
Proofs involving recursive definitions


Recursion
In mathematics
Recursive optimization


Recursion
In mathematics
The recursion theorem


Recursion
In mathematics
The recursion theorem
Proof of uniqueness


Additional details required to visualize or implement Turing machines


Turing machine
Overview
Physical description


Additional details required to visualize or implement Turing machines

Alternative definitions
The "state"
"State" diagrams


Turing machine
Choice c-machines, oracle o-machines


Turing machine
Universal Turing machines


Turing machine
Comparison with real machines


Turing machine
Comparison with real machines
Limitations

Computational complexity theory
Interaction


Turing machine
Comparison with real machines
Limitations
Computational complexity theory


Turing machine
Comparison with real machines
Limitations
Interaction


Historical background: computational machinery

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

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


Historical background: computational machinery


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


Turing machine
History
Alan Turing's a-machine


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


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


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


Category theory
Categories, objects, and morphisms


Category theory
Categories, objects, and morphisms
Categories


Category theory
Categories, objects, and morphisms
Morphisms


Universal constructions, limits, and colimits


Category theory
Other concepts
Equivalent categories


Category theory
Other concepts
Further concepts and results


Category theory
Other concepts
Higher-dimensional categories


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


Kolmogorov complexity
Invariance theorem

Informal treatment
A more formal treatment


Kolmogorov complexity
Invariance theorem
Informal treatment


Kolmogorov complexity
Invariance theorem
A more formal treatment


Kolmogorov complexity
History and context


Uncomputability of Kolmogorov complexity


Uncomputability of Kolmogorov complexity

A naive attempt at a program to compute K


A naive attempt at a program to compute K


Kolmogorov complexity
Basic results
Chain rule for Kolmogorov complexity


Kolmogorov complexity
Chaitin's incompleteness theorem


Kolmogorov complexity
Minimum message length


Kolmogorov complexity
Kolmogorov randomness


Kolmogorov complexity
Relation to entropy


Kolmogorov complexity
Conditional versions


Lambda calculus and programming languages


Lambda calculus
Explanation and applications


Lambda calculus
History
Origin of the λ symbol


Lambda calculus
Informal description
Motivation


Lambda calculus
Informal description
The lambda calculus

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


Lambda calculus
Informal description
The lambda calculus
Lambda terms


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


Lambda calculus
Informal description
The lambda calculus
Alpha equivalence


Lambda calculus
Informal description
The lambda calculus
Free variables


Lambda calculus
Informal description
The lambda calculus
Capture-avoiding substitutions


Lambda calculus
Informal description
The lambda calculus
β-reduction


Definition
Notation
Free and bound variables


Lambda calculus
Formal definition
Definition


Lambda calculus
Formal definition
Notation


Lambda calculus
Formal definition
Free and bound variables


Lambda calculus
Reduction
α-conversion
Substitution


Lambda calculus
Normal forms and confluence


Arithmetic in lambda calculus
Logic and predicates
Pairs


Lambda calculus
Encoding datatypes
Arithmetic in lambda calculus


Lambda calculus
Encoding datatypes
Logic and predicates


Lambda calculus
Encoding datatypes
Pairs


Lambda calculus
Additional programming techniques

Named constants
Recursion and fixed points
Standard terms
Abstraction elimination


Lambda calculus
Additional programming techniques
Named constants


Lambda calculus
Additional programming techniques
Recursion and fixed points


Lambda calculus
Additional programming techniques
Standard terms


Lambda calculus
Additional programming techniques
Abstraction elimination


Lambda calculus and programming languages

Anonymous functions
Parallelism and concurrency


Lambda calculus
Variations and extensions


Proof theory and constructive mathematics


Mathematical logic
History
Early history


Mathematical logic
History
19th century
Foundational theories


Set theory and paradoxes
Symbolic logic
Beginnings of the other branches


Mathematical logic
History
20th century
Set theory and paradoxes


Mathematical logic
History
20th century
Symbolic logic


Mathematical logic
History
20th century
Beginnings of the other branches


Mathematical logic
Formal logical systems

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


Mathematical logic
Formal logical systems
First-order logic


Mathematical logic
Formal logical systems
Other classical logics


Mathematical logic
Formal logical systems
Nonclassical and modal logic


Mathematical logic
Formal logical systems
Algebraic logic


Mathematical logic
Recursion theory
Algorithmically unsolvable problems


Mathematical logic
Connections with computer science


Mathematical logic
Foundations of mathematics


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


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


History of logic
Logic in the East
Logic in India


History of logic
Logic in the East
Hindu logic


History of logic
Logic in the East
Origin


History of logic
Logic in the East
Before Gautama


History of logic
Logic in the East
Dattatreya


History of logic
Logic in the East
Medhatithi Gautama


History of logic
Logic in the East
Panini


History of logic
Logic in the East
Nyaya-Vaisheshika


History of logic
Logic in the East
Jain Logic


History of logic
Logic in the East
Buddhist logic


History of logic
Logic in the East
Nagarjuna


History of logic
Logic in the East
Dignaga


History of logic
Logic in the East
Syllogism and influence


History of logic
Logic in the East
Logic in China


Prehistory of logic
Ancient Greece before Aristotle
Aristotle
Stoics


History of logic
Logic in the West
Prehistory of logic


History of logic
Logic in the West
Ancient Greece before Aristotle

Thales
Pythagoras
Heraclitus and Parmenides
Plato


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


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


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


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


History of logic
Logic in the West
Aristotle


History of logic
Logic in the West
Aristotle
The Organon


History of logic
Logic in the West
Stoics


Logic in the Middle East
Logic in medieval Europe


History of logic
Medieval logic
Logic in the Middle East


History of logic
Medieval logic
Logic in medieval Europe


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


History of logic
Traditional logic
The textbook tradition


History of logic
Traditional logic
Logic in Hegel's philosophy


History of logic
Traditional logic
Logic and psychology


Embryonic period
Algebraic period
Logicist period
Metamathematical period
Logic after WWII


History of logic
Modern logic
Embryonic period


History of logic
Modern logic
Algebraic period


History of logic
Modern logic
Logicist period


History of logic
Modern logic
Metamathematical period


History of logic
Modern logic
Logic after WWII


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


Category (mathematics)
Small and large categories


Category (mathematics)
Construction of new categories


Category (mathematics)
Construction of new categories
Dual category


Category (mathematics)
Construction of new categories
Product categories


Category (mathematics)
Types of morphisms


Category (mathematics)
Types of categories


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


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


P versus NP problem
Consequences of solution


P versus NP problem
Consequences of solution
P = NP


P versus NP problem
Consequences of solution
P ≠ NP


P versus NP problem
Results about difficulty of proof


P versus NP problem
Logical characterizations


P versus NP problem
Polynomial-time algorithms


P versus NP problem
Formal definitions
P and NP


P versus NP problem
Formal definitions
P and NP
Example


P versus NP problem
Formal definitions
NP-completeness


Formal statement
Use
Equivalent statements
Relation with other axioms
Notes
References


Axiom of dependent choice
Formal statement


Axiom of dependent choice
Equivalent statements


Axiom of dependent choice
Relation with other axioms


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

The unit class, impredicativity, and the vicious circle principle


Logicism
Intent, or goal, of logicism
History


Logicism
Epistemology, ontology and logicism


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

Preliminaries
The definition of the natural numbers
Criticism


The unit class, impredicativity, and the vicious circle principle

A solution to impredicativity: a hierarchy of types


A solution to impredicativity: a hierarchy of types


Interpretability of arithmetic
References


Axiom of adjunction
Interpretability of arithmetic


Timeline of mathematical logic
19th century


Timeline of mathematical logic
20th century


Timeline of mathematical logic
20th century
1950-1999


History
Major themes
Contemporary schools of thought
Arguments
Aesthetics
Journals


