Math 2.0

Mathematical logic and foundations Constructive mathematics Concept

Double-negation translation

Builds on Intuitionistic logicNegationPeano axiomsHeyting arithmetic

Summary

A double-negation translation is a map from formulas to formulas that embeds classical logic into intuitionistic logic. It inserts negations ¬¬\neg \neg at chosen places, so that a formula is classically provable exactly when its translation is intuitionistically provable. The best known versions are due to Kolmogorov, to Gödel and Gentzen, and to Kuroda. It shows that Peano arithmetic is consistent if Heyting arithmetic is.

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Intuitionistic logic, concept, no evidence levelIntuitionistic logicconceptPeano axioms, concept, no evidence levelPeano axiomsconceptProposition, concept, no evidence levelPropositionconceptHeyting arithmetic, concept, no evidence levelHeyting arithmeticconceptNegation, definition, no evidence levelNegationdefinitionDouble-negation translation, concept, no evidence levelDouble-negation trans…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. Intuitionistic logic concept
  2. Peano axioms concept
  3. Proposition concept
  4. Heyting arithmetic concept
  5. Negation definition
  6. Double-negation translation concept

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Double-negation_translation
About this page
TagHGJVM49PA4
URIm2://HGJVM49PA4
Kindconcept
WritingW0 stub
MSC03F25, 03F55, 03B20
UsesIntuitionistic logic, Negation, Peano axioms, Heyting arithmetic
Same as
Changed2 November 2026

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