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  "body": "\n## Summary\n\nHeyting arithmetic, HA, is the theory of the natural numbers with the axioms of [[Peano arithmetic|0QFT7Z76N5]] and the rules of [[intuitionistic logic|Q11N8X99FK]].\nIt is the standard formal system for constructive arithmetic.\nEvery existence theorem of HA comes with a way to compute a witness.\nIt proves the same formulas as Peano arithmetic up to a double-negation translation, so the two theories are equally consistent.\n\n## Explanations\n\n### L1 human\n\nHeyting arithmetic keeps the usual axioms of the natural numbers, including induction.\nIt only removes one rule of logic: the law that every statement is either true or false.\nSo it can prove \"$A$ or $B$\" only when it can tell which of the two holds.\nFor example, it proves that every number is even or odd, because one can compute which.\nIt cannot prove the statement \"every program halts or does not halt\", since no algorithm decides which.\n",
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