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  "body": "\n## Summary\n\nA theory is categorical in an infinite cardinal $\\kappa$ when it has, up to isomorphism, exactly one model of size $\\kappa$.\nBy the Löwenheim–Skolem theorems, a theory with an infinite model has models of many sizes.\nSo categoricity is always relative to a cardinal.\nCategoricity gives completeness through the [[Łoś–Vaught test|ZR7Y5C235P]].\nFor countable theories, [[Morley's categoricity theorem|V7N84P60SJ]] says that categoricity in one uncountable cardinal implies it in all of them.\nThe study of categorical theories was the starting point of stability theory.\n\n## Explanations\n\n### L1 human\n\nA theory is a list of axioms, and it usually has many different models.\nA theory is categorical in a size when all its models of that size are copies of one another.\nFor example, any two countable dense linear orders without endpoints, such as the rationals and the rationals between $0$ and $1$, are isomorphic as orders.\nSo the theory of such orders is categorical in the countable size, but it is not categorical in larger sizes.\n",
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