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  "body": "\n## Summary\n\nA structure is countably generated when some [[countable|01TQB88H4C]] subset of it generates the whole structure under its operations.\nIt is the countable analogue of a [[finitely generated algebra|WR4WF1K2KW]].\nIn a countable signature, a structure is countably generated exactly when its universe is countable.\n\n## Explanations\n\n### L1 human\n\nStart with a list of elements, one for each natural number, and close it up under the operations.\nIf this reaches every element, the structure is countably generated.\nThe rational numbers with addition, negation and $0$ are countably generated, by the list of all fractions $1/n$.\nThey are not finitely generated, so the countable notion is really weaker.\nThe real numbers with the same operations are not countably generated, since closing a countable set under these operations gives only a countable set.\n",
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