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  "body": "\n## Definition\n\nThe language of second-order arithmetic has number variables $n, m, x, y, \\ldots$ and set variables $X, Y, \\ldots$.\nIt has the constants $0$ and $1$, the operations $+$ and $\\cdot$, and the relation $\u003c$ on numbers.\nIt also has the membership relation $n \\in X$ between a number and a set.\n*Second-order arithmetic* $\\mathsf{Z}_2$ is the [[theory|92SWXST1ZM]] in this two-sorted first-order language with these axioms:\n\n1. the basic axioms: the axioms of a discretely ordered commutative semiring with least element $0$ and successor $n + 1$, in the style of [[Robinson arithmetic|3K1AKGXF8A]];\n2. the induction axiom: $\\forall X\\, \\big( (0 \\in X \\wedge \\forall n\\, (n \\in X \\to n + 1 \\in X)) \\to \\forall n\\, (n \\in X) \\big)$;\n3. the comprehension scheme: $\\exists X\\, \\forall n\\, (n \\in X \\leftrightarrow \\varphi(n))$ for every [[formula|HYJ10AF6RT]] $\\varphi(n)$ in which $X$ does not occur free.\n\n## Remarks\n\nExample: the full structure, with number domain $\\N$ and set domain the power set of $\\N$, is a model of $\\mathsf{Z}_2$.\nEach clause holds because every subset of $\\N$ is available, and ordinary induction holds in $\\N$.\n\nNon-example: the structure with number domain $\\N$ and set domain the computable subsets of $\\N$ satisfies clauses 1 and 2 but not clause 3.\nThe formula $\\varphi(e)$ that says \"program $e$ halts on input $e$\" uses only number quantifiers, and the set it defines is the halting problem, which is not computable.\n\nSubsystems are obtained by restricting comprehension and induction to classes of formulas, for example arithmetical comprehension in $\\mathsf{ACA}_0$.\nDespite the name, $\\mathsf{Z}_2$ is a first-order theory: its models may have a set domain smaller than the full power set.\nTogether with clause 2, the comprehension scheme proves induction for every formula, so the first-order [[Peano axioms|0QFT7Z76N5]] are provable in $\\mathsf{Z}_2$.\n",
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