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  "body": "\n## Statement\n\nThere is no [[set|0SJ3RX9TXA]] $R$ such that, for each set $x$, $x \\in R$ if and only if $x \\notin x$.\n\n## Remarks\n\nProof: if such an $R$ exists, apply the condition to $x = R$. Then $R \\in R$ if and only if $R \\notin R$, which is a contradiction.\nSo the collection $\\{x \\mid x \\notin x\\}$ is not a set. Unrestricted comprehension, which makes a set from each property, is inconsistent.\nIt follows that there is no set of all sets. Otherwise, separation would give $R$ as a subset of it.\n\n## History\n\nBertrand Russell found the paradox in May 1901.\nErnst Zermelo found it independently, by 1902 and possibly in 1899, but he did not publish it.\nIn a 1902 letter, Russell told Gottlob Frege that the paradox arises in the logical system of Frege.\nAxiomatic set theories, such as [[Zermelo–Fraenkel set theory|0HE5WDQ9FJ]], avoid it with a restricted form of comprehension.\n\n## Explanations\n\n### L1 human\n\nTry to form \"the set of all sets that do not contain themselves\".\nAsk whether this set contains itself: both answers lead to a contradiction.\nSo some collections are too large to be sets.\n",
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