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  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]] and let $\\mathcal{M}$ be an [[$L$-structure|983GZBQYT4]] with universe $M$.\nFor $X \\subseteq M$, let $\\operatorname{Sg}(X)$ denote the [[substructure generated by|2TQD8NMG9P]] $X$.\nIt is the smallest subset of $M$ that contains $X$ and the constants and is closed under the functions of $\\mathcal{M}$.\nThe structure $\\mathcal{M}$ is *countably generated* when there is a [[countable set|01TQB88H4C]] $X \\subseteq M$ with $\\operatorname{Sg}(X) = M$.\n\n## Remarks\n\nHere countable means finite or countably infinite, so every [[finitely generated algebra|WR4WF1K2KW]] is countably generated.\n\nExample: the group $(\\Q, +, -, 0)$ is countably generated.\nTake $X = \\set{1/n : n \\ge 1}$; then $p/q = p \\cdot (1/q)$ is a sum of copies of $1/q$ or of its negative, so $\\operatorname{Sg}(X) = \\Q$.\n\nNon-example: the group $(\\R, +, -, 0)$ is not countably generated.\nFor a countable $X$, each element of $\\operatorname{Sg}(X)$ is the value of a term at finitely many elements of $X$.\nThere are only countably many such values, so $\\operatorname{Sg}(X)$ is countable, while $\\R$ is not.\n\nThe same counting shows that, when $L$ is countable, an $L$-structure is countably generated if and only if its universe is countable.\n",
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