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  "body": "\n## Counterexample\n\nUse von Neumann [[ordinals|0YSCSX5YZC]]: an ordinal is a transitive set that is [[well-ordered|01CGSJ0GNQ]] by the membership relation $\\in$.\nSuppose that the collection $\\Omega$ of all ordinals is a [[set|0BSJ7RQ9BW]].\nEvery element of an ordinal is an ordinal, so $\\Omega$ is transitive.\nThe ordinals are well-ordered by $\\in$, so $\\Omega$ is a transitive set well-ordered by $\\in$.\nHence $\\Omega$ is an ordinal, and so $\\Omega \\in \\Omega$.\nThis contradicts the fact that no ordinal is an element of itself.\nSo the collection of all ordinals is not a set.\n\n## Remarks\n\nIn [[naive set theory|XGAMQBSE1G]] every property defines a set, so the argument there is a paradox.\nIn [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]] it is a theorem: the ordinals form a proper [[class|Y48FA69563]], not a set.\nIn the original form, the order type of all ordinals would be an ordinal greater than every ordinal, including itself.\nCesare Burali-Forti published the argument in 1897.\n\n## Explanations\n\n### L1 human\n\nEach ordinal counts the length of a well-ordered list.\nIf all ordinals formed one set, that set would itself be a well-ordered list, so it would have a length.\nThat length would be an ordinal bigger than all ordinals, which is absurd.\nThe way out is that \"all ordinals\" is too big to be a set.\n",
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