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  "body": "\n## Definition\n\nLet $L$ be the [[signature|WDVVEYD6AT]] with a constant symbol $0$, a unary function symbol $S$, and binary function symbols $+$ and $\\cdot$.\n*Robinson arithmetic* $\\mathsf{Q}$ is the first-order [[theory|92SWXST1ZM]] in $L$ whose axioms are the universal closures of these seven formulas:\n\n1. $S x \\ne 0$;\n2. $S x = S y \\to x = y$;\n3. $x \\ne 0 \\to \\exists y\\, (x = S y)$;\n4. $x + 0 = x$;\n5. $x + S y = S(x + y)$;\n6. $x \\cdot 0 = 0$;\n7. $x \\cdot S y = (x \\cdot y) + x$.\n\n## Remarks\n\nExample: the natural numbers $\\N$ with $0$, $S(n) = n + 1$, and the usual addition and multiplication satisfy all seven axioms, so they form a model of $\\mathsf{Q}$.\n\nNon-example: the integers $\\Z$ with $S(n) = n + 1$ and the usual operations are not a model of $\\mathsf{Q}$.\nAxiom 1 fails, because $S(-1) = 0$.\n\n$\\mathsf{Q}$ is the [[Peano axioms|0QFT7Z76N5]] without the induction scheme, with axiom 3 added in its place.\n$\\mathsf{Q}$ does not prove $\\forall x\\, \\forall y\\, (x + y = y + x)$; there are models of $\\mathsf{Q}$ in which addition is not commutative.\n$\\mathsf{Q}$ proves every true sentence of the form $\\exists x_1 \\cdots \\exists x_n\\, \\varphi$ with $\\varphi$ quantifier-free, and every computable function is representable in $\\mathsf{Q}$.\n",
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