Philosophy of mathematics
History
Contemporary philosophy


Mathematical realism
Mathematical anti-realism


Philosophy of mathematics
Major themes
Mathematical realism


Philosophy of mathematics
Major themes
Mathematical anti-realism


Philosophy of mathematics
Contemporary schools of thought

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


Philosophy of mathematics
Contemporary schools of thought
Artistic


Philosophy of mathematics
Contemporary schools of thought
Platonism


Philosophy of mathematics
Contemporary schools of thought
Mathematicism


Philosophy of mathematics
Contemporary schools of thought
Logicism


Philosophy of mathematics
Contemporary schools of thought
Formalism


Philosophy of mathematics
Contemporary schools of thought
Conventionalism


Philosophy of mathematics
Contemporary schools of thought
Intuitionism


Philosophy of mathematics
Contemporary schools of thought
Intuitionism
Constructivism


Philosophy of mathematics
Contemporary schools of thought
Intuitionism
Finitism


Philosophy of mathematics
Contemporary schools of thought
Structuralism


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories

Aristotelian realism
Psychologism
Empiricism


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


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories
Psychologism


Philosophy of mathematics
Contemporary schools of thought
Embodied mind theories
Empiricism


Philosophy of mathematics
Contemporary schools of thought
Fictionalism


Philosophy of mathematics
Contemporary schools of thought
Social constructivism


Philosophy of mathematics
Contemporary schools of thought
Beyond the traditional schools

Popper's two senses of number statements


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


Popper's two senses of number statements


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


Indispensability argument for realism
Epistemic argument against realism


Philosophy of mathematics
Arguments
Indispensability argument for realism


Philosophy of mathematics
Arguments
Epistemic argument against realism


Zeno
Thomson
Benacerraf
Modern literature


Supertask
History
Zeno
Achilles and the tortoise


Philosophy of mathematics
Physical possibility
Super Turing machines


Supertask
History
Modern literature
Philosophy of mathematics


Supertask
History
Modern literature
Physical possibility


Supertask
History
Modern literature
Super Turing machines


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


Supertask
Prominent supertasks
Ross–Littlewood paradox


Supertask
Prominent supertasks
Benardete's paradox


Supertask
Prominent supertasks
Grim Reaper paradox


Supertask
Prominent supertasks
Davies' super-machine


Incompatibility of the axiom of determinacy with the axiom of choice

Infinitary logic and the axiom of determinacy

Large cardinals and the axiom of determinacy


Axiom of determinacy
Types of game that are determined


Incompatibility of the axiom of determinacy with the axiom of choice


Infinitary logic and the axiom of determinacy


Large cardinals and the axiom of determinacy


Axiom of determinacy
Projective ordinals


Implications
In arithmetic
Significance
Statements true in L
References
External links


Axiom of constructibility
Statements true in L


Axiom of constructibility
External links


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


Axiom of extensionality
Formal statement


Axiom of extensionality
In predicate logic without equality


Axiom of extensionality
In set theory with ur-elements


Extracting the natural numbers from the infinite set


Axiom of infinity
Interpretation and consequences


Axiom of infinity
An apparently weaker version


Zermelo's models and the axiom of limitation of size


Axiom of limitation of size
Formal statement


Axiom of limitation of size
Implications of the axiom


Zermelo's models and the axiom of limitation of size

The models Vκ where κ is a strongly inaccessible cardinal


The models Vκ where κ is a strongly inaccessible cardinal


Axiom of limitation of size
Limitation of size doctrine


Axiom of limitation of size
Bibliography


Formal statement
Consequences
Alternatives
References


Axiom of pairing
Alternatives
Non-independence


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

Regularity, the cumulative hierarchy, and types

Regularity in the presence of urelements


Axiom of regularity
Elementary implications of regularity

No infinite descending sequence of sets exists

Simpler set-theoretic definition of the ordered pair

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


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


No infinite descending sequence of sets exists


Simpler set-theoretic definition of the ordered pair


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


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


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


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


Axiom of regularity
Regularity and Russell's paradox


Regularity, the cumulative hierarchy, and types


Regularity in the presence of urelements


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


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


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


Example
Set theory definition
Category theory point of view


Disjoint union
Category theory point of view


Statement
Applications
Relation to other axiom schemas
History
References


Axiom schema of replacement
Applications


Axiom schema of replacement
Relation to other axiom schemas


Axiom schema of replacement
Relation to other axiom schemas
Collection


Axiom schema of replacement
Relation to other axiom schemas
Separation


Relation to the axiom schema of replacement


Axiom schema of specification
Unrestricted comprehension


Axiom schema of specification
In NBG class theory


Axiom schema of specification
In higher-order settings


Axiom schema of specification
In Quine's New Foundations


Axiom schema of specification
References


Axiom schema of specification
References
Citations


History
Motivation
Formal definition
Cardinal arithmetic
The continuum hypothesis


Successor cardinal
Cardinal addition
Cardinal multiplication
Cardinal exponentiation


Cardinal number
Cardinal arithmetic
Successor cardinal


Cardinal number
Cardinal arithmetic
Cardinal addition


Cardinal number
Cardinal arithmetic
Cardinal addition
Subtraction


Cardinal number
Cardinal arithmetic
Cardinal multiplication


Cardinal number
Cardinal arithmetic
Cardinal multiplication
Division


Cardinal number
Cardinal arithmetic
Cardinal exponentiation


Cardinal number
Cardinal arithmetic
Cardinal exponentiation
Roots


Cardinal number
Cardinal arithmetic
Cardinal exponentiation
Logarithms


Cardinal number
The continuum hypothesis


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


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


Zermelo set theory
The axioms of Zermelo set theory


Zermelo set theory
Connection with standard set theory


Zermelo set theory
The aim of Zermelo's paper


Zermelo set theory
The axiom of separation


Induction by cases
Transfinite recursion
Relationship to the axiom of choice


Transfinite induction
Induction by cases


Transfinite induction
Transfinite recursion


Transfinite induction
Relationship to the axiom of choice


Preliminaries, notation, and basic notions

Set theoretic properties and constructions


Preliminaries, notation, and basic notions


Filter (set theory)
Filters and prefilters

Finer/coarser, subordination, and meshing


Filter (set theory)
Filters and prefilters
Basic examples


Filter (set theory)
Filters and prefilters
Ultrafilters


Filter (set theory)
Filters and prefilters
Kernels

Classifying families by their kernels
Characterizing fixed ultra prefilters


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


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


Finer/coarser, subordination, and meshing


Set theoretic properties and constructions

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


Subordination is preserved by images and preimages


Nets to prefilters
Prefilters to nets
Subordinate filters and subnets


Filter (set theory)
Filters and nets
Nets to prefilters


Filter (set theory)
Filters and nets
Prefilters to nets


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


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

Non–equivalence of subnets and subordinate filters


Non–equivalence of subnets and subordinate filters


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


Burali-Forti paradox
Stated in terms of von Neumann ordinals


Burali-Forti paradox
Stated more generally


Burali-Forti paradox
Resolutions of the paradox


Comparison with the usual definition of infinite set

Proof of equivalence to infinity, assuming axiom of countable choice


Comparison with the usual definition of infinite set


Dedekind-infinite set
Dedekind-infinite sets in ZF


Dedekind-infinite set
Relation to the axiom of choice


Proof of equivalence to infinity, assuming axiom of countable choice


Existence
Additional properties
Variations
References


Biography
Mathematician
Awards
Published works


Relationship with astrology and esotericism


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


Mathematics
Areas of mathematics
Number theory


Mathematics
Areas of mathematics
Geometry


Mathematics
Areas of mathematics
Algebra


Mathematics
Areas of mathematics
Calculus and analysis


Mathematics
Areas of mathematics
Discrete mathematics


Mathematics
Areas of mathematics
Mathematical logic and set theory


Mathematics
Areas of mathematics
Statistics and other decision sciences


Mathematics
Areas of mathematics
Computational mathematics


Mathematics
Symbolic notation and terminology


Pure and applied mathematics
Unreasonable effectiveness
Specific sciences


Mathematics
Relationship with sciences
Pure and applied mathematics


Mathematics
Relationship with sciences
Unreasonable effectiveness


Mathematics
Relationship with sciences
Specific sciences

Physics
Computing
Biology and chemistry
Earth sciences
Social sciences


Mathematics
Relationship with sciences
Specific sciences
Physics


Mathematics
Relationship with sciences
Specific sciences
Computing


Mathematics
Relationship with sciences
Specific sciences
Biology and chemistry


Mathematics
Relationship with sciences
Specific sciences
Earth sciences


Mathematics
Relationship with sciences
Specific sciences
Social sciences


Relationship with astrology and esotericism


Mathematics
Philosophy
Proposed definitions


Psychology (aesthetic, creativity and intuition)


