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  "body": "\n## Statement\n\nLet $M = (\\N, 0^M, S^M, \\oplus, \\otimes)$ be a [[model|MVCH6N9SNW]] of the [[Peano axioms|0QFT7Z76N5]] $\\mathsf{PA}$ whose domain is $\\N$, and suppose $M$ is not isomorphic to the standard model.\nThen neither $\\oplus \\colon \\N^2 \\to \\N$ nor $\\otimes \\colon \\N^2 \\to \\N$ is a [[computable function|KZSB5E471K]].\nIn particular, no [[countable|01TQB88H4C]] non-standard model of $\\mathsf{PA}$ is computable.\n\n## Remarks\n\nProof idea: take two computably enumerable sets $A$ and $B$ that are disjoint but cannot be separated by any computable set.\nLet $p_n$ be the $n$-th prime.\nUsing coding of finite sets inside $M$ and overspill, one finds an element $c$ of $M$ such that $X = \\set{n \\in \\N : M \\models p_n \\mid c}$ satisfies $A \\subseteq X$ and $X \\cap B = \\emptyset$.\nSuppose $\\oplus$ is computable.\nFor a standard number $p_n$, the product of $p_n$ and $q$ in $M$ is the sum of $p_n$ copies of $q$, so it is computable from $q$.\nIn $M$ there are unique $q$ and $r \u003c p_n$ with $c = p_n q + r$, and $r$ is standard, so a search for $q$ and $r$ finds them and decides whether $p_n$ divides $c$.\nThen $X$ would be a computable set separating $A$ and $B$, a contradiction.\nThe case of $\\otimes$ needs a further argument.\n\nThe theorem uses that the model is non-standard: the standard model $\\N$ has computable addition and multiplication.\n\n## Explanations\n\n### L1 human\n\nThe natural numbers can be built on a computer: addition and multiplication are computable.\nOther models of the axioms of arithmetic exist, but none of them can be built on a computer in this way.\nSo among the countable models of arithmetic, the standard one is the only one that is computable.\n\n### L2 human\n\nThe theorem gives a computational characterization of the standard model among countable [[non-standard models of arithmetic|1QWN5CV4CG]].\nIt contrasts with weaker theories: for Presburger arithmetic, or for Robinson arithmetic $\\mathsf{Q}$, there are computable non-standard models.\n",
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