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  "body": "\n## Definition\n\nWork in the language of [[second-order arithmetic|DC60FYCSPM]], which has number variables and set variables.\nIt is interpreted in the standard model, where set variables range over all subsets of $\\N$.\nA formula is *arithmetical* if it has no quantifiers over sets; it may have free set variables.\nThe classes of formulas $\\Sigma^1_n$ and $\\Pi^1_n$ are defined by induction on $n$.\n\n1. $\\Sigma^1_0 = \\Pi^1_0$ is the class of arithmetical formulas.\n2. A formula is $\\Sigma^1_{n+1}$ if it has the form $\\exists X_1 \\cdots \\exists X_k \\, \\psi$ with $\\psi$ in $\\Pi^1_n$.\n3. A formula is $\\Pi^1_{n+1}$ if it has the form $\\forall X_1 \\cdots \\forall X_k \\, \\psi$ with $\\psi$ in $\\Sigma^1_n$.\n\nA set $A \\subseteq \\N$ is $\\Sigma^1_n$, or $\\Pi^1_n$, if a formula of that class defines it in the standard model.\nThe formula must have one free number variable and no other free variables.\nA set is $\\Delta^1_n$ if it is both $\\Sigma^1_n$ and $\\Pi^1_n$, and *analytical* if it is $\\Sigma^1_n$ for some $n$.\n\n## Remarks\n\nExample: let $W$ be the set of $e$ such that program $e$ computes the characteristic function of a well-ordering of a subset of $\\N$.\n\"Every nonempty set $X$ in the field of the ordering has a least element\" is a universal set quantifier over an arithmetical formula.\nSo $W$ is $\\Pi^1_1$; in fact it is $\\Pi^1_1$-complete, so it is not $\\Sigma^1_1$.\n\nNon-example: the set of Gödel numbers of second-order sentences true in the standard model is not analytical, by Tarski's undefinability argument.\n\nEvery [[arithmetical|7CVJCQKY41]] set is $\\Delta^1_1$, and the inclusion is strict.\nBy a theorem of Kleene, the $\\Delta^1_1$ sets are exactly the [[hyperarithmetical sets|NS1HWGTD8E]].\n",
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