# List of mathematical logic topics

This is a list of mathematical logic topics.
For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more theory of algorithms.

## Working foundations

* Peano axioms
** Giuseppe Peano
* Mathematical induction
** Structural induction
** Recursive definition
* Naive set theory
** Element (mathematics)
*** Ur-element
** Singleton (mathematics)
** Simple theorems in the algebra of sets
** Algebra of sets
** Power set
** Empty set
** Non-empty set
** Empty function
* Universe (mathematics)
* Axiomatization
* Axiomatic system
** Axiom schema
* Axiomatic method
* Formal system
* Mathematical proof
** Direct proof
** Reductio ad absurdum
** Proof by exhaustion
** Constructive proof
** Nonconstructive proof
* Tautology
* Consistency proof
* Arithmetization of analysis
* Foundations of mathematics
* Formal language
* Principia Mathematica
* Hilbert's program
* Impredicative
* Definable real number
* Algebraic logic
** Boolean algebra (logic)
* Dialectica space
* categorical logic

## Model theory

* Finite model theory
** Descriptive complexity theory
** Model checking
** Trakhtenbrot's theorem
* Computable model theory
** Tarski's exponential function problem
** Undecidable problem
* Institutional model theory
** Institution (computer science)
* Non-standard analysis
** Non-standard calculus
** Hyperinteger
** Hyperreal number
** Transfer principle
** Overspill
** Elementary Calculus: An Infinitesimal Approach
** Criticism of non-standard analysis
** Standard part function
* Set theory
** Forcing (mathematics)
*** Boolean-valued model
* Kripke semantics
** General frame
* Predicate logic
** First-order logic
*** Infinitary logic
*** Many-sorted logic
** Higher-order logic
*** Lindström quantifier
*** Second-order logic
* Soundness theorem
* Gödel's completeness theorem
** Original proof of Gödel's completeness theorem
* Compactness theorem
* Löwenheim–Skolem theorem
** Skolem's paradox
* Gödel's incompleteness theorems
* Structure (mathematical logic)
* Interpretation (logic)
* Substructure (mathematics)
* Elementary substructure
** Skolem hull
* Non-standard model
* Atomic model (mathematical logic)
* Prime model
* Saturated model
* Existentially closed model
* Ultraproduct
* Age (model theory)
** Amalgamation property
** Hrushovski construction
* Potential isomorphism
* Theory (mathematical logic)
** Complete theory
*** Vaught's test
** Morley's categoricity theorem
*** Stability spectrum
**** Morley rank
**** Stable theory
***** Forking extension
***** Strongly minimal theory
***** Stable group
****** Tame group
*** o-minimal theory
*** Weakly o-minimal structure
*** C-minimal theory
*** Spectrum of a theory
**** Vaught conjecture
** Model complete theory
** List of first-order theories
** Conservative extension
** Elementary class
*** Pseudoelementary class
*** Strength (mathematical logic)
* Differentially closed field
* Exponential field
* Ax–Grothendieck theorem
* Ax–Kochen theorem
* Peano axioms
* Non-standard model of arithmetic
* First-order arithmetic
* Second-order arithmetic
* Presburger arithmetic
* Wilkie's theorem
* Functional predicate
* T-schema
* Back-and-forth method
* Barwise compactness theorem
* Skolemization
* Lindenbaum–Tarski algebra
* Löb's theorem
* Arithmetical set
* Definable set
* Ehrenfeucht–Fraïssé game
* Herbrand interpretation / Herbrand structure
* Imaginary element
* Indiscernibles
* Interpretation (model theory) / Interpretable structure
* Pregeometry (model theory)
* Quantifier elimination
* Reduct
* Signature (logic)
* Skolem normal form
* Type (model theory)
* Zariski geometry

## Set theory

* Algebra of sets
* Axiom of choice
** Axiom of countable choice
** Axiom of dependent choice
** Zorn's lemma
* Boolean algebra (structure)
* Boolean-valued model
* Burali-Forti paradox
* Cantor's back-and-forth method
* Cantor's diagonal argument
* Cantor's first uncountability proof
* Cantor's theorem
* Cantor–Bernstein–Schroeder theorem
* Cardinality
** Aleph number
*** Aleph-null
*** Aleph-one
** Beth number
** Cardinal number
** Hartogs number
* Cartesian product
* Class (set theory)
* Complement (set theory)
* Complete Boolean algebra
* Continuum (set theory)
** Suslin's problem
* Continuum hypothesis
* Countable set
* Descriptive set theory
** Analytic set
** Analytical hierarchy
** Borel equivalence relation
** Infinity-Borel set
** Lightface analytic game
** Perfect set property
** Polish space
** Prewellordering
** Projective set
** Property of Baire
** Uniformization (set theory)
** Universally measurable set
* Determinacy
** AD+
** Axiom of determinacy
** Axiom of projective determinacy
** Axiom of real determinacy
* Empty set
* Forcing (mathematics)
* Fuzzy set
* Internal set theory
* Intersection (set theory)
* L
* L(R)
* Large cardinal property
* Musical set theory
* Ordinal number
** Infinite descending chain
** Limit ordinal
** Successor ordinal
** Transfinite induction
*** ∈-induction
** Well-founded set
** Well-order
* Power set
* Russell's paradox
* Set theory
** Alternative set theory
** Axiomatic set theory
** Kripke–Platek set theory with urelements
** Morse–Kelley set theory
** Naive set theory
** New Foundations
** Positive set theory
** Zermelo–Fraenkel set theory
** Zermelo set theory
* Set (mathematics)
* Simple theorems in the algebra of sets
* Subset
* Θ (set theory)
* Tree (descriptive set theory)
* Tree (set theory)
* Union (set theory)
* Von Neumann universe
* Zero sharp

