List of mathematical logic topics

This is a list of mathematical logic topics.

For traditional syllogistic logic, see the

• 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
• 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 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
  • 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)
• 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
  • 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

See also

• Kurt Gödel
• Alfred Tarski
• Saharon Shelah