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  "body": "\n## Definition\n\nLet $\\sigma$ be a first-order [[signature|WDVVEYD6AT]].\nLet $K$ be a class of [[$\\sigma$-structures|983GZBQYT4]].\nThen $K$ is an *elementary class* if there is a first-order [[theory|92SWXST1ZM]] $T$ in $\\sigma$ whose [[models|MVCH6N9SNW]] are exactly the members of $K$.\n\n## Remarks\n\nIf $T$ can be chosen finite, or equivalently a single sentence, $K$ is called a *basic elementary class*.\nSome authors write $\\mathrm{EC}$ for basic elementary classes and $\\mathrm{EC}_\\Delta$ for elementary classes.\n\nExample: groups, in the signature $(\\cdot, {}^{-1}, e)$, form a basic elementary class, axiomatized by the three group axioms.\nFields of characteristic $0$ form an elementary class, using the infinitely many axioms $1 + \\cdots + 1 \\ne 0$; it is not a basic elementary class.\n\nNon-example: torsion groups do not form an elementary class.\nSuppose $T$ axiomatized them.\nAdd a new constant $c$ and the sentences $c^n \\ne e$ for all $n \\ge 1$.\nEach finite subset of this theory has a model, namely a finite cyclic group of large order with $c$ a generator.\nBy the [[compactness theorem|1NR3HAWG9D]] the whole theory has a model, which is a model of $T$ with an element of infinite order, a contradiction.\n",
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