Math 2.0

Mathematical logic and foundations Constructive mathematics Concept

Intuitionistic type theory

Builds on Type theoryIntuitionistic logicCurry–Howard correspondence

Summary

Intuitionistic type theory is the dependent type theory introduced by Per Martin-Löf from the 1970s on, also called Martin-Löf type theory. It is a foundation of constructive mathematics based on intuitionistic logic. Through the Curry–Howard correspondence, its types are also propositions, and its terms are both proofs and programs. It is the basis of proof assistants such as Agda, and it is close to the systems behind Coq and Lean.

Reading path

To understand this page from the ground up, read these 23 pages in order. The foundations come first.

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Element, definition, no evidence levelElementdefinitionIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptLambda calculus, concept, no evidence levelLambda calculusconceptProposition, concept, no evidence levelPropositionconceptSet, concept, no evidence levelSetconceptSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptModus ponens, concept, no evidence levelModus ponensconceptMathematical proof, concept, no evidence levelMathematical proofconceptPropositional logic, concept, no evidence levelPropositional logicconceptTautology, definition, no evidence levelTautologydefinitionLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptRussell's paradox, statement, no evidence levelRussell's paradoxstatementType theory, concept, no evidence levelType theoryconceptCurry–Howard correspondence, concept, no evidence levelCurry–Howard correspo…conceptIntuitionistic type theory, concept, no evidence levelIntuitionistic type t…concept
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Intuitionistic logic concept
  3. Lambda calculus concept
  4. Proposition concept
  5. Set concept
  6. Set definition
  7. Conjunction definition
  8. Disjunction definition
  9. Implication definition
  10. Negation definition
  11. Biconditional definition
  12. Truth table concept
  13. Modus ponens concept
  14. Mathematical proof concept
  15. Propositional logic concept
  16. Tautology definition
  17. Logical equivalence definition
  18. Double negation law statement
  19. Proof by contradiction concept
  20. Russell's paradox statement
  21. Type theory concept
  22. Curry–Howard correspondence concept
  23. Intuitionistic type theory concept

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Intuitionistic_type_theory
About this page
TagESVYD9CZ8C
URIm2://ESVYD9CZ8C
Kindconcept
WritingW0 stub
MSC03B38, 03F55, 03F50
SpecializesType theory
UsesType theory, Intuitionistic logic, Curry–Howard correspondence
Same as
Changed2 November 2026

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