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  "body": "\n## Summary\n\nA double-negation translation is a map from formulas to formulas that embeds classical logic into [[intuitionistic logic|Q11N8X99FK]].\nIt inserts [[negations|0XR7A7RSXB]] $\\neg \\neg$ at chosen places, so that a formula is classically provable exactly when its translation is intuitionistically provable.\nThe best known versions are due to Kolmogorov, to Gödel and Gentzen, and to Kuroda.\nIt shows that [[Peano arithmetic|0QFT7Z76N5]] is consistent if [[Heyting arithmetic|CK8ZQRN183]] is.\n\n## Explanations\n\n### L1 human\n\nIntuitionistic logic rejects some classical reasoning, such as \"$A$ or not $A$\".\nBut it accepts \"it is not the case that neither $A$ nor not $A$\", that is, $\\neg \\neg (A \\vee \\neg A)$.\nThe translation rewrites each classical claim in such a doubly negated, weaker form.\nThen every classical proof becomes an intuitionistic proof of the weaker claim.\nSo a contradiction in classical reasoning would give a contradiction in intuitionistic reasoning too.\n",
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