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  "body": "\n## Definition\n\nLet $L$ be a signature and let $T$ be an $L$-[[theory|92SWXST1ZM]].\nAn $L$-[[structure|983GZBQYT4]] $\\mathcal{M}$ is a model of $T$ when $\\mathcal{M} \\models \\varphi$ for every [[sentence|DKFGQTRYZE]] $\\varphi$ in $T$.\nHere $\\mathcal{M} \\models \\varphi$ means that $\\varphi$ is true in $\\mathcal{M}$ under the usual inductive definition of truth.\nThe notation is $\\mathcal{M} \\models T$.\n\n## Remarks\n\nExample: let $T$ be the theory of groups in the signature $\\set{\\cdot, e, {}^{-1}}$.\nThe structure $(\\Z, +, 0, -)$, with $\\cdot$ read as $+$, $e$ as $0$ and ${}^{-1}$ as $-$, is a model of $T$.\nIndeed addition of integers is associative, $0$ is neutral and $-x$ is an inverse of $x$.\n\nNon-example: the structure $(\\N, +, 0, \\id)$, with ${}^{-1}$ read as the identity map, is not a model of $T$.\nThe sentence $\\forall x \\, (x \\cdot x^{-1} = e)$ fails at $x = 1$, because $1 + 1 \\ne 0$.\n\nA theory is satisfiable when it has a model.\nThe class of all models of $T$ is often written $\\mathrm{Mod}(T)$.\n\n## Explanations\n\n### L1 human\n\nTo check that a structure is a model of a theory, you check each sentence of the theory in the structure.\nEvery group is a model of the group axioms, and every model of the group axioms is a group.\n",
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