{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $\\kappa$ be an [[inaccessible cardinal|NW4DREQ5DQ]].\nThe cardinal $\\kappa$ is *Mahlo* if the set\n$$\\set{\\lambda \u003c \\kappa : \\lambda \\text{ is a regular cardinal}}$$\nis a [[stationary subset|YE6WZR46RQ]] of $\\kappa$, that is, it meets every closed unbounded subset of $\\kappa$.\n\nA weakly inaccessible cardinal $\\kappa$ with the same property is *weakly Mahlo*.\n\n## Remarks\n\nFor inaccessible $\\kappa$, the uncountable strong limit cardinals below $\\kappa$ form a closed unbounded set.\nSo $\\kappa$ is Mahlo if and only if the inaccessible cardinals below $\\kappa$ form a stationary set.\nThe least inaccessible cardinal $\\kappa$ is not Mahlo.\nIndeed, a regular cardinal among the uncountable strong limit cardinals below $\\kappa$ would be an inaccessible cardinal below $\\kappa$.\nSo the set of regular cardinals below $\\kappa$ misses this closed unbounded set.\nIf $\\kappa$ is Mahlo, then $\\kappa$ is the $\\kappa$-th inaccessible cardinal.\nEvery weakly compact cardinal is Mahlo.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "defines": "CJFEPPP9KA",
    "kind": "definition",
    "msc": [
      "03E55"
    ],
    "name": "Mahlo cardinal",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Mahlo_cardinal"
      }
    ],
    "tag": "PGJNYGAN75",
    "uses": [
      "NW4DREQ5DQ",
      "YE6WZR46RQ",
      "13ZV2F8JAR"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PGJNYGAN75/",
  "links": [
    {
      "field": "defines",
      "target": "CJFEPPP9KA"
    },
    {
      "field": "uses",
      "target": "NW4DREQ5DQ"
    },
    {
      "field": "uses",
      "target": "YE6WZR46RQ"
    },
    {
      "field": "uses",
      "target": "13ZV2F8JAR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PGJNYGAN75.md",
  "msc": [
    "03E55"
  ],
  "name": "Mahlo cardinal",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PGJ/PGJNYGAN75.md",
  "status": null,
  "tag": "PGJNYGAN75",
  "uri": "m2://PGJNYGAN75",
  "url": "https://math2.tamnd.com/wiki/mahlo-cardinal-definition/",
  "writing": {
    "tag": "PGJNYGAN75",
    "kind": "definition",
    "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
      }
    ]
  }
}