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  "body": "\n## Definition\n\nLet $\\mathrm{T}$ be Gödel's system of [[primitive recursive|5NVA2JRHPV]] functionals of finite type.\nTo each formula $A$ of [[Heyting arithmetic|RXHJPVMPJH]], the *Dialectica interpretation* assigns a formula $A^D = \\exists x\\, \\forall y\\, A_D(x, y)$.\nHere $x$ and $y$ are finite tuples of variables of finite type, and $A_D$ is quantifier-free.\nWrite $A^D = \\exists x\\, \\forall y\\, A_D(x, y)$ and $B^D = \\exists u\\, \\forall v\\, B_D(u, v)$.\nThe clauses are these.\n\n1. If $A$ is atomic, then $A^D = A_D = A$, with empty tuples $x$ and $y$.\n2. $(A \\wedge B)^D = \\exists x, u\\, \\forall y, v\\, \\bigl( A_D(x, y) \\wedge B_D(u, v) \\bigr)$.\n3. $(A \\vee B)^D = \\exists z, x, u\\, \\forall y, v\\, \\bigl( (z = 0 \\to A_D(x, y)) \\wedge (z \\ne 0 \\to B_D(u, v)) \\bigr)$, where $z$ is a number variable.\n4. $(A \\to B)^D = \\exists f, g\\, \\forall x, v\\, \\bigl( A_D(x, f x v) \\to B_D(g x, v) \\bigr)$.\n5. $(\\exists z\\, A(z))^D = \\exists z, x\\, \\forall y\\, A_D(x, y, z)$.\n6. $(\\forall z\\, A(z))^D = \\exists f\\, \\forall z, y\\, A_D(f z, y, z)$.\n\nNegation $\\neg A$ is read as $A \\to (0 = 1)$.\n\n## Remarks\n\nSoundness theorem (Gödel): if Heyting arithmetic proves a closed formula $A$, then there is a tuple $t$ of closed terms of $\\mathrm{T}$ such that $\\mathrm{T}$ proves $A_D(t, y)$.\nExample: for $A = \\forall n\\, \\exists m\\, (n \u003c m)$, clauses 5 and 6 give $A^D = \\exists f\\, \\forall n\\, (n \u003c f n)$, and the term $f = \\lambda n.\\, n + 1$ is a witness.\nNon-example: let $P(n, k)$ be the primitive recursive relation, given by [[Kleene's T predicate|S9DK13K5PS]], which says that the program $n$ on input $n$ halts within $k$ steps.\nThe formula $\\forall n\\, (\\exists k\\, P(n, k) \\vee \\neg \\exists k\\, P(n, k))$ has no witness in $\\mathrm{T}$.\nA witness would give a computable function $n \\mapsto z$ that decides the halting problem.\nCombined with a [[double-negation translation|HGJVM49PA4]], it interprets Peano arithmetic in system $\\mathrm{T}$.\n",
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