Mathematics
Training and practice
Education


Mathematics
Cultural impact
Artistic expression


Mathematics
Cultural impact
Popularization


Life
Work
Bibliography
References
External links


Biography
Work
In popular culture
Bibliography
Filmography


Willard Van Orman Quine
Biography
Education


Willard Van Orman Quine
Biography
World War II


Willard Van Orman Quine
Biography
Personal


Willard Van Orman Quine
Biography
Harvard


Logic
Set theory
Metaphysics
Epistemology


Willard Van Orman Quine
Work
Metaphysics

Rejection of the analytic–synthetic distinction

Confirmation holism and ontological relativity

Indispensability argument for mathematical realism


Rejection of the analytic–synthetic distinction


Confirmation holism and ontological relativity


Willard Van Orman Quine
Work
Metaphysics
Existence and its contrary


Willard Van Orman Quine
Work
Metaphysics
Ontological commitment


Indispensability argument for mathematical realism


Willard Van Orman Quine
Work
Epistemology


Willard Van Orman Quine
In popular culture


Willard Van Orman Quine
Bibliography
Selected books


Willard Van Orman Quine
Bibliography
Important articles


Life
Research in set theory
Zermelo's navigation problem
Publications


Ernst Zermelo
Zermelo's navigation problem


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


Definition
Legal matters
Usage
Open content


Free content
Legal matters
Public domain


Media
Software
Engineering and technology
Academia
Legislation


Free content
Usage
Engineering and technology


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


Free content
Open content
"Open content" definition


Free content
Open content
Open content and education


Biography
Mathematical work
Publications
Notes
References
External links


Life and education
Career and private life
Mathematics
Physics
Economics
Computer science
Defense work
Personality
Recognition
Legacy
Selected works


Family background
Child prodigy
University studies


John von Neumann
Life and education
Family background


John von Neumann
Life and education
Child prodigy


John von Neumann
Life and education
University studies


John von Neumann
Career and private life


John von Neumann
Career and private life
Illness and death


Set theory
Proof theory
Ergodic theory
Measure theory
Topological groups
Functional analysis
Operator algebras
Lattice theory
Mathematical statistics
Other work


John von Neumann
Mathematics
Set theory
Von Neumann paradox


John von Neumann
Mathematics
Proof theory


John von Neumann
Mathematics
Ergodic theory


John von Neumann
Mathematics
Measure theory


John von Neumann
Mathematics
Topological groups


John von Neumann
Mathematics
Functional analysis


John von Neumann
Mathematics
Operator algebras


John von Neumann
Mathematics
Lattice theory


John von Neumann
Mathematics
Mathematical statistics


Quantum mechanics
Fluid dynamics
Other work


John von Neumann
Physics
Quantum mechanics

Von Neumann entropy
Quantum mutual information
Density matrix
Von Neumann measurement scheme
Quantum logic


John von Neumann
Physics
Quantum mechanics
Von Neumann entropy


John von Neumann
Physics
Quantum mechanics
Quantum mutual information


John von Neumann
Physics
Quantum mechanics
Density matrix


John von Neumann
Physics
Quantum mechanics
Von Neumann measurement scheme


John von Neumann
Physics
Quantum mechanics
Quantum logic


Game theory
Mathematical economics
Linear programming


John von Neumann
Economics
Mathematical economics


John von Neumann
Economics
Linear programming


Cellular automata, DNA and the universal constructor

Scientific computing and numerical analysis


Cellular automata, DNA and the universal constructor


Scientific computing and numerical analysis


John von Neumann
Computer science
Weather systems and global warming


John von Neumann
Computer science
Technological singularity hypothesis


Manhattan Project
Post war
Atomic Energy Commission
Mutual assured destruction
Consultancies


John von Neumann
Defense work
Manhattan Project


John von Neumann
Defense work
Atomic Energy Commission


John von Neumann
Defense work
Mutual assured destruction


John von Neumann
Defense work
Consultancies


John von Neumann
Personality
Mathematical style


John von Neumann
Recognition
Cognitive abilities


John von Neumann
Recognition
Eidetic memory


Mastery of mathematics
Honors and awards


John von Neumann
Legacy
Mastery of mathematics


John von Neumann
Legacy
Honors and awards


Books authored / coauthored
Scholarly articles
Popular articles
Collected works


John von Neumann
Selected works
Books authored / coauthored


John von Neumann
Selected works
Scholarly articles


John von Neumann
Selected works
Popular articles


John von Neumann
Selected works
Collected works


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


Kurt Gödel
Early life and education
Childhood


Kurt Gödel
Early life and education
Studies in Vienna


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


Kurt Gödel
Career
Incompleteness theorems


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


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


Kurt Gödel
Bibliography
Important publications


Etymology
Characteristics
Pre-modern encyclopedias
Printed encyclopedias
Digital encyclopedias


Differences between encyclopedias and dictionaries


Encyclopedia
Characteristics
Four major elements


Encyclopedia
Characteristics
Encyclopedic dictionaries


Encyclopédie
Encyclopædia Britannica
Brockhaus Enzyklopädie
Encyclopedias in the US


Encyclopedia
Printed encyclopedias
Encyclopédie


Encyclopedia
Printed encyclopedias
Encyclopædia Britannica


Encyclopedia
Printed encyclopedias
Brockhaus Enzyklopädie


Encyclopedia
Printed encyclopedias
Encyclopedias in the US


Encyclopedia
Digital encyclopedias
Physical media


Encyclopedia
Digital encyclopedias
Online


Encyclopedia
Digital encyclopedias
Online
Free encyclopedias


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


Tropical cyclone scales
Atlantic, Eastern and Central Pacific


Tropical cyclone scales
North Indian Ocean


Tropical cyclone scales
South-West Indian Ocean


Tropical cyclone scales
Australia and Fiji


Tropical cyclone scales
Alternative scales


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


Martin's axiom
Equivalent forms of MA(𝛋)


Formal definition
Discussion
Axiomatizations
Graph models


Hereditarily finite set
Formal definition


Hereditarily finite set
Discussion
Ackermann coding


Hereditarily finite set
Discussion
Representation


Hereditarily finite set
Axiomatizations
Theories of finite sets


Hereditarily finite set
Axiomatizations
ZF


Galileo's paradox
Galileo on infinite sets


Preliminaries, notation, and basic notions

Set theoretic properties and constructions


Preliminaries, notation, and basic notions


Filter (set theory)
Filters and prefilters

Finer/coarser, subordination, and meshing


Filter (set theory)
Filters and prefilters
Basic examples


Filter (set theory)
Filters and prefilters
Ultrafilters


Filter (set theory)
Filters and prefilters
Kernels

Classifying families by their kernels
Characterizing fixed ultra prefilters


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


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


Finer/coarser, subordination, and meshing


Set theoretic properties and constructions

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


Subordination is preserved by images and preimages


Nets to prefilters
Prefilters to nets
Subordinate filters and subnets


Filter (set theory)
Filters and nets
Nets to prefilters


Filter (set theory)
Filters and nets
Prefilters to nets


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


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

Non–equivalence of subnets and subordinate filters


Non–equivalence of subnets and subordinate filters


Notable theories on the structure of concepts


Concepts as mental representations
Concepts as abstract objects


Concept
Ontology of concepts
Concepts as mental representations

The psychological view of concepts
The physicalist view of concepts


Concept
Ontology of concepts
Concepts as mental representations
The psychological view of concepts


Concept
Ontology of concepts
Concepts as mental representations
The physicalist view of concepts


Concept
Ontology of concepts
Concepts as abstract objects

A priori concepts
Embodied content
Realist universal concepts
Sense and reference
Concepts in calculus


