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  "body": "\n## Definition\n\nLet $L$ be a first-order language and let $T$ be a [[complete theory|655K8Y7VS8]] in $L$ that has infinite [[models|MVCH6N9SNW]].\nLet $M$ be an infinite $L$-structure.\nA subset of $M$ is [[definable|CY1TXHF6KQ]] in one variable if it has the form $\\set{a \\in M : M \\models \\varphi(a, \\bar b)}$ for a formula $\\varphi(x, \\bar b)$ with parameters $\\bar b$ in $M$.\nThe structure $M$ is *minimal* if every such subset is finite or cofinite in $M$.\nThe theory $T$ is *strongly minimal* if every model of $T$ is minimal.\n\n## Remarks\n\nExample: the theory of infinite sets in the empty language is strongly minimal.\nBy quantifier elimination, a formula $\\varphi(x, \\bar b)$ is equivalent to a Boolean combination of equations $x = b_i$, so it defines a finite or a cofinite set.\nThe theory of algebraically closed fields of a fixed characteristic is strongly minimal for a similar reason: a nonzero polynomial in one variable has finitely many roots.\n\nNon-example: the theory of dense linear orders without endpoints is not strongly minimal.\nIn $(\\Q, \u003c)$ the formula $x \u003e 0$ defines a set that is infinite and has an infinite complement.\n\nA structure can be minimal while its theory is not strongly minimal, since the condition must hold in every model of the theory.\nIn a strongly minimal theory, algebraic closure satisfies the exchange property, so every model has a dimension.\nIf $L$ is countable, a strongly minimal theory is categorical in every uncountable cardinality.\n",
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