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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]].\nAn $L$-theory is a set $T$ of [[sentences|DKFGQTRYZE]] of [[first-order logic|086QKPF0A6]] in the signature $L$.\n\n## Remarks\n\nExample: in the signature $\\set{\\cdot, e, {}^{-1}}$, the set of the three sentences $\\forall x \\, \\forall y \\, \\forall z \\, ((x \\cdot y) \\cdot z = x \\cdot (y \\cdot z))$, $\\forall x \\, (x \\cdot e = x \\land e \\cdot x = x)$ and $\\forall x \\, (x \\cdot x^{-1} = e \\land x^{-1} \\cdot x = e)$ is a theory, the theory of groups.\nEach of its elements is a formula without free variables, so it is a set of sentences.\n\nNon-example: the set $\\set{x \\cdot e = x}$ is not a theory, because the variable $x$ is free in its only formula.\n\nSome authors require a theory to be closed under logical consequence; a set of sentences is then called a set of axioms for the theory it generates.\nFor an $L$-structure $\\mathcal{M}$, the set $\\mathrm{Th}(\\mathcal{M})$ of all $L$-sentences true in $\\mathcal{M}$ is the theory of $\\mathcal{M}$.\n\n## Explanations\n\n### L1 human\n\nA theory is any set of sentences; it may be finite or infinite, and it may even be contradictory.\nThe theory of groups has three sentences.\nPeano arithmetic is a theory with infinitely many sentences, one induction axiom for each formula.\n",
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