Concept
Ontology of concepts
Concepts as abstract objects
A priori concepts


Concept
Ontology of concepts
Concepts as abstract objects
Embodied content


Concept
Ontology of concepts
Concepts as abstract objects
Realist universal concepts


Concept
Ontology of concepts
Concepts as abstract objects
Sense and reference


Concept
Ontology of concepts
Concepts as abstract objects
Concepts in calculus


Notable theories on the structure of concepts

Classical theory
Prototype theory
Theory-theory


Definition
Conceptions of deduction
Rules of inference
Validity and soundness
Difference from ampliative reasoning
In various fields
History
Related concepts and theories


Deductive reasoning
Conceptions of deduction


Prominent rules of inference
Fallacies
Definitory and strategic rules


Deductive reasoning
Rules of inference
Prominent rules of inference

Modus ponens
Modus tollens
Hypothetical syllogism


Deductive reasoning
Rules of inference
Prominent rules of inference
Modus ponens


Deductive reasoning
Rules of inference
Prominent rules of inference
Modus tollens


Deductive reasoning
Rules of inference
Prominent rules of inference
Hypothetical syllogism


Deductive reasoning
Rules of inference
Fallacies


Deductive reasoning
Rules of inference
Definitory and strategic rules


Deductive reasoning
Validity and soundness


Deductive reasoning
Difference from ampliative reasoning


Cognitive psychology
Epistemology
Probability logic


Deductive reasoning
In various fields
Cognitive psychology

Psychological theories of deductive reasoning


Psychological theories of deductive reasoning


Deductive reasoning
In various fields
Cognitive psychology
Intelligence


Deductive reasoning
In various fields
Epistemology


Deductive reasoning
In various fields
Probability logic


Deductive reasoning
Related concepts and theories

Deductivism
Natural deduction
Geometrical method


Deductive reasoning
Related concepts and theories
Deductivism


Deductive reasoning
Related concepts and theories
Natural deduction


Deductive reasoning
Related concepts and theories
Geometrical method


Heuristic mathematics and experimental mathematics


Mathematical proof
History and etymology


Direct proof
Proof by mathematical induction
Proof by contraposition
Proof by contradiction
Proof by construction
Proof by exhaustion
Probabilistic proof
Combinatorial proof
Nonconstructive proof
Statistical proofs in pure mathematics
Computer-assisted proofs


Mathematical proof
Methods of proof
Direct proof


Mathematical proof
Methods of proof
Proof by mathematical induction


Mathematical proof
Methods of proof
Proof by contraposition


Mathematical proof
Methods of proof
Proof by contradiction


Mathematical proof
Methods of proof
Proof by construction


Mathematical proof
Methods of proof
Proof by exhaustion


Mathematical proof
Methods of proof
Probabilistic proof


Mathematical proof
Methods of proof
Combinatorial proof


Mathematical proof
Methods of proof
Nonconstructive proof


Mathematical proof
Methods of proof
Statistical proofs in pure mathematics


Mathematical proof
Methods of proof
Computer-assisted proofs


Mathematical proof
Undecidable statements


Inductive logic proofs and Bayesian analysis

Influence of mathematical proof methods outside mathematics


Mathematical proof
Related concepts
Visual proof


Mathematical proof
Related concepts
Elementary proof


Mathematical proof
Related concepts
Two-column proof


Mathematical proof
Related concepts
Colloquial use of "mathematical proof"


Mathematical proof
Related concepts
Statistical proof using data


Inductive logic proofs and Bayesian analysis


Mathematical proof
Related concepts
Proofs as mental objects


Influence of mathematical proof methods outside mathematics


Life cycle of Cladius difformis
Taxonomy
References
Bibliography
External links


Tenthredinidae
Life cycle of Cladius difformis


Tenthredinidae
Taxonomy
Subfamilies and genera


Events
Births
Deaths
Holidays and observances
References
External links


History
Main classification
Subclasses of the integers
Subclasses of the complex numbers
Extensions of the concept


Numerals
First use of numbers
Zero
Negative numbers
Rational numbers
Irrational numbers
Transcendental numbers and reals
Infinity and infinitesimals
Complex numbers
Prime numbers


Number
History
Transcendental numbers and reals


Number
History
Infinity and infinitesimals


Natural numbers
Integers
Rational numbers
Real numbers
Complex numbers


Number
Main classification
Natural numbers


Number
Main classification
Rational numbers


Number
Main classification
Complex numbers


Even and odd numbers
Prime numbers
Other classes of integers


Number
Subclasses of the integers
Even and odd numbers


Number
Subclasses of the integers
Prime numbers


Number
Subclasses of the integers
Other classes of integers


Number
Subclasses of the complex numbers

Algebraic, irrational and transcendental numbers


Algebraic, irrational and transcendental numbers


Number
Subclasses of the complex numbers
Constructible numbers


Number
Subclasses of the complex numbers
Computable numbers


p-adic numbers
Hypercomplex numbers
Transfinite numbers
Nonstandard numbers


Number
Extensions of the concept
p-adic numbers


Number
Extensions of the concept
Hypercomplex numbers


Number
Extensions of the concept
Transfinite numbers


Number
Extensions of the concept
Nonstandard numbers


History
Taxonomy of spaces
Types of spaces
Spaces and structure
List of mathematical spaces


The golden age of geometry and afterwards


Space (mathematics)
History
Before the golden age of geometry


Three taxonomic ranks
Relations between species of spaces


Space (mathematics)
Taxonomy of spaces
Three taxonomic ranks


Space (mathematics)
Taxonomy of spaces
Relations between species of spaces


Normed, Banach, inner product, and Hilbert spaces

Measurable, measure, and probability spaces


Space (mathematics)
Types of spaces
Linear and topological spaces


Space (mathematics)
Types of spaces
Affine and projective spaces


Space (mathematics)
Types of spaces
Metric and uniform spaces


Normed, Banach, inner product, and Hilbert spaces


Space (mathematics)
Types of spaces
Smooth and Riemannian manifolds


Measurable, measure, and probability spaces


Space (mathematics)
Types of spaces
Non-commutative geometry


Space (mathematics)
Types of spaces
Schemes


Space (mathematics)
Types of spaces
Topoi


Space (mathematics)
Spaces and structure


Space (mathematics)
List of mathematical spaces


Heuristic mathematics and experimental mathematics


Mathematical proof
History and etymology


Direct proof
Proof by mathematical induction
Proof by contraposition
Proof by contradiction
Proof by construction
Proof by exhaustion
Probabilistic proof
Combinatorial proof
Nonconstructive proof
Statistical proofs in pure mathematics
Computer-assisted proofs


Mathematical proof
Methods of proof
Direct proof


Mathematical proof
Methods of proof
Proof by mathematical induction


Mathematical proof
Methods of proof
Proof by contraposition


Mathematical proof
Methods of proof
Proof by contradiction


Mathematical proof
Methods of proof
Proof by construction


Mathematical proof
Methods of proof
Proof by exhaustion


Mathematical proof
Methods of proof
Probabilistic proof


Mathematical proof
Methods of proof
Combinatorial proof


Mathematical proof
Methods of proof
Nonconstructive proof


Mathematical proof
Methods of proof
Statistical proofs in pure mathematics


Mathematical proof
Methods of proof
Computer-assisted proofs


Mathematical proof
Undecidable statements


Inductive logic proofs and Bayesian analysis

Influence of mathematical proof methods outside mathematics


Mathematical proof
Related concepts
Visual proof


Mathematical proof
Related concepts
Elementary proof


Mathematical proof
Related concepts
Two-column proof


Mathematical proof
Related concepts
Colloquial use of "mathematical proof"


Mathematical proof
Related concepts
Statistical proof using data


Inductive logic proofs and Bayesian analysis


Mathematical proof
Related concepts
Proofs as mental objects


Influence of mathematical proof methods outside mathematics


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


Restrictions
Review of changes
Vandalism
Edit warring


Wikipedia
Policies and laws
Content policies and guidelines


Wikipedia
Governance
Dispute resolution
Arbitration Committee


Wikipedia
Language editions
English Wikipedia editor numbers


Accuracy of content
Discouragement in education
Coverage of topics and systemic bias
Explicit content
Privacy
Sexism


