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  "body": "\n## Definition\n\nLet $L$ be a countable signature.\nThe *age* of an [[$L$-structure|983GZBQYT4]] $M$ is the class of all finitely generated $L$-structures that embed into $M$.\nLet $\\mathcal{K}$ be a class of finitely generated $L$-structures, closed under isomorphism.\nA *Fraïssé limit* of $\\mathcal{K}$ is an $L$-structure $M$ such that:\n\n1. $M$ is [[countable|01TQB88H4C]];\n2. the age of $M$ is $\\mathcal{K}$;\n3. $M$ is ultrahomogeneous: every isomorphism between two finitely generated [[substructures|676H7EGTKJ]] of $M$ extends to an [[automorphism|0K1SCPBNKJ]] of $M$.\n\n## Remarks\n\nFraïssé's theorem: suppose $\\mathcal{K}$ is nonempty and has countably many isomorphism types.\nSuppose it has the hereditary property, the joint embedding property and the [[amalgamation property|XSCSX39GEZ]].\nThen $\\mathcal{K}$ has a Fraïssé limit, and any two Fraïssé limits of $\\mathcal{K}$ are isomorphic.\nConversely, the age of a countable ultrahomogeneous structure has these properties.\n\nExample: $(\\Q, \u003c)$ is the Fraïssé limit of the class of finite linear orders.\nEvery finite linear order embeds in $\\Q$, so clause 2 holds.\nFor clause 3, an order isomorphism between finite subsets of $\\Q$ extends piecewise linearly to an order automorphism of $\\Q$.\n\nNon-example: $(\\Z, \u003c)$ is countable and its age is again the class of finite linear orders, but it fails clause 3.\nThe map $0 \\mapsto 0$, $1 \\mapsto 2$ is an isomorphism between the substructures $\\set{0, 1}$ and $\\set{0, 2}$.\nIt has no extension to an automorphism, because no integer lies strictly between $0$ and $1$, while $1$ lies strictly between $0$ and $2$.\n",
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