{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2ENTSSHEPZ"
    },
    {
      "field": "uses",
      "target": "PGJNYGAN75"
    },
    {
      "field": "about",
      "target": "WQ967Q2XJD"
    },
    {
      "field": "defines",
      "target": "ZCE333FAWC"
    }
  ],
  "body": "\n## Summary\n\nA regular cardinal is an infinite [[cardinal|0EBACQFAZW]] that cannot be reached by fewer smaller steps than itself: its [[cofinality|BAWHE1W6MH]] equals itself.\nAn infinite cardinal that is not regular is singular.\nRegular cardinals are the natural setting for clubs, stationary sets and much of infinite combinatorics, and the [[precise definition|ZCE333FAWC]] is short.\n\n## Explanations\n\n### L1 human\n\nThe cardinal $\\aleph_0$ is regular: a finite union of finite sets is finite, so finitely many finite steps never reach $\\omega$.\nThe cardinal $\\aleph_1$ is regular too, because a countable union of countable sets is countable.\nBut $\\aleph_\\omega$ is singular: the countably many smaller cardinals $\\aleph_0, \\aleph_1, \\aleph_2, \\ldots$ already climb up to it.\nSo regularity says a cardinal is not the limit of a short sequence of smaller cardinals.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "03E10",
      "03E04"
    ],
    "name": "Regular cardinal",
    "same_as": {
      "wikidata": "Q1193137",
      "wikipedia": "Regular cardinal"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Regular_cardinal"
      }
    ],
    "specializes": [
      "0EBACQFAZW"
    ],
    "tag": "13ZV2F8JAR"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/13ZV2F8JAR/",
  "links": [
    {
      "field": "specializes",
      "target": "0EBACQFAZW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/13ZV2F8JAR.md",
  "msc": [
    "03E10",
    "03E04"
  ],
  "name": "Regular cardinal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/13Z/13ZV2F8JAR.md",
  "status": null,
  "tag": "13ZV2F8JAR",
  "uri": "m2://13ZV2F8JAR",
  "url": "https://math2.tamnd.com/wiki/regular-cardinal/",
  "writing": {
    "tag": "13ZV2F8JAR",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}