Wikipedia
Reception
Discouragement in education


Wikipedia
Reception
Discouragement in education
Medical information


Wikipedia
Reception
Coverage of topics and systemic bias

Coverage of topics and bias
Systemic biases


Wikipedia
Reception
Coverage of topics and systemic bias
Coverage of topics and bias


Wikipedia
Reception
Coverage of topics and systemic bias
Systemic biases


Wikimedia Foundation and affiliate movements

Internal research and operational development


Wikimedia Foundation and affiliate movements


Wikipedia
Operation
Software operations and support


Wikipedia
Operation
Hardware operations and support


Internal research and operational development


Wikipedia
Operation
Internal news publications


Wikipedia
Operation
The Wikipedia Library


Content licensing
Methods of access
Chinese access


Wikipedia
Access to content
Content licensing


Wikipedia
Access to content
Methods of access


Wikipedia
Access to content
Methods of access
Mobile access


Wikipedia
Access to content
Chinese access


Trusted source to combat fake news
Readership
Cultural significance
Sister projects – Wikimedia
Publishing
Research use


Wikipedia
Cultural influence
Trusted source to combat fake news


Wikipedia
Cultural influence
Readership
COVID-19 pandemic


Wikipedia
Cultural influence
Cultural significance


Wikipedia
Cultural influence
Cultural significance
Awards


Wikipedia
Cultural influence
Cultural significance
Satire


Wikipedia
Cultural influence
Sister projects – Wikimedia


Wikipedia
Cultural influence
Research use


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


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


English language
History
Proto-Germanic to Old English


English language
History
Early Modern English


English language
History
Spread of Modern English


English language
Geographical distribution

Three circles of English-speaking countries


Three circles of English-speaking countries


English language
Geographical distribution
Pluricentric English


English language
Geographical distribution
English as a global language


Consonants
Vowels
Phonotactics
Stress, rhythm and intonation
Regional variation


English language
Phonology
Stress, rhythm and intonation


English language
Phonology
Regional variation


Nouns and noun phrases
Verbs and verb phrases
Syntax


English language
Grammar
Nouns and noun phrases

Adjectives
Determiners
Pronouns, case, and person
Prepositions


English language
Grammar
Nouns and noun phrases
Adjectives


English language
Grammar
Nouns and noun phrases
Determiners


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


English language
Grammar
Nouns and noun phrases
Prepositions


English language
Grammar
Verbs and verb phrases

Tense, aspect and mood
Phrasal verbs
Adverbs


English language
Grammar
Verbs and verb phrases
Tense, aspect and mood


English language
Grammar
Verbs and verb phrases
Phrasal verbs


English language
Grammar
Verbs and verb phrases
Adverbs


Basic constituent order
Clause syntax
Auxiliary verb constructions
Questions
Discourse level syntax


English language
Grammar
Syntax
Basic constituent order


English language
Grammar
Syntax
Clause syntax


English language
Grammar
Syntax
Auxiliary verb constructions


English language
Grammar
Syntax
Questions


English language
Grammar
Syntax
Discourse level syntax


English loanwords and calques in other languages


English language
Vocabulary
Word formation processes


English language
Vocabulary
Word origins


English language
Dialects, accents and varieties

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


English language
Dialects, accents and varieties
Britain and Ireland


English language
Dialects, accents and varieties
North America


English language
Dialects, accents and varieties
Australia and New Zealand


English language
Dialects, accents and varieties
Southeast Asia


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


English language
Dialects, accents and varieties
Non-native varieties


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


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


Tropical cyclone
Intensity
Factors that influence intensity


Tropical cyclone
Intensity
Rapid intensification


Tropical cyclone
Intensity
Methods for assessing intensity


Tropical cyclone
Intensity
Intensity metrics


Tropical cyclone
Classification and naming


Tropical cyclone
Classification and naming
Classification


Tropical cyclone
Classification and naming
Naming


Tropical cyclone
Structure
Eye and center


Interaction with the mid-latitude westerlies


Tropical cyclone
Movement
Environmental steering


Tropical cyclone
Movement
Multiple storm interaction


Tropical cyclone
Formation regions and warning centers


Natural phenomena caused or worsened by tropical cyclones


Tropical cyclone
Effects
Effect on property and human life


Tropical cyclone
Effects
Environmental effects


Tropical cyclone
Climatology
Influence of climate change


Tropical cyclone
Observation and forecasting

Observation
Forecasting
Geopotential height


Tropical cyclone
Observation and forecasting
Observation


Tropical cyclone
Observation and forecasting
Forecasting


Tropical cyclone
Observation and forecasting
Geopotential height


Conic section analysis of the hyperbolic appearance of circles

Hyperbolas as plane sections of quadrics


Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form

As an affine image of the unit hyperbola x² − y² = 1

As an affine image of the hyperbola y = 1/x


Hyperbola
Definitions
As locus of points


Hyperbola
Definitions
Hyperbola with equation y = A/x


Hyperbola
Definitions
By the directrix property


Hyperbola
Definitions
As plane section of a cone


Hyperbola
Definitions
Pin and string construction


Hyperbola
Definitions
Steiner generation of a hyperbola


Inscribed angles for hyperbolas y = a/(x − b) + c and the 3-point-form


As an affine image of the unit hyperbola x² − y² = 1


As an affine image of the hyperbola y = 1/x

Tangent construction
Point construction
Tangent–asymptotes triangle


Hyperbola
Definitions
Reciprocation of a circle


Hyperbola
Definitions
Quadratic equation


Parametric representation with hyperbolic sine/cosine


Hyperbola
In Cartesian coordinates
Equation


Hyperbola
In Cartesian coordinates
Equation
Eccentricity


Hyperbola
In Cartesian coordinates
Asymptotes


Hyperbola
In Cartesian coordinates
Semi-latus rectum


Hyperbola
In Cartesian coordinates
Tangent


Hyperbola
In Cartesian coordinates
Rectangular hyperbola


Hyperbola
In Cartesian coordinates
Conjugate hyperbola


The tangent bisects the angle between the lines to the foci


Hyperbola
Properties
Midpoints of parallel chords


Hyperbola
Properties
Orthogonal tangents – orthoptic


Hyperbola
Properties
Pole-polar relation for a hyperbola


Conic section analysis of the hyperbolic appearance of circles


Sundials
Multilateration
Path followed by a particle
Korteweg–de Vries equation
Angle trisection
Efficient portfolio frontier
Biochemistry


Hyperbola
Applications
Path followed by a particle


Hyperbola
Applications
Korteweg–de Vries equation


Hyperbola
Applications
Efficient portfolio frontier


Hyperbolas as plane sections of quadrics


History
Main subdivisions
Other subfields
Applications
Prizes


Origins
Early modern number theory
Maturity and division into subfields


Classical Greece and the early Hellenistic period


Number theory
History
Origins
Dawn of arithmetic


Number theory
History
Origins
Diophantus


Number theory
History
Origins
Āryabhaṭa, Brahmagupta, Bhāskara


Number theory
History
Origins
Arithmetic in the Islamic golden age


Number theory
History
Origins
Western Europe in the Middle Ages


Number theory
History
Early modern number theory

Fermat
Euler
Lagrange, Legendre, and Gauss


Number theory
History
Early modern number theory
Fermat


Number theory
History
Early modern number theory
Euler


Number theory
History
Early modern number theory
Lagrange, Legendre, and Gauss


Number theory
History
Maturity and division into subfields


Elementary number theory
Analytic number theory
Algebraic number theory
Diophantine geometry


Number theory
Main subdivisions
Elementary number theory


Number theory
Main subdivisions
Analytic number theory


Number theory
Main subdivisions
Algebraic number theory


Number theory
Main subdivisions
Diophantine geometry


Probabilistic number theory
Arithmetic combinatorics
Computational number theory


Number theory
Other subfields
Probabilistic number theory


Number theory
Other subfields
Arithmetic combinatorics


Number theory
Other subfields
Computational number theory


Characterizations
Properties
Coordinates and equations
Construction
Symmetry
Squares inscribed in triangles
Squaring the circle
Non-Euclidean geometry
Crossed square
Graphs


