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  "body": "\n## Statement\n\nLet $T$ be a satisfiable theory in a first-order signature $L$, and let $\\kappa$ be an infinite [[cardinal|0EBACQFAZW]] with $\\kappa \\ge \\abs{L}$.\nSuppose that $T$ has no finite models and that $T$ is [[$\\kappa$-categorical|1JJCGNX240]].\nThen $T$ is [[complete|655K8Y7VS8]].\n\n## Remarks\n\nSuppose $T$ is not complete, and pick a sentence $\\varphi$ such that neither $\\varphi$ nor $\\neg \\varphi$ follows from $T$.\nThen $T \\cup \\set{\\varphi}$ and $T \\cup \\set{\\neg \\varphi}$ both have models, and these are infinite.\nBy the [[Löwenheim–Skolem theorem|484M0TZ1S7]], both have models of size $\\kappa$.\nThese two models are not isomorphic, since one satisfies $\\varphi$ and the other does not, which contradicts $\\kappa$-categoricity.\n\nExample: the theory of dense linear orders without endpoints has no finite models and is $\\aleph_0$-categorical, so it is complete.\nSimilarly, the theory of algebraically closed fields of a fixed characteristic is complete, since it is categorical in every uncountable cardinal.\n\nThe hypothesis on finite models is necessary.\nTake the empty theory $T = \\emptyset$ in the empty signature, whose models are the nonempty sets.\nFor each infinite $\\kappa$, any two sets of size $\\kappa$ are in bijection, so $T$ is $\\kappa$-categorical.\nBut $T$ is not complete: the sentence $\\forall x \\, \\forall y \\, (x = y)$ holds in a model of size $1$ and fails in a model of size $2$.\n\n## Explanations\n\n### L1 human\n\nA theory is complete when it decides every statement in its language.\nThe test says that if all infinite models of one size look the same and there are no finite models, the theory decides everything.\nIt is a quick way to prove completeness for natural theories such as dense orders or algebraically closed fields.\n\n### L2 human\n\nThe test turns a structural fact, uniqueness of models of one cardinality, into a logical fact, completeness.\nCompleteness of a recursively axiomatized theory implies that it is decidable, so the test also gives decision procedures.\nIt is often combined with a back-and-forth argument for $\\aleph_0$-categoricity, or with a dimension argument, such as transcendence degree, for uncountable categoricity.\nIts converse fails: the theory of $(\\Z, \u003c)$ is complete but categorical in no infinite cardinal.\n",
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