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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Table of 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

Working foundations

• Peano axioms
• Mathematical induction
  • Three forms of mathematical induction
  • Structural induction
• Naive set theory
  • Simple theorems in set theory
  • Power set
  • Empty set
  • Empty function
• Universe (mathematics)
• Axiomatization
• Axiomatic system
• Mathematical proof
• Tautology
• Consistency
• Arithmetization of analysis
• Foundations of mathematics
• Generating arithmetic
• Formal language
• Giuseppe Peano

• Functional predicate
• First-order predicate calculus
  • Prenex normal form
  • Skolem normal form
• Compactness theorem
• Löwenheim-Skolem theorem
• Soundness theorem
• Gödel's completeness theorem
  • Original proof of Gödel's completeness theorem
• Gödel's incompleteness theorem
• Ultraproduct
• Non-standard analysis
  • Hyperreal number
• Second-order logic
• Whitehead problem
• Kurt Gödel
• Alfred Tarski
• Saharon Shelah

• 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
  • 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
• Zermelo set theory
• Class (set theory)
• Continuum hypothesis
• Forcing (mathematics)
• Freiling's Axiom of Symmetry
• Goodstein's theorem
• Gödel's constructible universe
• Von Neumann universe
• Hereditarily finite set
• Normal function
  • Fixed-point lemma for normal functions
• Suslin's problem
• Ernst Zermelo
• Paul Cohen

• Erdös cardinal
• Extendible cardinal
• Huge cardinal
• Inaccessible cardinal
• Ineffable cardinal
• Mahlo cardinal
• Measurable cardinal
• N-huge cardinal
• N-Mahlo cardinal
• Ramsey cardinal
• Rank-into-rank
• Strong cardinal
• Strongly inaccessible cardinal
• Subtle cardinal
• Supercompact cardinal
• Superstrong cardinal
• Totally indescribable cardinal
• Weakly compact cardinal
• Weakly inaccessible cardinal
• Woodin cardinal

• Entscheidungsproblem
• Decision problem
• Existence theorem
• Church-Turing thesis
• Recursive function
  • Algorithm
  • Recursion
  • Primitive recursive function
  • Ackermann function
  • Turing machine
  • Halting problem
  • Computability theory, computation
  • Herbrand Universe
  • Markov algorithm
  • Lambda calculus
    • Church-Rosser theorem
  • Combinatory logic
  • Post correspondence problem
• Kleene's recursion theorem
• Recursively enumerable set
  • Recursively enumerable language
• Decidable language
• Undecidable language
• Rice's theorem
• 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 classes P and NP
  • NP-complete
  • Time hierarchy theorem
• Hypercomputation
  • Oracle machine
• Alonzo Church
• Emil Post
• Alan Turing
• Jacques Herbrand
• Haskell Curry
• Stephen Cole Kleene

• Metamathematics
• Sequent
• Sequent calculus
• Substructural logics
  • Relevant logic
• Gerhard Gentzen