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  "body": "\n## Definition\n\nLet $A$ be a [[set|0SJ3RX9TXA]].\nThe cardinality of $A$, written $|A|$, is an object assigned to $A$ such that, for all sets $A$ and $B$,\n$$|A| = |B| \\iff A \\text{ and } B \\text{ are } \\text{[[equinumerous|08EJHHMB5R]]}.$$\nFor a finite set $A$, $|A|$ is the number $n$ of its elements, that is, the unique $n \\in \\mathbb{N}$ with $A \\approx \\{1, \\dots, n\\}$.\nFor an infinite set, $|A|$ is a [[cardinal number|0QKPNZKNEA]].\n\n## Remarks\n\nOther notations are $\\#A$, $\\operatorname{card}(A)$ and $\\overline{\\overline{A}}$.\nWith the [[axiom of choice|0HTGB13FE2]], one can take $|A|$ to be the least [[ordinal|0RDJK7EQYY]] equinumerous with $A$.\n$|A| \\le |B|$ means that there is an [[injection|0PVG1CRAJN]] from $A$ to $B$. See [[order of cardinals|01GTP8M5RG]].\n\n## Explanations\n\n### L1 human\n\nThe cardinality of $\\{a, b, c\\}$ is $3$.\nFor infinite sets, cardinality is a new kind of number that records which sets can be put in bijection.\nFor example, $|\\mathbb{N}| = |\\mathbb{Z}|$, but $|\\mathbb{N}| \u003c |\\mathbb{R}|$.\n",
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