Editing List of mathematical logic topics

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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 [[algorithm]]s.

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

==Discovery systems==

*[[Automated Mathematician]]
*[[Eurisko]]

==Historical==

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

==See also==
{{Portal|Philosophy}}
*[[Kurt Gödel]]
*[[Alfred Tarski]]
*[[Saharon Shelah]]

{{Logic}}

[[Category:Mathematics-related lists|Logic]]
[[Category:Mathematical logic| L]]
[[Category:Outlines of mathematics and logic|Mathematical logic]]
[[Category:Outlines|Mathematical logic]]
[[Category:Lists of topics|Mathematical logic]]