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  "body": "\n## Summary\n\nNaive set theory is [[set theory|RTHHCX84HN]] done in ordinary language, without a formal list of axioms.\nIn its original form, every property defines a [[set|0BSJ7RQ9BW]], the set of all objects with that property.\nThis principle leads to contradictions, the best known being [[Russell's paradox|0HESMMSYEG]].\nThe axiomatic theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], restrict how sets may be formed to avoid them.\nThe name is also used for the informal set theory taught at the start of most courses, which is consistent with those axioms.\n\n## Explanations\n\n### L1 human\n\nNaive set theory treats a set as any collection you can describe.\nThis works well for everyday sets, such as the even numbers or the points of a circle.\nBut consider the set of all sets that are not members of themselves.\nIf it is a member of itself, then it is not, and if it is not, then it is.\nSo \"any collection you can describe\" is too generous, and the axioms of modern set theory say more carefully which sets exist.\n",
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