Definition
Characteristics
Convex polyhedra
Symmetries
Other important families of polyhedra
Generalisations of polyhedra
Alternative usages
History
In nature


Number of faces
Topological classification
Duality
Vertex figures
Surface area and distances
Volume
Dehn invariant


Polyhedron
Characteristics
Number of faces


Polyhedron
Characteristics
Topological classification


Polyhedron
Characteristics
Vertex figures


Polyhedron
Characteristics
Surface area and distances


Polyhedron
Characteristics
Dehn invariant


Regular polyhedra
Uniform polyhedra and their duals
Isohedra
Symmetry groups


Polyhedron
Symmetries
Uniform polyhedra and their duals


Polyhedron
Other important families of polyhedra

Polyhedra with regular faces
Pyramids
Stellations and facettings
Zonohedra
Space-filling polyhedra
Lattice polyhedra
Flexible polyhedra
Compounds
Orthogonal polyhedra
Embedded regular maps with planar faces
Canonical polyhedra


Polyhedron
Other important families of polyhedra
Polyhedra with regular faces


Polyhedron
Other important families of polyhedra
Polyhedra with regular faces
Equal regular faces


Polyhedron
Other important families of polyhedra
Polyhedra with regular faces
Johnson solids


Polyhedron
Other important families of polyhedra
Pyramids


Polyhedron
Other important families of polyhedra
Stellations and facettings


Polyhedron
Other important families of polyhedra
Zonohedra


Polyhedron
Other important families of polyhedra
Space-filling polyhedra


Polyhedron
Other important families of polyhedra
Lattice polyhedra


Polyhedron
Other important families of polyhedra
Flexible polyhedra


Polyhedron
Other important families of polyhedra
Compounds


Polyhedron
Other important families of polyhedra
Orthogonal polyhedra


Polyhedron
Other important families of polyhedra
Embedded regular maps with planar faces


Polyhedron
Other important families of polyhedra
Canonical polyhedra


Apeirohedra
Complex polyhedra
Curved polyhedra
Ideal polyhedra
Skeletons and polyhedra as graphs


Polyhedron
Generalisations of polyhedra
Apeirohedra


Polyhedron
Generalisations of polyhedra
Complex polyhedra


Polyhedron
Generalisations of polyhedra
Curved polyhedra

Spherical polyhedra
Curved spacefilling polyhedra


Polyhedron
Generalisations of polyhedra
Curved polyhedra
Spherical polyhedra


Polyhedron
Generalisations of polyhedra
Curved polyhedra
Curved spacefilling polyhedra


Polyhedron
Generalisations of polyhedra
Ideal polyhedra


Polyhedron
Generalisations of polyhedra
Skeletons and polyhedra as graphs


Higher-dimensional polyhedra
Topological polyhedra
Abstract polyhedra


Polyhedron
Alternative usages
Higher-dimensional polyhedra


Polyhedron
Alternative usages
Topological polyhedra


Polyhedron
Alternative usages
Abstract polyhedra


Before the Greeks
Ancient Greece
Ancient China
Medieval Islam
Renaissance
17th–19th centuries
20th–21st centuries


Approaches and subfields of combinatorics


Approaches and subfields of combinatorics

Enumerative combinatorics
Analytic combinatorics
Partition theory
Graph theory
Design theory
Finite geometry
Order theory
Matroid theory
Extremal combinatorics
Probabilistic combinatorics
Algebraic combinatorics
Combinatorics on words
Geometric combinatorics
Topological combinatorics
Arithmetic combinatorics
Infinitary combinatorics


Combinatorial optimization
Coding theory
Discrete and computational geometry
Combinatorics and dynamical systems
Combinatorics and physics


Combinatorics
Related fields
Combinatorial optimization


Combinatorics
Related fields
Coding theory


Combinatorics
Related fields
Discrete and computational geometry


Combinatorics
Related fields
Combinatorics and dynamical systems


Combinatorics
Related fields
Combinatorics and physics


Regular tetrahedron
Irregular tetrahedra
General properties
Integer tetrahedra
Related polyhedra and compounds
Applications
Tetrahedral graph


Orthogonal projections of the regular tetrahedron


Tetrahedron
Regular tetrahedron
Coordinates for a regular tetrahedron


Tetrahedron
Regular tetrahedron
Angles and distances


Tetrahedron
Regular tetrahedron
Isometries of the regular tetrahedron


Tetrahedron
Regular tetrahedron
Cross section of regular tetrahedron


Tetrahedron
Regular tetrahedron
Spherical tiling


Tetrahedron
Regular tetrahedron
Helical stacking


Trirectangular tetrahedron
Disphenoid
Orthoschemes
Space-filling tetrahedra
Fundamental domains
Isometries of irregular tetrahedra


Tetrahedron
Irregular tetrahedra
Trirectangular tetrahedron


Tetrahedron
Irregular tetrahedra
Disphenoid


Tetrahedron
Irregular tetrahedra
Orthoschemes


Tetrahedron
Irregular tetrahedra
Space-filling tetrahedra


Tetrahedron
Irregular tetrahedra
Fundamental domains


Tetrahedron
Irregular tetrahedra
Isometries of irregular tetrahedra


Properties analogous to those of a triangle

A law of sines for tetrahedra and the space of all shapes of tetrahedra


Heron-type formula for the volume of a tetrahedron


Tetrahedron
General properties
Volume
Volume divider


Tetrahedron
General properties
Volume
Non-Euclidean volume


Tetrahedron
General properties
Distance between the edges


Properties analogous to those of a triangle


Tetrahedron
General properties
Geometric relations


A law of sines for tetrahedra and the space of all shapes of tetrahedra


Tetrahedron
General properties
Law of cosines for tetrahedra


Tetrahedron
General properties
Interior point


Tetrahedron
General properties
Circumradius


Tetrahedron
General properties
Circumcenter


Tetrahedron
Related polyhedra and compounds


Numerical analysis
Structural engineering
Aviation
Chemistry
Electricity and electronics
Color space
Games
Geology
Popular culture


Tetrahedron
Applications
Numerical analysis


Tetrahedron
Applications
Structural engineering


Tetrahedron
Applications
Electricity and electronics


Tetrahedron
Applications
Popular culture


Etymology
Overview
Types of ontologies
History
Other ontological topics


Particulars and universals
Abstract and concrete
Ontological dependence
Identity
Modality
Substances
Properties and relations
States of affairs and events
Reality of things


Ontology
Overview
Particulars and universals


Ontology
Overview
Ontological dependence


Ontology
Overview
Properties and relations


Ontology
Overview
States of affairs and events


Information science and natural sciences


Ontology
Types of ontologies
Flat vs polycategorical vs hierarchical


Ontology
Types of ontologies
Thing ontologies vs fact ontologies


Ontology
Types of ontologies
Constituent ontologies vs blob theories


Ancient Greek
Hindu philosophy
Medieval
Modern
20th century


Ontological and epistemological certainty

Body and environment, questioning the meaning of being


Ontology
Other ontological topics
Ontological formations


Ontology
Other ontological topics
Ontology of fictional characters


Ontological and epistemological certainty


Body and environment, questioning the meaning of being


Ontology
Other ontological topics
Ontology and language


Ontology
Other ontological topics
Ontology and human geography


Ontology
Other ontological topics
Anthropology


Ontology
Other ontological topics
Reality and actuality


Ontology
Other ontological topics
Microcosmic ontology


Ontology
Other ontological topics
Ontological argument


Ontology
Other ontological topics
Hintikka's locution for existence


Point masses and the Dirac delta function


Point (geometry)
Points in Euclidean geometry


Vector space dimension
Topological dimension
Hausdorff dimension


Point (geometry)
Dimension of a point
Vector space dimension


Point (geometry)
Dimension of a point
Topological dimension


Point (geometry)
Dimension of a point
Hausdorff dimension


Point (geometry)
Geometry without points


Inscription in or circumscription about other figures


Circumference
Area enclosed
Equations
Tangent lines


Cartesian coordinates
Polar coordinates
Complex plane


Circle
Analytic results
Equations
Cartesian coordinates


Circle
Analytic results
Equations
Polar coordinates


Circle
Analytic results
Equations
Complex plane


Chord
Tangent
Theorems
Inscribed angles
Sagitta


Circle
Compass and straightedge constructions

Construction through three noncollinear points


Circle
Compass and straightedge constructions
Construction with given diameter


Construction through three noncollinear points


Circle
Circle of Apollonius
Cross-ratios


Circle
Circle of Apollonius
Generalised circles


Circle
Significance in art and symbolism


Points, lines, and circles associated with a triangle

Further formulas for general Euclidean triangles

Specifying the location of a point in a triangle


Triangle
Types of triangle
By lengths of sides


Triangle
Types of triangle
By internal angles


Similarity and congruence
Right triangles


Triangle
Basic facts
Similarity and congruence


Condition on the sides
Conditions on the angles


Triangle
Existence of a triangle
Condition on the sides


Triangle
Existence of a triangle
Conditions on the angles


Triangle
Existence of a triangle
Conditions on the angles
Trigonometric conditions


