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Foundations

Mathematical logic

Axiomatic systems

Computability theory

Category theory

Adjoint functors

Relationship between two functors abstracting many common constructions

Infinite cardinal number

Amalgamation property

Property of a class of models in universal algebra and model theory

Analytical hierarchy

Concept in mathematical logic and set theory

Arithmetical hierarchy

Hierarchy of complexity classes for formulas defining sets

Mathematical logic concept

Statement that is taken to be true

Axiom of choice

Axiom of set theory

Axiom of extensionality

Concept of axiomatic set theory.

Axiom of infinity

Axiom of the Zermelo-Fraenkel set theory

Axiom of pairing

Concept in axiomatic set theory

Axiom of power set

Concept in axiomatic set theory

Axiom of regularity

Axiom of set theory

Concept in axiomatic set theory

A formula in the metalanguage of an axiomatic system in which one or more schematic variables appear

Axiom schema of replacement

Concept in set theory

Axiom schema of specification

Concept in axiomatic set theory

Axiomatic system

Mathematical term; concerning axioms used to derive theorems

Banach–Tarski paradox

Taking apart an object and constructing two identical copies of it from the pieces

Binary relation

Relationship between two sets, defined by a set of ordered pairs

Boolean algebra

Manipulation of variables taking only "true" and "false" as values

Cantor's diagonal argument

Proof in set theory

Cantor's theorem

Theorem in set theory

Size of a set in mathematics

Cardinality of the continuum

Cardinality of the set of real numbers

Cartesian closed category

Type of category in category theory

Cartesian product

Mathematical set formed from two given sets

Church–Turing thesis

Thesis on the nature of computability

Class (set theory)

Collection of sets in mathematics that can be defined based on a property of its members

Combinatory logic

Logical formalism using combinators instead of variables

Mathematics construct

Commutative diagram

Collection of maps which give the same result

Compactness theorem

Complement (set theory)

Set theory concept

Computability theory

Branch of mathematical logic, computer science, and the theory of computation studying computable functions and Turing degrees

Computable function

Mathematical function that can be computed by a program

Set where an algorithm can take a number as an input and can decide whether the number belongs to the set

Computably enumerable set

Mathematical logic concept

Computably inseparable

Concept in computability theory

Concrete category

Category equipped with a faithful functor to the category of sets

Continuum hypothesis

Proposition in mathematical logic

Inference that says that a conditional statement is logically equivalent to its contrapositive

Category-theoretic construction

Decision problem

Yes/no problem in computer science

Element (mathematics)

Any one of the distinct objects that make up a set in set theory

Elementary equivalence

Concept in model theory

Mathematical set containing no elements

Entscheidungsproblem

Impossible task in computing

Equivalence relation

Mathematical concept for comparing objects

Existential quantification

Logical quantification stating that a statement holds for at least one object

First-order logic

Type of logical system

Lemma in category theory about commutative diagrams

Forcing (mathematics)

Technique invented by Paul Cohen for proving consistency and independence results

Forgetful functor

Drops some or all of the input's structure or properties

Foundations of mathematics

Study of the basic mathematical concepts

Value of a left adjoint functor to a forgetful functor

Free variables and bound variables

Classification of variables in a logic formula based on whether or not they are inside the scope of a quantifier

Mapping between categories in mathematical category theory

General recursive function

One of several equivalent definitions of a computable function

Gödel numbering

Function in mathematical logic

Gödel's completeness theorem

Fundamental theorem in mathematical logic

Gödel's incompleteness theorems

Limitative results in mathematical logic

Halting problem

Problem of determining whether a given program will finish running or continue forever

Higher-order logic

Form of predicate logic that is distinguished from first-order logic by additional quantifiers and, sometimes, stronger semantics

Hilbert's axioms

Functor mapping hom objects to an underlying category

Logical connective

Collection of objects, each associated with an element from some index set

Initial and terminal objects

Special objects used in (mathematical) category theory

Interpretation (model theory)

Concept in model theory

Intersection (set theory)

Concept in mathematics specific to the field of set theory

Intuitionistic logic

Various systems of symbolic logic

Category theory constructs

Graphical method to simplify Boolean expressions

Lambda calculus

Formal mathematical logic system centered on function abstractions and applications

Limit (category theory)

Terminology used in theoretical mathematics involving categories.

