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  "body": "\n## Definition\n\nFix a first-order language, and read $\\neg A$ as $A \\to \\bot$.\nThe *Gödel–Gentzen negative translation* is the map $A \\mapsto A^N$ on formulas defined by these clauses.\n\n1. $\\bot^N = \\bot$, and $P^N = \\neg \\neg P$ for every other atomic formula $P$.\n2. $(A \\wedge B)^N = A^N \\wedge B^N$.\n3. $(A \\vee B)^N = \\neg (\\neg A^N \\wedge \\neg B^N)$.\n4. $(A \\to B)^N = A^N \\to B^N$.\n5. $(\\forall x\\, A)^N = \\forall x\\, A^N$.\n6. $(\\exists x\\, A)^N = \\neg \\forall x\\, \\neg A^N$.\n\nFor a set of formulas $\\Gamma$, let $\\Gamma^N = \\set{A^N : A \\in \\Gamma}$.\nIts main property is this: $\\Gamma \\vdash A$ in classical logic if and only if $\\Gamma^N \\vdash A^N$ in [[intuitionistic logic|Q11N8X99FK]].\nA *double-negation translation* is any map with this property obtained by inserting double [[negations|0XR7A7RSXB]], such as the translations of Kolmogorov and Kuroda.\n\n## Remarks\n\nExample: for the [[law of excluded middle|0RYYK8PG4S]] with $P$ atomic, $(P \\vee \\neg P)^N = \\neg (\\neg \\neg \\neg P \\wedge \\neg \\neg \\neg \\neg P)$.\nSince $\\neg \\neg \\neg P$ is equivalent to $\\neg P$ in intuitionistic logic, this is equivalent to $\\neg (\\neg P \\wedge \\neg \\neg P)$, which is intuitionistically provable.\nNon-example: the identity map $A \\mapsto A$ is not a double-negation translation, because $P \\vee \\neg P$ is classically provable but not intuitionistically provable.\nFor arithmetic, [[Peano arithmetic|0QFT7Z76N5]] proves $A$ if and only if [[Heyting arithmetic|RXHJPVMPJH]] proves $A^N$.\nIn particular, if HA is consistent, then PA is consistent, because $\\bot^N = \\bot$.\nIn arithmetic, atomic formulas are equations, which are decidable in HA, so one may take $P^N = P$.\n",
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