{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2ENTSSHEPZ"
    },
    {
      "field": "uses",
      "target": "3SZADQXVZJ"
    },
    {
      "field": "uses",
      "target": "4F1ABY39W1"
    },
    {
      "field": "uses",
      "target": "PGJNYGAN75"
    },
    {
      "field": "specializes",
      "target": "CJFEPPP9KA"
    }
  ],
  "body": "\n## Summary\n\nAn inaccessible cardinal is an uncountable [[cardinal|0EBACQFAZW]] $\\kappa$ that cannot be reached from smaller cardinals.\nPower sets of smaller sets, and unions of fewer than $\\kappa$ smaller sets, stay smaller than $\\kappa$.\nIt is the smallest kind of [[large cardinal|X1FG7PSCTY]].\nIf $\\kappa$ is inaccessible, the sets of rank less than $\\kappa$ form a model of ZFC.\nSo ZFC cannot prove that inaccessible cardinals exist, if ZFC is consistent.\nIn category theory, inaccessible cardinals give [[Grothendieck universes|SMMK9E0MTM]].\n\n## Explanations\n\n### L1 human\n\nThe natural numbers cannot be reached from finite numbers: sums, products and powers of finite numbers stay finite.\nAn inaccessible cardinal is an uncountable size with the same property.\nTaking power sets of smaller sets, or unions of fewer than $\\kappa$ sets each smaller than $\\kappa$, never gives a set of size $\\kappa$.\nSo everything built from below stays below, and the sets below $\\kappa$ form a whole universe of their own.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "03E55"
    ],
    "name": "Inaccessible cardinal",
    "same_as": {
      "wikidata": "Q1331373",
      "wikipedia": "Inaccessible cardinal"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Inaccessible_cardinal"
      }
    ],
    "specializes": [
      "X1FG7PSCTY"
    ],
    "tag": "NW4DREQ5DQ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/NW4DREQ5DQ/",
  "links": [
    {
      "field": "specializes",
      "target": "X1FG7PSCTY"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NW4DREQ5DQ.md",
  "msc": [
    "03E55"
  ],
  "name": "Inaccessible cardinal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NW4/NW4DREQ5DQ.md",
  "status": null,
  "tag": "NW4DREQ5DQ",
  "uri": "m2://NW4DREQ5DQ",
  "url": "https://math2.tamnd.com/wiki/inaccessible-cardinal/",
  "writing": {
    "tag": "NW4DREQ5DQ",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}