javascript is disabled, enable it otherwise the site will not work properly!

javascript est désactivé, activez-le sinon le site ne fonctionnera pas correctement !

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