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  "body": "\n## Statement\n\nThere is no [[set|0SJ3RX9TXA]] $S$ with $\\aleph_0 \u003c |S| \u003c 2^{\\aleph_0}$.\nEquivalently, with the axiom of choice, $2^{\\aleph_0} = \\aleph_1$.\n\n## Remarks\n\nIn words: each infinite subset of $\\mathbb{R}$ is countable or equinumerous with $\\mathbb{R}$.\nThe hypothesis is independent of ZFC: if ZFC is consistent, then ZFC proves neither the hypothesis nor its negation.\nThe generalized continuum hypothesis says that $2^{\\aleph_\\alpha} = \\aleph_{\\alpha + 1}$ for each ordinal $\\alpha$.\n\n## History\n\nGeorg Cantor proposed the hypothesis in 1878.\nIt was the first of the problems that David Hilbert presented in 1900.\nKurt Gödel showed in 1940 that ZFC does not refute it, and Paul Cohen showed in 1963 that ZFC does not prove it.\n\n## Explanations\n\n### L1 human\n\nCantor showed that $\\mathbb{R}$ is larger than $\\mathbb{N}$.\nThe continuum hypothesis says that no size lies strictly between them.\nThe usual axioms of set theory cannot decide whether this is true.\n",
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