List of mathematical logic topics explained

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
    • • 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

See also

• Kurt Gödel
• Alfred Tarski
• Saharon Shelah