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  "body": "\n## Summary\n\nA saturated model is a model that realizes as many [[types|9V71Q3SD1G]] as its size allows.\nMore precisely, $M$ is $\\kappa$-saturated when every complete type over fewer than $\\kappa$ parameters of $M$ is realized in $M$.\nIt is saturated when it is $\\abs{M}$-saturated.\nSaturated models of a complete theory are unique up to isomorphism in each cardinality.\nEvery model of the theory of at most the same size embeds elementarily into a saturated one.\nThey serve as large \"universal\" models in which most model-theoretic work is done.\n\n## Explanations\n\n### L1 human\n\nA saturated model has no gaps that its own language can describe.\nIf a description of an element mentions only a few given elements and is consistent, such an element already exists in the model.\nFor example, the rational numbers with their order are saturated.\nGiven finitely many rationals, a consistent wish such as \"between $1/3$ and $1/2$\" or \"larger than all of them\" is met by some rational number.\nThe integers with their order are not: no integer is \"larger than $0$ and with infinitely many integers between it and $0$\".\n",
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