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  "body": "\n## Summary\n\nThe Elementary Theory of the Category of Sets (ETCS) is an axiomatic set theory proposed by Lawvere in 1964.\nIt describes sets and functions through the way functions compose, instead of through a membership relation.\nIts models are the nondegenerate well-pointed [[toposes|6837RS7N82]] with a natural numbers object in which every epimorphism splits.\nIt is weaker than [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] with choice, but strong enough for most ordinary mathematics.\n\n## Explanations\n\n### L1 human\n\nIn ETCS a set is not a bag of members; it is an object of a [[category|A2DE66937D]], and functions are its arrows.\nAn element of a set $X$ is a function from a one-point set into $X$.\nThe axioms say what one can build: products, function sets, subsets classified by truth values, the natural numbers, and sections of surjections.\nFor example, the subsets of $X$ correspond to functions from $X$ to a two-element set of truth values.\n",
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