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  "body": "\n## Definition\n\nLet $T$ be a [[theory|92SWXST1ZM]] in a first-order signature $L$.\nThe theory $T$ is *model complete* if the following holds for all [[models|MVCH6N9SNW]] $M$ and $N$ of $T$.\nIf $M$ is a [[substructure|676H7EGTKJ]] of $N$, then $M$ is an [[elementary substructure|8WZF9SBP39]] of $N$.\n\n## Remarks\n\nEquivalently, every embedding between models of $T$ is elementary.\nBy Robinson's test, $T$ is model complete if and only if every $L$-formula is equivalent modulo $T$ to an existential formula $\\exists \\bar{y} \\, \\chi(\\bar{x}, \\bar{y})$ with $\\chi$ quantifier-free.\nA theory with [[quantifier elimination|7FDDXV3JK7]] is model complete.\n\nExample: the theory of dense linear orders without endpoints is model complete, since it has quantifier elimination.\nLet $M \\subseteq N$ be two such orders and let $\\bar{a}$ be a tuple of $M$.\nA formula $\\varphi(\\bar{a})$ is equivalent to a quantifier-free formula.\nThat formula only compares elements of $\\bar{a}$, so it has the same truth value in $M$ and in $N$.\n\nNon-example: the theory of dense linear orders with a least and a greatest element is not model complete.\nThe orders $[0, 1] \\cap \\Q$ and $[0, 2] \\cap \\Q$ are models, and the first is a substructure of the second.\nThe element $1$ satisfies $\\forall y \\, (y \\le x)$ in $[0, 1] \\cap \\Q$ but not in $[0, 2] \\cap \\Q$.\n\nThe theory of real closed fields in the signature $\\set{+, -, \\cdot, 0, 1}$ is model complete but does not have quantifier elimination.\n",
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