{
  "aliases": [
    "independent-axiom"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nAn axiom $P$ is **independent** of a theory $T$, i.e., a collection of sentences, if and only if neither $P$ nor its negation $\\neg P$ is provable from $T$. Equivalently, $P$ and $\\neg P$ are each consistent with $T$, i.e., some models of $T$ make $P$ true and others make $\\neg P$ true. Proving the independence of $P$ from $T$ is usually done by demonstrating the existence of such models.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "aliases": [
      "independent-axiom"
    ],
    "inventory": [
      "1JM6WZKD3B"
    ],
    "kind": "statement",
    "name": "Axiom independence",
    "role": "proposition",
    "same_as": {
      "wikidata": "Q4830547",
      "wikipedia": "Axiom independence"
    },
    "sources": [
      {
        "locator": "Axiom independence",
        "mention": "wikipedia://en/6118589",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:6118589@r1358356173"
      }
    ],
    "tag": "QKCAEG2Y0E"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/QKCAEG2Y0E/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/QKCAEG2Y0E.md",
  "msc": null,
  "name": "Axiom independence",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/QKC/QKCAEG2Y0E.md",
  "status": null,
  "tag": "QKCAEG2Y0E",
  "uri": "m2://QKCAEG2Y0E",
  "url": "https://math2.tamnd.com/wiki/axiom-independence/",
  "writing": {
    "tag": "QKCAEG2Y0E",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}