Logical connective

Symbol connecting sentential formulas in logic

Löwenheim–Skolem theorem

Mathematical theorem

Many-one reduction

Type of Turing reduction

Material conditional

Logical connective

Mathematical induction

Form of mathematical proof

Mathematical logic

Subfield of mathematics

Model complete theory

Concept in model theory

Model of computation

Mathematical model describing how an output of a function is computed given an input

Area of mathematical logic

Rule of logical inference

Rule of logical inference

Monad (category theory)

Operation in algebra and mathematics

Map (arrow) between two objects of a category

Mathematical set with repetitions allowed

Naive set theory

Informal set theories

Natural transformation

Central object of study in category theory

New Foundations

Axiomatic set theory devised by W.V.O. Quine

Category theory lemma about commutative diagrams

Opposite category

Category constructed from another category C, whose objects are the same as those of C, whose morphisms from X to Y are the same as the morphisms in C from Y to X

Abstract machine used to study decision problems

Pair of mathematical objects

Order type of a well-ordered set

Partition of a set

Mathematical ways to group elements of a set

Axioms for the natural numbers

A set equipped with a choice of a specific element

Theorem in computability theory

Mathematical set containing all subsets of a given set

Prenex normal form

First-order formula consisting of a string of quantifiers and bound variables followed by a quantifier-free part

Primitive recursive function

Function that can be computed with loops of bounded length

Principle of explosion

Theorem which states that any statement can be proven from a contradiction

Product (category theory)

Generalized object in category theory

Proof by contradiction

Form of indirect proof that establishes the truth or validity of a proposition

Propositional calculus

Logical study of propositions (whether they are true or false) that are formed by other propositions with the use of logical connectives

Pullback (category theory)

Most general completion of a commutative square given two morphisms with same codomain

Pushout (category theory)

Category-theoretic colimit of a diagram of the form 𝑋←𝑍→𝑌

Quantifier (logic)

Mathematical use of "for all" and "there exists"

Quantifier elimination

Technique to simplify formulas

Omission of operations and relations of a structure

Reflexive relation

A binary relation over a set in which every element is related to itself

Theorem in computability theory

Rule of inference

Method of deriving conclusions

Russell's paradox

Paradox in the foundations of mathematics

Saturated model

Model for mathematical theories

Set (mathematics)

Collection of mathematical objects

Branch of mathematics that studies sets

Logical operation

Signature (logic)

Description of non-logical symbols

Skolem normal form

Formalism of first-order logic

Skolem's paradox

Mathematical logic concept

Tool used in mathematics

Concerned with the notion of stability in model theory

Structure (mathematical logic)

Mapping of mathematical formulas to a particular meaning, in universal algebra and in model theory

Category whose objects and morphisms are inside a bigger category

Mathematical set contained in another set

Symmetric relation

Type of binary relation

Tarski's axioms

First-order axiomatization of a fragment of Euclidean geometry

Theory (mathematical logic)

Set of sentences in a formal language

Transfer principle

That all statements of some language that are true for some structure are true for another structure

Transfinite induction

Mathematical concept

Transitive relation

Type of binary relation

Mathematical table used in logic

Finite ordered list of elements

Measurement for the level of algorithmic unsolvability of a set

Computation model defining an abstract machine

Turing reduction

Concept in computability theory

Proof by Alan Turing

Type (model theory)

Term in model theory and related areas of mathematics

Concept in mathematical logic

Mathematical construction

Union (set theory)

Mathematical operation where sets combine or relate

Universal property

Central object of study in category theory

Universal quantification

Mathematical use of "for all"

Universal Turing machine

Type of Turing machine

Universe (mathematics)

Collection that contains all the entities one wishes to consider in a given situation in mathematics

Affirms the existence of a computable universal function

Diagram that shows all possible logical relations between a collection of sets

Von Neumann universe

Set theory concept

Well-formed formula

Finite sequence of symbols from a given alphabet that is part of a formal language

Well-ordering theorem

Theoretic principle in mathematics stating every set can be well-ordered.

Theorem in category theory

Zermelo–Fraenkel set theory

Standard system of axiomatic set theory

Mathematical proposition equivalent to the axiom of choice

Concept in computability theory