# ⓘ List of mathematical logic topics

## ⓘ 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.

## 1. Working foundations

• Giuseppe Peano
• Peano axioms
• Recursive definition
• Mathematical induction
• Structural induction
• Element mathematics
• Ur-element
• Naive set theory
• Empty function
• Power set
• Simple theorems in the algebra of sets
• Non-empty set
• Empty set
• Algebra of sets
• Singleton mathematics
• Axiom schema
• Axiomatization
• Axiomatic system
• Universe mathematics
• Nonconstructive proof
• Mathematical proof
• Axiomatic method
• Constructive proof
• Formal system
• Direct proof
• Proof by exhaustion
• Formal language
• Foundations of mathematics
• Arithmetization of analysis
• Principia Mathematica
• Algebraic logic
• Impredicative
• Definable real number
• Consistency proof
• Tautology
• Boolean algebra logic
• Hilberts program
• categorical logic
• Dialectica space

## 2. Model theory

• Finite model theory
• Model checking
• Descriptive complexity theory
• Trakhtenbrots theorem
• Undecidable problem
• Tarskis exponential function problem
• Computable model theory
• Institution computer science
• Institutional model theory
• Hyperinteger
• Overspill
• Criticism of non-standard analysis
• Non-standard analysis
• Transfer principle
• Standard part function
• Elementary Calculus: An Infinitesimal Approach
• Non-standard calculus
• Hyperreal number
• Boolean-valued model
• Set theory
• Forcing mathematics
• General frame
• Kripke semantics
• Predicate logic
• Infinitary logic
• First-order logic
• Many-sorted logic
• Lindstrom quantifier
• Higher-order logic
• Second-order logic
• Original proof of Godels completeness theorem
• Soundness theorem
• Godels completeness theorem
• Compactness theorem
• Lowenheim–Skolem theorem
• Substructure mathematics
• Skolem hull
• Structure mathematical logic
• Elementary substructure
• Godels incompleteness theorems
• Interpretation logic
• Hrushovski construction
• Ultraproduct
• Saturated model
• Non-standard model
• Age model theory
• Atomic model mathematical logic
• Amalgamation property
• Prime model
• Existentially closed model
• Theory mathematical logic
• Potential isomorphism
• Vaughts test
• Complete theory
• Morley rank
• Stable group
• Forking extension
• Tame group
• Morleys categoricity theorem
• Strongly minimal theory
• Stable theory
• Stability spectrum
• Weakly o-minimal structure
• o-minimal theory
• Spectrum of a theory
• C-minimal theory
• Vaught conjecture
• Model complete theory
• Pseudoelementary class
• Elementary class
• List of first-order theories
• Strength mathematical logic
• Conservative extension
• Ax–Kochen theorem
• Non-standard model of arithmetic
• Herbrand interpretation / Herbrand structure
• Pregeometry model theory
• Wilkies theorem
• Type model theory
• Signature logic
• Arithmetical set
• Ehrenfeucht–Fraïsse game
• Barwise compactness theorem
• Peano axioms
• Functional predicate
• Skolemization
• Differentially closed field
• Skolem normal form
• Lobs theorem
• T-schema
• First-order arithmetic
• Presburger arithmetic
• Reduct
• Definable set
• Interpretation model theory / Interpretable structure
• Quantifier elimination
• Back-and-forth method
• Indiscernibles
• Second-order arithmetic
• Imaginary element
• Lindenbaum–Tarski algebra
• Ax–Grothendieck theorem
• Exponential field
• Zariski geometry

## 3. Set theory

• Axiom of choice
• Axiom of countable choice
• Axiom of dependent choice
• Zorns lemma
• Algebra of sets
• Cantors diagonal argument
• Cantors back-and-forth method
• Aleph-null
• Cardinality
• Boolean algebra structure
• Aleph-one
• Cantors first uncountability proof
• Boolean-valued model
• Cantors theorem
• Cantor–Bernstein–Schroeder theorem
• Aleph number
• Beth number
• Cardinal number
• Hartogs number
• Complement set theory
• Continuum set theory
• Suslins problem
• Class set theory
• Cartesian product
• Complete Boolean algebra
• Uniformization set theory
• Perfect set property
• Prewellordering
• Infinity-Borel set
• Borel equivalence relation
• Countable set
• Continuum hypothesis
• Property of Baire
• Universally measurable set
• Analytical hierarchy
• Projective set
• Descriptive set theory
• Polish space
• Lightface analytic game
• Analytic set
• Determinacy
• Axiom of determinacy
• Axiom of projective determinacy
• Axiom of real determinacy
• LR
• Intersection set theory
• ∈-induction
• Empty set
• Ordinal number
• Large cardinal property
• Musical set theory
• Successor ordinal
• Limit ordinal
• Infinite descending chain
• L
• Fuzzy set
• Forcing mathematics
• Transfinite induction
• Internal set theory
• Well-order
• Well-founded set
• Kripke–Platek set theory with urelements
• New Foundations
• Naive set theory
• Positive set theory
• Set theory
• Axiomatic set theory
• Zermelo–Fraenkel set theory
• Alternative set theory
• Morse–Kelley set theory
• Power set
• Zermelo set theory
• Set mathematics
• Θ set theory
• Subset
• Tree set theory
• Tree descriptive set theory
• Von Neumann universe
• Union set theory
• Simple theorems in the algebra of sets
• Zero sharp

