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  "body": "\n## Definition\n\nLet $L$ be a signature and let $T$ be an $L$-[[theory|92SWXST1ZM]].\nFor an $L$-[[sentence|DKFGQTRYZE]] $\\varphi$, write $T \\models \\varphi$ when every [[model|MVCH6N9SNW]] of $T$ satisfies $\\varphi$.\nThe theory $T$ is complete when the following hold.\n\n1. $T$ has a model.\n2. For every $L$-sentence $\\varphi$, $T \\models \\varphi$ or $T \\models \\neg\\varphi$.\n\n## Remarks\n\nExample: for any $L$-structure $\\mathcal{M}$, the theory $\\mathrm{Th}(\\mathcal{M})$ of all sentences true in $\\mathcal{M}$ is complete.\nIt has the model $\\mathcal{M}$, and for each sentence $\\varphi$ either $\\varphi$ or $\\neg\\varphi$ is true in $\\mathcal{M}$, hence lies in $\\mathrm{Th}(\\mathcal{M})$.\n\nNon-example: the theory of groups has a model but is not complete.\nFor $\\varphi = \\forall x \\, \\forall y \\, (x \\cdot y = y \\cdot x)$, the group $\\Z$ satisfies $\\varphi$ and the symmetric group $S_3$ satisfies $\\neg\\varphi$.\n\nA satisfiable theory $T$ is complete if and only if all its models are [[elementarily equivalent|P3B1V182N4]].\nBy the completeness theorem, $T \\models \\varphi$ can be replaced by $T \\vdash \\varphi$, provability from $T$.\nSome authors do not require clause 1.\n\n## Explanations\n\n### L1 human\n\nA complete theory answers every question that its language can ask.\nEach structure has a complete theory, its full list of true sentences.\nA natural list of axioms is complete only in special cases, such as the axioms of dense linear orders without endpoints, or of algebraically closed fields of a fixed characteristic.\n",
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