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  "body": "\n## Summary\n\nThe axiom of regularity, also called the axiom of foundation, is an axiom of [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]].\nIt says that every nonempty [[set|0BSJ7RQ9BW]] has an element that shares no element with it.\nIn other words, the membership relation $\\in$ is [[well-founded|0DQPNJNPYK]].\nIt rules out sets such as $x = \\set{x}$, and it implies that every set appears at some stage of the cumulative hierarchy $V_\\alpha$.\n\n## Explanations\n\n### L1 human\n\nThink of building sets in stages: first the empty set, then sets of what you have, and so on.\nRegularity says that every set is built this way, from the bottom up.\nSo no set can contain itself, and there is no endless chain $x_1 \\ni x_2 \\ni x_3 \\ni \\cdots$ going down.\nFor example, in the set $\\set{\\emptyset, \\set{\\emptyset}}$, the element $\\emptyset$ has no element in common with the set.\nThe axiom is almost never used in ordinary mathematics, but it makes the universe of sets much easier to describe.\n",
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