{
  "aliases": [
    "countable-transitive-model",
    "set-model",
    "standard-transitive-model",
    "transitive-model"
  ],
  "backlinks": null,
  "body": "\n## Summary\n\nIn set theory, a **standard model** for a theory $T$ (in the language of set theory) is a model $M$ for $T$ where the membership relation $\\in_{M}$ is the same as the membership relation $\\in$ of a set theoretical universe $V$ (restricted to the domain of $M$). In other words, $M$ is a substructure of $V$. A standard model $M$ that satisfies the additional transitivity condition that $x \\in y \\in M$ implies $x \\in M$ is a **standard transitive model** (or simply a **transitive model**).\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "aliases": [
      "countable-transitive-model",
      "set-model",
      "standard-transitive-model",
      "transitive-model"
    ],
    "inventory": [
      "5KGJHE89G5"
    ],
    "kind": "concept",
    "name": "Standard model (set theory)",
    "same_as": {
      "wikidata": "Q117715071",
      "wikipedia": "Standard model (set theory)"
    },
    "sources": [
      {
        "locator": "Standard model (set theory)",
        "mention": "wikipedia://en/18391755",
        "note": "adopted",
        "sections": [
          "Summary"
        ],
        "wikipedia": "en:18391755@r1367617467"
      }
    ],
    "tag": "WP6KSK5ZD9"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/WP6KSK5ZD9/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/WP6KSK5ZD9.md",
  "msc": null,
  "name": "Standard model (set theory)",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/WP6/WP6KSK5ZD9.md",
  "status": null,
  "tag": "WP6KSK5ZD9",
  "uri": "m2://WP6KSK5ZD9",
  "url": "https://math2.tamnd.com/wiki/standard-model-set-theory/",
  "writing": {
    "tag": "WP6KSK5ZD9",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}