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  "body": "\n## Definition\n\nWork in the first-order language of arithmetic with symbols $0$, $S$, $+$, $\\cdot$ and $\u003c$, interpreted in the standard model $\\N$ of the [[natural numbers|0MWKJ44V5A]].\nA bounded quantifier has the form $\\exists x \u003c t$ or $\\forall x \u003c t$, where $t$ is a term not containing $x$.\nThe classes of [[formulas|HYJ10AF6RT]] $\\Sigma_n$ and $\\Pi_n$ are defined by induction on $n$.\n\n1. $\\Sigma_0 = \\Pi_0$ is the class of formulas in which every quantifier is bounded.\n2. A formula is $\\Sigma_{n+1}$ if it has the form $\\exists x_1 \\cdots \\exists x_k \\, \\psi$ with $\\psi$ in $\\Pi_n$.\n3. A formula is $\\Pi_{n+1}$ if it has the form $\\forall x_1 \\cdots \\forall x_k \\, \\psi$ with $\\psi$ in $\\Sigma_n$.\n\nA set $A \\subseteq \\N^m$ is $\\Sigma^0_n$ if $A = \\set{\\bar a : \\N \\models \\varphi(\\bar a)}$ for some $\\Sigma_n$ formula $\\varphi$, and $\\Pi^0_n$ if the same holds with a $\\Pi_n$ formula.\nA set is $\\Delta^0_n$ if it is both $\\Sigma^0_n$ and $\\Pi^0_n$.\nA set is *arithmetical* if it is $\\Sigma^0_n$ for some $n$.\n\n## Remarks\n\nExample: the halting set $K = \\set{e : \\exists c \\, T(e, e, c)}$ is $\\Sigma^0_1$, where $T$ is [[Kleene's T predicate|S9DK13K5PS]].\nEvery primitive recursive relation is $\\Delta^0_1$, so $T$ is defined by a $\\Sigma_1$ formula, and so is $K$.\nThe set $\\set{e : \\forall x \\, \\exists c \\, T(e, x, c)}$ of programs computing total functions is $\\Pi^0_2$.\n\nNon-example: the set of Gödel numbers of sentences true in $\\N$ is not arithmetical, by [[Tarski's undefinability theorem|YRV6SYK2BY]].\n\nA set is $\\Sigma^0_n$ if and only if its complement is $\\Pi^0_n$.\nThe inclusions $\\Sigma^0_n \\cup \\Pi^0_n \\subseteq \\Delta^0_{n+1}$ hold, and each is strict.\nThe classes $\\Sigma^0_1$ and $\\Delta^0_1$ are the computably enumerable and the computable sets.\n",
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    "name": "Arithmetical hierarchy",
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