Points, lines, and circles associated with a triangle


Trigonometric ratios in right triangles
Sine, cosine and tangent rules
Solution of triangles


Triangle
Computing the sides and angles
Trigonometric ratios in right triangles

Sine, cosine and tangent
Inverse functions


Triangle
Computing the sides and angles
Trigonometric ratios in right triangles
Sine, cosine and tangent


Triangle
Computing the sides and angles
Trigonometric ratios in right triangles
Inverse functions


Triangle
Computing the sides and angles
Sine, cosine and tangent rules


Triangle
Computing the sides and angles
Solution of triangles


Further formulas for general Euclidean triangles

Medians, angle bisectors, perpendicular side bisectors, and altitudes


Medians, angle bisectors, perpendicular side bisectors, and altitudes


Triangle
Figures inscribed in a triangle

Conics
Convex polygon
Hexagon
Squares
Triangles


Triangle
Figures inscribed in a triangle
Conics


Triangle
Figures inscribed in a triangle
Convex polygon


Triangle
Figures inscribed in a triangle
Hexagon


Triangle
Figures inscribed in a triangle
Squares


Triangle
Figures inscribed in a triangle
Triangles


Triangle
Figures circumscribed about a triangle


Specifying the location of a point in a triangle


Properties
Definition
Other representations
Generalizations of the Euclidean line
Extensions


In higher dimensions
Types
In axiomatic systems


Line (geometry)
Properties
In higher dimensions


Line (geometry)
Properties
In higher dimensions
Collinear points


Line (geometry)
Properties
In axiomatic systems


Linear equation
Parametric equation
Hesse normal form


Line (geometry)
Definition
Linear equation


Line (geometry)
Definition
Parametric equation


Line (geometry)
Definition
Hesse normal form


Line (geometry)
Other representations
Vectors


Line (geometry)
Other representations
Polar coordinates


Line (geometry)
Generalizations of the Euclidean line


Line (geometry)
Generalizations of the Euclidean line
Projective geometry


A geometrical construction to find a sector area

Focal length and radius of curvature at the vertex

As the affine image of the unit parabola


Parabola
Definition as a locus of points


Parabola
In a Cartesian coordinate system

Axis of symmetry parallel to the y axis
General position


Parabola
In a Cartesian coordinate system
Axis of symmetry parallel to the y axis


Parabola
In a Cartesian coordinate system
General position


Parabola
Similarity to the unit parabola


Parabola
Conic section and quadratic form

Diagram, description, and definitions
Derivation of quadratic equation
Focal length
Position of the focus
Alternative proof with Dandelin spheres


Parabola
Conic section and quadratic form
Diagram, description, and definitions


Parabola
Conic section and quadratic form
Derivation of quadratic equation


Parabola
Conic section and quadratic form
Focal length


Parabola
Conic section and quadratic form
Position of the focus


Parabola
Conic section and quadratic form
Alternative proof with Dandelin spheres


Parabola
Proof of the reflective property

Construction and definitions
Deductions
Other consequences
Alternative proofs


Parabola
Proof of the reflective property
Construction and definitions


Parabola
Proof of the reflective property
Deductions


Parabola
Proof of the reflective property
Other consequences

Intersection of a tangent and perpendicular from focus

Reflection of light striking the convex side


Parabola
Proof of the reflective property
Other consequences
Tangent bisection property


Intersection of a tangent and perpendicular from focus


Reflection of light striking the convex side


Parabola
Proof of the reflective property
Alternative proofs


Parabola
Properties related to Pascal's theorem

4-points property
3-points–1-tangent property
2-points–2-tangents property
Axis direction


Parabola
Properties related to Pascal's theorem
4-points property


Parabola
Properties related to Pascal's theorem
3-points–1-tangent property


Parabola
Properties related to Pascal's theorem
2-points–2-tangents property


Parabola
Properties related to Pascal's theorem
Axis direction


Parabola
Steiner generation
Dual parabola


Parabola
Inscribed angles and the 3-point form


Two tangent properties related to the latus rectum


Parabola
Tangent properties
Orthoptic property


Parabola
Tangent properties
Lambert's theorem


Parabola
Facts related to chords and arcs

Focal length calculated from parameters of a chord

Area enclosed between a parabola and a chord

Corollary concerning midpoints and endpoints of chords


Focal length calculated from parameters of a chord


Area enclosed between a parabola and a chord


Corollary concerning midpoints and endpoints of chords


Parabola
Facts related to chords and arcs
Arc length


A geometrical construction to find a sector area


Focal length and radius of curvature at the vertex


As the affine image of the unit parabola


Etymology
Classification
Properties and formulas
Generalizations
Naming
History
In nature
Computer graphics


Number of sides
Convexity and intersection
Equality and symmetry
Miscellaneous


Polygon
Classification
Convexity and intersection


Polygon
Classification
Equality and symmetry


Simple polygons
Regular polygons
Self-intersecting


Polygon
Properties and formulas
Area
Simple polygons


Polygon
Properties and formulas
Area
Regular polygons


Polygon
Properties and formulas
Area
Self-intersecting


Polygon
Properties and formulas
Centroid


Equilateral triangles on the sides of an arbitrary hexagon

Gallery of natural and artificial hexagons


Hexagon
Related polygons and tilings
Self-crossing hexagons


Hexagon
Hexagon inscribed in a conic section


Hexagon
Hexagon inscribed in a conic section
Cyclic hexagon


Hexagon
Hexagon tangential to a conic section


Equilateral triangles on the sides of an arbitrary hexagon


Hexagon
Convex equilateral hexagon
Polyhedra with hexagons


Gallery of natural and artificial hexagons


Regular pentagons
Equilateral pentagons
Cyclic pentagons
General convex pentagons
Pentagons in tiling
Pentagons in polyhedra
Pentagons in nature
Other examples


Chords from the circumscribed circle to the vertices


Pentagon
Regular pentagons
Derivation of the area formula


Pentagon
Regular pentagons
Point in plane


Pentagon
Regular pentagons
Geometrical constructions

Richmond's method
Carlyle circles
Euclid's method


Pentagon
Regular pentagons
Geometrical constructions
Richmond's method


Pentagon
Regular pentagons
Geometrical constructions
Carlyle circles


Pentagon
Regular pentagons
Geometrical constructions
Euclid's method


Pentagon
Regular pentagons
Physical construction methods


Pentagon
Regular pentagons
Regular pentagram


Definition as locus of points
In Cartesian coordinates
Parametric representation
Polar forms
Eccentricity and the directrix property
Focus-to-focus reflection property
Conjugate diameters
Orthogonal tangents
Drawing ellipses
Inscribed angles and three-point form
Pole-polar relation
Metric properties
In triangle geometry
As plane sections of quadrics
Applications


