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  "body": "\n## Statement\n\nLet $A$ be a [[set|0SJ3RX9TXA]] and let $f \\colon A \\to \\mathcal{P}(A)$ be a [[function|0XHDTE5YMZ]] to the [[power set|0BC7KX51Z8]] of $A$.\nThen $f$ is not [[surjective|0XKH881BZ7]].\nSo $|A| \u003c |\\mathcal{P}(A)|$.\n\n## Remarks\n\nProof: let $D = \\{a \\in A \\mid a \\notin f(a)\\}$. If $D = f(d)$ for some $d \\in A$, then $d \\in D$ if and only if $d \\notin D$, a contradiction.\nThe map $a \\mapsto \\{a\\}$ is an injection, so $|A| \\le |\\mathcal{P}(A)|$, and the inequality is strict.\nIt follows that there is no largest cardinal. The proof is a form of the [[diagonal argument|0XCH8QVDCQ]], and it is close to [[Russell's paradox|0HESMMSYEG]].\n\n## Explanations\n\n### L1 human\n\nFor each set, its set of subsets is strictly larger.\nFor a set with $3$ elements, this is $3 \u003c 8$.\nFor infinite sets, it gives an endless tower of larger and larger infinities.\n",
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