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  "body": "\n## Definition\n\nLet $T$ be a [[theory|92SWXST1ZM]] in a first-order signature $L$, and let $\\kappa$ be an infinite [[cardinal|0EBACQFAZW]].\nThe theory $T$ is *$\\kappa$-categorical* if two conditions hold.\n\n1. $T$ has a [[model|MVCH6N9SNW]] whose domain has [[cardinality|0G0ASKNA6Z]] $\\kappa$.\n2. Any two models of $T$ of cardinality $\\kappa$ are isomorphic.\n\n## Remarks\n\nA theory is *uncountably categorical* if it is $\\kappa$-categorical for every uncountable $\\kappa$.\nSome authors drop the existence of a model of size $\\kappa$ from the definition.\n\nExample: the theory $\\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is $\\kappa$-categorical for every uncountable $\\kappa$.\nBy [[Steinitz's theorem|CAEHMSX9RT]], such a field is determined up to isomorphism by its transcendence degree over $\\Q$, and a field of uncountable size $\\kappa$ has transcendence degree $\\kappa$.\n\nNon-example: $\\mathrm{ACF}_0$ is not $\\aleph_0$-categorical.\nThe fields $\\overline{\\Q}$ and the algebraic closure of $\\Q(t)$ are both countable models, of transcendence degree $0$ and $1$, so they are not isomorphic.\n\nThe theory of dense linear orders without endpoints is $\\aleph_0$-categorical, by Cantor's back-and-forth argument, but not $\\kappa$-categorical for any uncountable $\\kappa$.\n",
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