Standard equation
Parameters
Tangent
Shifted ellipse
General ellipse


Ellipse
In Cartesian coordinates
Standard equation


Ellipse
In Cartesian coordinates
Parameters

Principal axes
Linear eccentricity
Eccentricity
Semi-latus rectum


Ellipse
In Cartesian coordinates
Parameters
Principal axes


Ellipse
In Cartesian coordinates
Parameters
Linear eccentricity


Ellipse
In Cartesian coordinates
Parameters
Eccentricity


Ellipse
In Cartesian coordinates
Parameters
Semi-latus rectum


Ellipse
In Cartesian coordinates
Tangent


Ellipse
In Cartesian coordinates
Shifted ellipse


Ellipse
In Cartesian coordinates
General ellipse


Standard parametric representation
Rational representation
Tangent slope as parameter
General ellipse


Ellipse
Parametric representation
Standard parametric representation


Ellipse
Parametric representation
Rational representation


Ellipse
Parametric representation
Tangent slope as parameter


Ellipse
Parametric representation
General ellipse


Polar form relative to center
Polar form relative to focus


Ellipse
Polar forms
Polar form relative to center


Ellipse
Polar forms
Polar form relative to focus


Ellipse
Eccentricity and the directrix property


Ellipse
Focus-to-focus reflection property


Theorem of Apollonios on conjugate diameters


Ellipse
Conjugate diameters
Definition of conjugate diameters


de La Hire's point construction
Pins-and-string method
Paper strip methods
Approximation by osculating circles
Steiner generation
As hypotrochoid


Ellipse
Drawing ellipses
de La Hire's point construction


Ellipse
Drawing ellipses
Pins-and-string method


Ellipse
Drawing ellipses
Paper strip methods


Ellipse
Drawing ellipses
Approximation by osculating circles


Ellipse
Drawing ellipses
Steiner generation


Ellipse
Drawing ellipses
As hypotrochoid


Ellipse
Inscribed angles and three-point form


Ellipse
Inscribed angles and three-point form
Circles

Inscribed angle theorem for circles
Three-point form of circle equation


Ellipse
Inscribed angles and three-point form
Circles
Inscribed angle theorem for circles


Ellipse
Inscribed angles and three-point form
Circles
Three-point form of circle equation


Ellipse
Inscribed angles and three-point form
Ellipses

Inscribed angle theorem for ellipses
Three-point form of ellipse equation


Ellipse
Inscribed angles and three-point form
Ellipses
Inscribed angle theorem for ellipses


Ellipse
Inscribed angles and three-point form
Ellipses
Three-point form of ellipse equation


Physics
Statistics and finance
Computer graphics
Optimization theory


Elliptical reflectors and acoustics
Planetary orbits
Harmonic oscillators
Phase visualization
Elliptical gears
Optics


Ellipse
Applications
Physics
Elliptical reflectors and acoustics


Ellipse
Applications
Physics
Planetary orbits


Ellipse
Applications
Physics
Harmonic oscillators


Ellipse
Applications
Physics
Phase visualization


Ellipse
Applications
Physics
Elliptical gears


Ellipse
Applications
Statistics and finance


Ellipse
Applications
Optimization theory


Regular dodecahedron
Other pentagonal dodecahedra
Rhombic dodecahedron
Other dodecahedra
Practical usage


Dodecahedron
Other pentagonal dodecahedra

Pyritohedron
Tetartoid
Dual of triangular gyrobianticupola


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron

Crystal pyrite
Cartesian coordinates
Geometric freedom


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron
Crystal pyrite


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron
Cartesian coordinates


Dodecahedron
Other pentagonal dodecahedra
Pyritohedron
Geometric freedom


Dodecahedron
Other pentagonal dodecahedra
Tetartoid


Dodecahedron
Other pentagonal dodecahedra
Tetartoid
Cartesian coordinates


Dodecahedron
Other pentagonal dodecahedra
Tetartoid
Geometric freedom


Dodecahedron
Other pentagonal dodecahedra
Dual of triangular gyrobianticupola


Orthogonal projections
Spherical tiling
Cartesian coordinates
Equation in three dimensional space
Formulas
Doubling the cube
Uniform colorings and symmetry
Geometric relations
In theology
Other dimensions
Related polyhedra
Cubical graph


Cube
Equation in three dimensional space


Cube
Related polyhedra
In uniform honeycombs and polychora


Regular octahedron
Irregular octahedra
Octahedra in the physical world
Related polyhedra


Dimensions
Orthogonal projections
Spherical tiling
Cartesian coordinates
Area and volume
Geometric relations
Characteristic orthoscheme
Topology
Nets
Faceting
Uniform colorings and symmetry


Octahedron
Regular octahedron
Dimensions


Octahedron
Regular octahedron
Orthogonal projections


Octahedron
Regular octahedron
Spherical tiling


Octahedron
Regular octahedron
Cartesian coordinates


Octahedron
Regular octahedron
Area and volume


Octahedron
Regular octahedron
Geometric relations

Dual
Stellation
Snub octahedron
Tessellations


Octahedron
Regular octahedron
Geometric relations
Dual


Octahedron
Regular octahedron
Geometric relations
Stellation


Octahedron
Regular octahedron
Geometric relations
Snub octahedron


Octahedron
Regular octahedron
Geometric relations
Tessellations


Octahedron
Regular octahedron
Characteristic orthoscheme


Octahedron
Regular octahedron
Uniform colorings and symmetry


Octahedron
Octahedra in the physical world

Octahedra in nature
Octahedra in art and culture
Tetrahedral octet truss


Octahedron
Octahedra in the physical world
Octahedra in nature


Octahedron
Octahedra in the physical world
Octahedra in art and culture


Octahedron
Octahedra in the physical world
Tetrahedral octet truss


Tetratetrahedron
Trigonal antiprism
Square bipyramid
Other related polyhedra


Octahedron
Related polyhedra
Tetratetrahedron


Octahedron
Related polyhedra
Trigonal antiprism


Octahedron
Related polyhedra
Square bipyramid


Octahedron
Related polyhedra
Other related polyhedra


Regular icosahedra
Stellated icosahedra
Pyritohedral symmetry
Other icosahedra


Convex regular icosahedron
Great icosahedron


Icosahedron
Regular icosahedra
Convex regular icosahedron


Icosahedron
Regular icosahedra
Great icosahedron


Cartesian coordinates
Jessen's icosahedron
Cuboctahedron


Icosahedron
Pyritohedral symmetry
Cartesian coordinates


Icosahedron
Pyritohedral symmetry
Jessen's icosahedron


Icosahedron
Pyritohedral symmetry
Cuboctahedron


Rhombic icosahedron
Pyramid and prism symmetries
Johnson solids


Icosahedron
Other icosahedra
Rhombic icosahedron


Icosahedron
Other icosahedra
Pyramid and prism symmetries


Icosahedron
Other icosahedra
Johnson solids


Cylinder between pencils of elliptic and hyperbolic paraboloids

Geometric representation of multiplication table


Elliptic paraboloid
Hyperbolic paraboloid


Paraboloid
Properties and applications
Elliptic paraboloid


Paraboloid
Properties and applications
Elliptic paraboloid
Parabolic reflector


Paraboloid
Properties and applications
Hyperbolic paraboloid


Paraboloid
Properties and applications
Hyperbolic paraboloid
Examples in architecture


Cylinder between pencils of elliptic and hyperbolic paraboloids


Geometric representation of multiplication table


Paraboloid
Dimensions of a paraboloidal dish


Basic terminology
Equations
Properties
Treatment by area of mathematics
Curves on a sphere
Generalizations
History
Gallery
Regions


Enclosed volume
Surface area
Other geometric properties
Properties of the sphere


Sphere
Properties
Other geometric properties


Sphere
Properties
Other geometric properties
Pencil of spheres


Sphere
Properties
Properties of the sphere


Spherical geometry
Differential geometry
Topology


Sphere
Treatment by area of mathematics
Spherical geometry


Sphere
Treatment by area of mathematics
Differential geometry


Sphere
Treatment by area of mathematics
Topology


Intersection of a sphere with a more general surface


Sphere
Curves on a sphere
Spherical conics


History
Main concepts
Contemporary geometry
Applications


Axioms
Objects
Length, area, and volume
Congruence and similarity
Compass and straightedge constructions
Dimension
Symmetry


Points Lines Planes Angles Curves Surfaces Manifolds


Geometry
Main concepts
Objects
Manifolds