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  "body": "\n## Definition\n\nLet $L$ be a signature, let $M$ be an $L$-structure, and let $\\kappa$ be an infinite [[cardinal|0EBACQFAZW]].\nThe structure $M$ is *$\\kappa$-saturated* if the following holds for every set $A$ of elements of $M$ with $\\abs{A} \u003c \\kappa$.\nEvery complete $1$-[[type|9V71Q3SD1G]] over $A$ is realized in $M$.\nThe structure $M$ is *saturated* if it is infinite and $\\abs{M}$-saturated, where $\\abs{M}$ is the [[cardinality|0G0ASKNA6Z]] of its domain.\n\n## Remarks\n\nA $\\kappa$-saturated structure also realizes every complete $n$-type over fewer than $\\kappa$ parameters, for every $n \\ge 1$.\n\nExample: $(\\Q, \u003c)$ is saturated.\nA complete $1$-type over a finite set $A \\subseteq \\Q$ says where $x$ lies relative to the elements of $A$.\nEither $x$ equals some $a \\in A$, or $x$ lies in one of the open intervals cut out by $A$, including the two unbounded ones.\nEach of these intervals contains a rational number, so the type is realized.\n\nNon-example: the field $\\overline{\\Q}$ of algebraic numbers is a model of the theory of algebraically closed fields of characteristic $0$, but it is not $\\aleph_0$-saturated.\nThe set $\\set{f(x) \\ne 0 : f \\in \\Z[x], f \\ne 0}$ is consistent with its theory over $A = \\emptyset$, as $\\C$ shows, and it extends to a complete type.\nNo element of $\\overline{\\Q}$ realizes it, since every algebraic number is a root of a nonzero integer polynomial.\n\nEvery structure has a $\\kappa$-saturated elementary extension for each $\\kappa$.\nSaturated models of the same complete theory and the same cardinality are isomorphic.\n",
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