{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "TEWH75TX8Y"
    }
  ],
  "body": "\n## Summary\n\nThe aleph numbers $\\aleph_\\alpha$, indexed by the [[ordinals|0YSCSX5YZC]], are the infinite [[cardinal numbers|0EBACQFAZW]] of well-orderable sets, listed in increasing order.\nThe first is [[aleph-null|0V2H5PGJ9B]], the size of the natural numbers, and $\\aleph_1$ is the next infinite cardinal.\nWith the axiom of choice, every infinite set has an aleph as its cardinality, so the alephs measure all infinite sizes.\nThe [[precise definition|TEWH75TX8Y]] is by transfinite recursion.\n\n## Explanations\n\n### L1 human\n\nThe alephs are the sizes of infinity, lined up in order.\nThe smallest infinite size is $\\aleph_0$, the size of $\\set{0, 1, 2, \\ldots}$.\nThe next one is $\\aleph_1$, then $\\aleph_2$, and so on, and after all $\\aleph_n$ for finite $n$ comes $\\aleph_\\omega$.\nThe real numbers have size $2^{\\aleph_0}$, which is at least $\\aleph_1$.\nWhich aleph it is cannot be decided from the usual axioms; the [[continuum hypothesis|0MN2MER9WN]] says it is $\\aleph_1$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "03E10"
    ],
    "name": "Aleph number",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Aleph_number"
      }
    ],
    "specializes": [
      "0EBACQFAZW"
    ],
    "tag": "6Q9Z2BM1XR"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/6Q9Z2BM1XR/",
  "links": [
    {
      "field": "specializes",
      "target": "0EBACQFAZW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/6Q9Z2BM1XR.md",
  "msc": [
    "03E10"
  ],
  "name": "Aleph number",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6Q9/6Q9Z2BM1XR.md",
  "status": null,
  "tag": "6Q9Z2BM1XR",
  "uri": "m2://6Q9Z2BM1XR",
  "url": "https://math2.tamnd.com/wiki/aleph-number/",
  "writing": {
    "tag": "6Q9Z2BM1XR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}