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  "body": "\n## Definition\n\n*Heyting arithmetic*, written HA, is the first-order theory in the language $0, S, +, \\cdot, =$ with these parts.\n\n1. Its logic is first-order [[intuitionistic logic|Q11N8X99FK]] with equality.\n2. Its axioms are those of first-order [[Peano arithmetic|0QFT7Z76N5]]. They are $S x \\ne 0$, $S x = S y \\to x = y$, and $x + 0 = x$, $x + S y = S(x + y)$, $x \\cdot 0 = 0$, $x \\cdot S y = x \\cdot y + x$. They also include the induction scheme $A(0) \\wedge \\forall x\\, (A(x) \\to A(S x)) \\to \\forall x\\, A(x)$ for every formula $A$.\n\nThus HA has the same axioms as Peano arithmetic PA.\nThe theory PA is HA together with the [[law of excluded middle|0RYYK8PG4S]] $A \\vee \\neg A$ for all formulas $A$.\n\n## Remarks\n\nExample: HA proves $\\forall x\\, \\forall y\\, (x = y \\vee \\neg x = y)$, by induction on $x$ and $y$.\nSo excluded middle holds in HA for equations, and, by induction on formulas, for all formulas without unbounded quantifiers.\nNon-example: HA does not prove $\\forall x\\, (\\exists y\\, T(x, x, y) \\vee \\neg \\exists y\\, T(x, x, y))$, where $T$ is Kleene's predicate for \"program $x$ on input $x$ halts within $y$ steps\".\nThis formula is provable in PA, so the two theories differ.\nHA has the disjunction property: if HA proves a closed formula $A \\vee B$, then it proves $A$ or it proves $B$.\nIt has the existence property: if HA proves a closed formula $\\exists x\\, A(x)$, then it proves $A(\\overline{n})$ for some numeral $\\overline{n}$.\n",
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