## 4. Large cardinals

• Strongly inaccessible cardinal
• Ramsey cardinal
• Erdos cardinal
• Hyper-Woodin cardinal
• Unfoldable cardinal
• Measurable cardinal
• Ineffable cardinal
• Shelah cardinal
• Inaccessible cardinal
• Rank-into-rank
• Weakly compact cardinal
• N-huge cardinal
• Woodin cardinal
• Huge cardinal
• Mahlo cardinal
• Weakly hyper-Woodin cardinal
• Strong cardinal
• Weakly inaccessible cardinal
• Subtle cardinal
• Superstrong cardinal
• Supercompact cardinal
• Extendible cardinal
• Remarkable cardinal
• Almost Ramsey cardinal
• Totally indescribable cardinal

## 5. Recursion theory

• Primitive recursive function
• Decision problem
• Ackermann function
• Church-Rosser theorem
• Herbrand Universe
• Mu operator
• Halting problem
• Entscheidungsproblem
• Computability theory, computation
• Recursion
• Calculus of constructions
• Markov algorithm
• Lambda calculus
• Church-Turing thesis
• Turing machine
• Algorithm
• Computable function
• Decidability logic
• Post correspondence problem
• Combinatory logic
• Kleenes recursion theorem
• Recursively enumerable language
• Recursively enumerable set
• Diophantine set
• Effective results in number theory
• Arithmetical hierarchy
• Computational complexity theory
• Polynomial hierarchy
• Rices theorem
• Exponential time
• Undecidable language
• Decidable language
• Word problem for groups
• Subrecursion theory
• Exponential hierarchy
• Posts theorem
• Turing degree
• Complexity class
• Matiyasevichs theorem
• Polynomial time
• Presburger arithmetic
• Cooks theorem
• Complexity classes P and NP
• List of complexity classes
• NP-complete
• Space hierarchy theorem
• Time hierarchy theorem
• Natural proof
• Oracle machine
• Hypercomputation
• Alonzo Church
• Stephen Cole Kleene
• Definable real number
• Emil Post
• Alan Turing
• Jacques Herbrand
• Rozsa Peter

## 6. Proof theory

• Contraction
• Sequent
• Tarskis undefinability theorem
• Structural rule
• Cut-elimination
• Diagonal lemma
• Sequent calculus
• Structural proof theory
• Substructural logics
• Self-verifying theories
• Metamathematics
• Interpretability logic
• Provability logic
• Analytic proof
• Weakening
• Proof net
• Linear logic
• Intuitionistic linear logic
• Relevant logic
• Affine logic
• Strict logic
• Cotolerant sequence
• System F
• Reverse mathematics
• Nonfirstorderizability
• Gentzens consistency proof
• Gerhard Gentzen
• Tolerant sequence
• Deduction theorem
• Cirquent calculus
• Ludics
• Weak interpretability
• Proof-theoretic semantics
• Cointerpretability
• Interpretability

## 7. Mathematical constructivism

• Church–Rosser theorem
• Lambda calculus
• Existence theorem
• Nonconstructive proof
• Intuitionistic type theory
• Type theory
• Intuitionistic logic
• Luitzen Egbertus Jan Brouwer
• Computability logic
• System F
• Ultraintuitionism
• Constructivist analysis
• Calculus of constructions
• Typed lambda calculus
• Curry–Howard isomorphism
• Introduction to topos theory
• Simply typed lambda calculus
• Computable measure theory
• Finitism
• Lambda cube
• LF logical framework

## 8. Theorem provers

• Isabelle theorem prover
• E equational theorem prover
• Coq
• HOL theorem prover
• First-order resolution
• Vampire theorem prover
• LCF theorem prover
• Gandalf theorem prover
• QED project
• Automated theorem proving
• Otter theorem prover
• ACL2 theorem prover
• Interactive proof system
• Mizar system

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