Math 2.0

Mathematical logic and foundations Proof theory Concept

Realizability

Builds on Heyting arithmeticComputable functionBrouwer–Heyting–Kolmogorov interpretation

Summary

Realizability is a way to read intuitionistic statements as statements about programs. A realizer of a formula is a program, often the code of a computable function, that carries the computational content of a proof of it. Kleene introduced number realizability for Heyting arithmetic in 1945, as a precise form of the Brouwer–Heyting–Kolmogorov interpretation. Realizability shows that intuitionistic proofs of existence yield algorithms, and it gives models in which classical principles fail.

Reading path

To understand this page from the ground up, read these 25 pages in order. The foundations come first.

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Computable function, concept, no evidence levelComputable functionconceptIntuitionistic logic, concept, no evidence levelIntuitionistic logicconceptPeano axioms, concept, no evidence levelPeano axiomsconceptProposition, concept, no evidence levelPropositionconceptHeyting arithmetic, concept, no evidence levelHeyting arithmeticconceptConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionNegation, definition, no evidence levelNegationdefinitionPredicate, definition, no evidence levelPredicatedefinitionBiconditional, definition, no evidence levelBiconditionaldefinitionTruth table, concept, no evidence levelTruth tableconceptLaw of excluded middle, statement, no evidence levelLaw of excluded middlestatementModus ponens, concept, no evidence levelModus ponensconceptExistential quantifier, definition, no evidence levelExistential quantifierdefinitionPropositional logic, concept, no evidence levelPropositional logicconceptMathematical proof, concept, no evidence levelMathematical proofconceptTautology, definition, no evidence levelTautologydefinitionConstructive proof, concept, no evidence levelConstructive proofconceptLogical equivalence, definition, no evidence levelLogical equivalencedefinitionDouble negation law, statement, no evidence levelDouble negation lawstatementProof by contradiction, concept, no evidence levelProof by contradictionconceptConstructivism, concept, no evidence levelConstructivismconceptBrouwer–Heyting–Kolmogorov interpretation, concept, no evidence levelBrouwer–Heyting–Kolmo…conceptRealizability, concept, no evidence levelRealizabilityconcept
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. Computable function concept
  2. Intuitionistic logic concept
  3. Peano axioms concept
  4. Proposition concept
  5. Heyting arithmetic concept
  6. Conjunction definition
  7. Disjunction definition
  8. Implication definition
  9. Negation definition
  10. Predicate definition
  11. Biconditional definition
  12. Truth table concept
  13. Law of excluded middle statement
  14. Modus ponens concept
  15. Existential quantifier definition
  16. Propositional logic concept
  17. Mathematical proof concept
  18. Tautology definition
  19. Constructive proof concept
  20. Logical equivalence definition
  21. Double negation law statement
  22. Proof by contradiction concept
  23. Constructivism concept
  24. Brouwer–Heyting–Kolmogorov interpretation concept
  25. Realizability concept

Further reading

  1. Web https://en.wikipedia.org/wiki/Realizability
About this page
Tag4H35VQ16DC
URIm2://4H35VQ16DC
Kindconcept
WritingW0 stub
MSC03F50, 03F55, 03D80
UsesHeyting arithmetic, Computable function, Brouwer–Heyting–Kolmogorov interpretation
Same as
Changed2 November 2026

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