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  "body": "\n## Statement\n\nLet $L$ be a countable signature, let $T$ be a satisfiable $L$-[[theory|92SWXST1ZM]], and let $p(x_1, \\dots, x_n)$ be an $n$-[[type|9V71Q3SD1G]] of $T$.\nCall $p$ isolated if there is an $L$-formula $\\varphi(x_1, \\dots, x_n)$ such that $T \\cup \\set{\\varphi}$ is satisfiable and $T \\models \\forall \\bar{x} \\, (\\varphi(\\bar{x}) \\to \\psi(\\bar{x}))$ for every $\\psi \\in p$.\nIf $p$ is not isolated, then $T$ has a [[countable|01TQB88H4C]] [[model|MVCH6N9SNW]] that omits $p$.\n\n## Remarks\n\nThe proof is a Henkin construction.\nOne adds countably many new constants, lists all tuples of constants, and builds a complete theory with witnesses step by step.\nAt the step for a tuple $\\bar{c}$, one adds the negation of some formula of $p$ at $\\bar{c}$.\nThis is possible because no formula consistent with what has been built so far implies all of $p$.\nThe canonical model of the final theory is countable and no tuple in it realizes $p$.\nThe same argument omits countably many non-isolated types at once.\n\nExample: let $T$ be the theory of $(\\N, 0, 1, +, \\cdot, \u003c)$ and $p(x) = \\set{x \\ne 0, x \\ne 1, x \\ne 1 + 1, \\dots}$.\nThe type $p$ is not isolated, and $\\N$ is a countable model of $T$ that omits it.\n\nIf $T$ is complete, an isolated type is realized in every model of $T$, so the hypothesis cannot be dropped.\nCountability of $L$ is also needed.\nTake constants $c_\\alpha$ for $\\alpha \u003c \\omega_1$ and $d_k$ for $k \\in \\N$, let $T$ say that the $c_\\alpha$ are pairwise distinct, and let $p(x) = \\set{x \\ne d_k : k \\in \\N}$.\nThen $p$ is not isolated, but every model of $T$ is uncountable and so has an element different from all $d_k$.\n\n## Explanations\n\n### L1 human\n\nA type is a wish list of properties for an element.\nSome wish lists are forced: a single property, consistent with the theory, already implies the whole list.\nFor a complete theory, every model then has an element that meets the whole list.\nThe theorem says that, in a countable language, every list that is not forced in this way can be avoided by some countable model.\n\n### L2 human\n\nThe omitting types theorem is the main tool for building countable models with prescribed properties.\nIt shows that a complete theory in a countable language has an atomic model, that is, a prime model, when the isolated types are dense.\nIt is used in the proof of the Ryll-Nardzewski theorem and in the study of the number of countable models, the subject of the Vaught conjecture.\nIt is a counterpart of the compactness theorem: compactness produces models that realize types, and this theorem produces models that omit them.\n",
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