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  "body": "\n## Statement\n\nLet $T$ be a [[complete theory|655K8Y7VS8]] in a countable first-order signature, with infinite models.\nIf $T$ is [[$\\kappa$-categorical|1JJCGNX240]] for some [[uncountable|0W7FBYS1F1]] [[cardinal|0EBACQFAZW]] $\\kappa$, then $T$ is $\\kappa$-categorical for every uncountable cardinal $\\kappa$.\n\n## Remarks\n\nMichael Morley proved the theorem in 1965, answering a question of Jerzy Łoś.\nA key step shows that a theory categorical in an uncountable cardinal is $\\omega$-stable: over each countable set of parameters there are only countably many complete types.\nSuch a theory has a strongly minimal formula, possibly with parameters.\nEach model is determined up to isomorphism by the dimension of the set that this formula defines, as a vector space is determined by its dimension.\n\nExample: the theory of [[algebraically closed fields|MT8TDVPBXB]] of a fixed characteristic is categorical in every uncountable cardinal, with transcendence degree as the dimension.\nThe theory of infinite vector spaces over $\\Q$ is another example.\n\nThe countability of the signature matters in this form.\nSaharon Shelah extended the theorem: a complete theory $T$ categorical in some cardinal $\\kappa \u003e \\abs{T}$ is categorical in every cardinal $\\kappa \u003e \\abs{T}$.\nThe theorem says nothing about $\\aleph_0$.\nThe theory of algebraically closed fields of characteristic $0$ is uncountably categorical but not $\\aleph_0$-categorical.\nThe theory of dense linear orders without endpoints is $\\aleph_0$-categorical but not uncountably categorical.\n\n## Explanations\n\n### L1 human\n\nSome theories have only one model of each uncountable size, up to isomorphism.\nMorley's theorem says that, for a countable language, having only one model of a single uncountable size already forces this for every uncountable size.\nAlgebraically closed fields are the typical case: such a field is fixed by its characteristic and how many independent transcendental elements it has.\n\n### L2 human\n\nThe proof introduced tools that became the core of stability theory: Morley rank, $\\omega$-stability, and prime models over sets.\nA countable complete theory with infinite models has one of four patterns of categoricity.\nIt is categorical in all infinite cardinals, only in $\\aleph_0$, only in the uncountable ones, or in none.\nBaldwin and Lachlan later showed that an uncountably categorical theory has either one countable model or countably infinitely many.\n",
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