This is a list of mathematical logic topics, by Wikipedia page.

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.

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Contents

1 Working foundations 2 Model theory 3 Set theory 4 Descriptive set theory 5 Large cardinals 6 Recursion theory 7 Proof theory 8 Mathematical constructivism 9 Modal logic 10 Theorem provers 11 Discovery systems 12 Historical

Working foundations

• Peano axioms
  • Giuseppe Peano
• Mathematical induction
  • Three forms of mathematical induction
  • Structural induction
  • Recursive definition
• Naive set theory
  • Element (mathematics)
    • Ur-element
  • Singleton (mathematics)
  • Simple theorems in the algebra of sets
  • The algebra of sets
  • Power set
  • Empty set
  • Non-empty set
  • Empty function
• Universe (mathematics)
• Axiomatization
• Axiomatic system
  • Axiom schema
• Axiomatic method
• Mathematical proof
  • Direct proof
  • Reductio ad absurdum
  • Proof by exhaustion
  • Constructive proof
  • Nonconstructive proof
• Tautology
• Consistency
• Arithmetization of analysis
• Foundations of mathematics
• Generating arithmetic
• Formal language
• Principia Mathematica
• Hilbert's program

Model theory

• Functional predicate, monadic
• First-order predicate calculus
  • Prenex normal form
  • Skolem normal form
• Lindenbaum-Tarski algebra
• T-schema
• Maximal consistent set
• Compactness theorem
• Löwenheim-Skolem theorem
• Soundness theorem
• Gödel's completeness theorem
  • Original proof of Gödel's completeness theorem
• Deduction theorem
• Elementary embedding
• Elementary substructure
• Löb's theorem
• Gödel's incompleteness theorem
• Ultraproduct
• Saturated model
• Non-standard analysis
  • Hyperreal number
  • Non-standard calculus
  • Transfer principle
  • Overspill
• Tolerant sequence
• Cotolerant sequence
• Interpretability
• Weak interpretability
• Cointerpretability
• Second-order logic
• Whitehead problem
• Thompson groups
• Kurt Gödel
• Alfred Tarski
• Saharon Shelah

Set theory

• Axiomatic set theory
• Well-founded relation
• Transfinite number
  • Hilbert's paradox of the Grand Hotel
• Ordinal number
  • Successor ordinal
  • Burali-Forti paradox
  • Limit ordinal
• Equinumerous
• Cardinal number
  • Successor cardinal
  • Countable set
  • Uncountable set
  • Cantor's first uncountability proof
  • Cantor's diagonal argument
  • Cantor's theorem
  • König's theorem
  • Aleph number
  • Beth number
  • Gimel function
  • 2 to the power of C
  • Regular cardinal
  • Limit cardinal
  • Zero sharp
  • Cardinal assignment
  • Von Neumann cardinal assignment
• Cantor-Bernstein-Schroeder theorem
• Zermelo-Fraenkel axioms
  • Axiom of regularity
  • Axiom of extensionality
  • Axiom of pairing
  • Axiom of empty set
  • Axiom of power set
  • Axiom of union
  • Axiom of infinity
  • Axiom schema of specification
  • Axiom schema of replacement
  • Axiom of choice
    • Zorn's lemma
    • Well-ordering theorem
    • Well-ordering principle
    • Transfinite induction
    • Hausdorff maximality theorem
    • Axiom of dependent choice
    • Axiom of countable choice
    • Boolean prime ideal theorem
    • Ultrafilter lemma
  • List of statements undecidable in ZFC
• Zermelo set theory
• Von Neumann-Bernays-Gödel axioms
• Class (set theory)
• Continuum hypothesis
• Forcing (mathematics)
• Freiling's axiom of symmetry
• Goodstein's theorem
• Internal set theory
• Gödel's constructible universe
  • Axiom of constructibility
• Von Neumann universe
• Hereditarily finite set
• Normal function
  • Fixed-point lemma for normal functions
• Covering lemma
• Stratification (mathematics)
• Suslin's problem
• Clubsuit
• Diamondsuit
• Ernst Zermelo
• Paul Cohen
• New Foundations

Descriptive set theory

• Analytical hierarchy

Large cardinals

• Almost Ramsey cardinal
• Erdös cardinal
• Extendible cardinal
• Huge cardinal
• Hyper-Woodin cardinal
• Inaccessible cardinal
• Ineffable cardinal
• Mahlo cardinal
• Measurable cardinal
• N-huge cardinal
• N-Mahlo 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
• Recursive 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
• Alonzo Church
• Emil Post
• Alan Turing
• Jacques Herbrand
• Haskell Curry
• Stephen Cole Kleene

Proof theory

• Metamathematics
• Cut-elimination
• Tarski's indefinability 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
• Reverse mathematics
• Nonfirstorderizability

Mathematical constructivism

• Nonconstructive proof
• Existence theorem
• Intuitionistic logic
  • Type theory
  • Introduction to topos theory
  • LF (logical framework)
  • Lambda cube
• Constructivist analysis
• 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

• Autoclass
• Automated Mathematician
• Eurisko

Historical

• Begriffschrift