## Descriptive set theory

* Analytical hierarchy

## Large cardinals

* Almost Ramsey cardinal
* Erdős cardinal
* Extendible cardinal
* Huge cardinal
* Hyper-Woodin cardinal
* Inaccessible cardinal
* Ineffable cardinal
* Mahlo cardinal
* Measurable cardinal
* N-huge cardinal
* Ramsey cardinal
* Rank-into-rank
* Remarkable cardinal
* Shelah cardinal
* Strong cardinal
* Strongly inaccessible cardinal
* Subtle cardinal
* Supercompact cardinal
* Superstrong cardinal
* Totally indescribable cardinal
* Weakly compact cardinal
* Weakly hyper-Woodin cardinal
* Weakly inaccessible cardinal
* Woodin cardinal
* Unfoldable cardinal

## Recursion theory

* Entscheidungsproblem
* Decision problem
* Decidability (logic)
* Church–Turing thesis
* Computable function
** Algorithm
** Recursion
** Primitive recursive function
** Mu operator
** Ackermann function
** Turing machine
** Halting problem
** Computability theory, computation
** Herbrand Universe
** Markov algorithm
** Lambda calculus
*** Church-Rosser theorem
*** Calculus of constructions
** Combinatory logic
** Post correspondence problem
* Kleene's recursion theorem
* Recursively enumerable set
** Recursively enumerable language
* Decidable language
* Undecidable language
* Rice's theorem
* Post's theorem
* Turing degree
* Effective results in number theory
* Diophantine set
* Matiyasevich's theorem
* Word problem for groups
* Arithmetical hierarchy
* Subrecursion theory
** Presburger arithmetic
** Computational complexity theory
** Polynomial time
** Exponential time
** Complexity class
*** Complexity classes P and NP
*** Cook's theorem
*** List of complexity classes
*** Polynomial hierarchy
*** Exponential hierarchy
** NP-complete
** Time hierarchy theorem
** Space hierarchy theorem
* Natural proof
* Hypercomputation
** Oracle machine
* Rózsa Péter
* Alonzo Church
* Emil Post
* Alan Turing
* Jacques Herbrand
* Haskell Curry
* Stephen Cole Kleene
* Definable real number

## Proof theory

* Metamathematics
* Cut-elimination
* Tarski's undefinability theorem
* Diagonal lemma
* Provability logic
* Interpretability logic
* Sequent
* Sequent calculus
* Analytic proof
* Structural proof theory
* Self-verifying theories
* Substructural logics
** Structural rule
*** Weakening
*** Contraction
** Linear logic
*** Intuitionistic linear logic
*** Proof net
** Affine logic
** Strict logic
** Relevant logic
* Proof-theoretic semantics
* Ludics
* System F
* Gerhard Gentzen
* Gentzen's consistency proof
* Reverse mathematics
* Nonfirstorderizability
* Interpretability
* Weak interpretability
* Cointerpretability
* Tolerant sequence
* Cotolerant sequence
* Deduction theorem
* Cirquent calculus

## Mathematical constructivism

* Nonconstructive proof
* Existence theorem
* Intuitionistic logic
* Intuitionistic type theory
* Type theory
* Lambda calculus
** Church–Rosser theorem
* Simply typed lambda calculus
* Typed lambda calculus
* Curry–Howard isomorphism
* Calculus of constructions
* Constructivist analysis
* Lambda cube
* System F
* Introduction to topos theory
* LF (logical framework)
* Computability logic
* Computable measure theory
* Finitism
* Ultraintuitionism
* Luitzen Egbertus Jan Brouwer

## Modal logic

* Kripke semantics
* Sahlqvist formula
* Interior algebra

## Theorem provers

* First-order resolution
* Automated theorem proving
* ACL2 theorem prover
* E equational theorem prover
* Gandalf theorem prover
* HOL theorem prover
* Isabelle theorem prover
* LCF theorem prover
* Otter theorem prover
* Paradox theorem prover
* Vampire theorem prover
* Interactive proof system
* Mizar system
* QED project
* Coq

## Discovery systems

* Automated Mathematician
* Eurisko

## Historical

* Begriffsschrift
* Systems of Logic Based on Ordinals – Alan Turing's Ph.D. thesis