Math 2.0

Mathematical logic and foundations Definition

Realizability

Builds on Heyting arithmeticHeyting arithmeticComputable functionBrouwer–Heyting–Kolmogorov interpretation

Definition

Let {e}\{e\} denote the partial computable function with code e∈ℕe \in \N, and write {e}(n)↓\{e\}(n)\downarrow when it is defined at nn. Fix a computable pairing of natural numbers with projections (e)0(e)_0 and (e)1(e)_1. Number realizability is the relation "ee realizes AA", written e⊩Ae \Vdash A, between a number ee and a closed formula AA of Heyting arithmetic. It is defined by induction on AA.

  1. e⊩(s=t)e \Vdash (s = t) if s=ts = t is true. No number realizes 0=10 = 1.
  2. e⊩A∧Be \Vdash A \wedge B if (e)0⊩A(e)_0 \Vdash A and (e)1⊩B(e)_1 \Vdash B.
  3. e⊩A∨Be \Vdash A \vee B if either (e)0=0(e)_0 = 0 and (e)1⊩A(e)_1 \Vdash A, or (e)0≠0(e)_0 \ne 0 and (e)1⊩B(e)_1 \Vdash B.
  4. e⊩A→Be \Vdash A \to B if for every nn with n⊩An \Vdash A, {e}(n)↓\{e\}(n)\downarrow and {e}(n)⊩B\{e\}(n) \Vdash B.
  5. e⊩∃x A(x)e \Vdash \exists x\, A(x) if (e)1⊩A((e)0‾)(e)_1 \Vdash A(\overline{(e)_0}), where k‾\overline{k} is the numeral of kk.
  6. e⊩∀x A(x)e \Vdash \forall x\, A(x) if for every nn, {e}(n)↓\{e\}(n)\downarrow and {e}(n)⊩A(n‾)\{e\}(n) \Vdash A(\overline{n}).

A closed formula is realizable if some number realizes it. Negation ¬A\neg A is read as A→(0=1)A \to (0 = 1).

Remarks

Soundness theorem (Kleene): if Heyting arithmetic proves a closed formula AA, then AA is realizable. Example: ∀x ∃y (y=x+x)\forall x\, \exists y\, (y = x + x) is realized by a code ee of a total function with {e}(n)=⟨2n,0⟩\{e\}(n) = \langle 2n, 0 \rangle. Indeed, {e}(n)\{e\}(n) is defined, its first part is 2n2n, and its second part realizes the true equation 2n=n+n2n = n + n by clause 1. Non-example: let H(x)H(x) say that the program xx halts on input xx. A realizer of ∀x (H(x)∨¬H(x))\forall x\, (H(x) \vee \neg H(x)) would compute, from xx, a bit telling whether H(x)H(x) holds, and so solve the halting problem. So this instance of the law of excluded middle is not realizable, and hence not provable in Heyting arithmetic. Many variants exist, such as realizability by functions, by terms of a typed calculus, and in toposes.

Reading path

To understand this page from the ground up, read these 26 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 quantifierdefinitionHeyting arithmetic, definition, no evidence levelHeyting arithmeticdefinitionPropositional 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, definition, no evidence levelRealizabilitydefinition
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. Heyting arithmetic definition
  17. Propositional logic concept
  18. Mathematical proof concept
  19. Tautology definition
  20. Constructive proof concept
  21. Logical equivalence definition
  22. Double negation law statement
  23. Proof by contradiction concept
  24. Constructivism concept
  25. Brouwer–Heyting–Kolmogorov interpretation concept
  26. Realizability definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Realizability
About this page
TagE6MB6B3YQ0
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Kinddefinition
WritingW0 stub
MSC03F50, 03F55, 03D80
DefinesRealizability
UsesHeyting arithmetic, Heyting arithmetic, Computable function, Brouwer–Heyting–Kolmogorov interpretation
Changed2 November 2026

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