{
  "aliases": [
    "aleph-zero",
    "aleph-nought"
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  "body": "\n## Description\n\nAleph-null, written $\\aleph_0$, is the [[cardinality|0JTGRV4WT4]] of the set $\\mathbb{N}$ of natural numbers.\nIt is the cardinality of each [[countably infinite set|0SEZ3JGZP7]].\nIt is the least infinite [[cardinal number|0QKPNZKNEA]]: each infinite set has cardinality at least $\\aleph_0$, with the axiom of choice.\n\n## Remarks\n\nOther names are aleph-zero and aleph-nought.\n$\\aleph_0 + \\aleph_0 = \\aleph_0 \\cdot \\aleph_0 = \\aleph_0$, but $2^{\\aleph_0} \u003e \\aleph_0$ by [[Cantor's theorem|0WTEMGYV2C]].\nAs an ordinal, $\\aleph_0$ is $\\omega$, the set of the finite ordinals.\n\n## Explanations\n\n### L1 human\n\n$\\aleph_0$ is the size of the infinity of counting.\nThe sets $\\mathbb{N}$, $\\mathbb{Z}$ and $\\mathbb{Q}$ all have this size.\n",
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        "what": "section Invariants",
        "note": "a table of the main invariants",
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