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  "body": "\n## Statement\n\nThe set $\\mathbb{R}$ of real numbers has [[cardinality|0JTGRV4WT4]] $2^{\\aleph_0}$:\n$$|\\mathbb{R}| = \\mathfrak{c} = |\\mathcal{P}(\\mathbb{N})|.$$\n\n## Remarks\n\nOne proof uses the [[Schröder–Bernstein theorem|0A9V6RKQG6]]. A real number gives a set of rationals, the rationals less than it, so $|\\mathbb{R}| \\le |\\mathcal{P}(\\mathbb{Q})| = 2^{\\aleph_0}$. Binary sequences give an injection in the other direction, for example through decimal expansions with digits $0$ and $1$.\nWith [[Cantor's theorem|0WTEMGYV2C]], this gives another proof that $\\mathbb{R}$ is [[uncountable|0518MM6BE1]].\n\n## Explanations\n\n### L1 human\n\nThe real numbers and the subsets of $\\mathbb{N}$ have the same size.\nRoughly, a real number is an infinite sequence of digits, and an infinite sequence of $0$ and $1$ is a subset of $\\mathbb{N}$.\n",
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