{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $T$ be a [[complete theory|655K8Y7VS8]] in a signature $L$, with infinite models.\nA [[model|MVCH6N9SNW]] $M$ of $T$ is a *prime model* of $T$ if $M$ has an [[elementary embedding|CGH8C8EA9N]] into every model $N$ of $T$.\n\n## Remarks\n\nExample: the field $\\overline{\\Q}$ of algebraic numbers is a prime model of $\\mathrm{ACF}_0$, the theory of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$.\nEvery algebraically closed field $K$ of characteristic $0$ contains $\\Q$, and hence a copy of the algebraic closure of $\\Q$.\nThis field embedding is elementary, because $\\mathrm{ACF}_0$ has quantifier elimination.\n\nNon-example: $\\C$ is a model of $\\mathrm{ACF}_0$, but it is not prime, since it is uncountable and so cannot embed into the countable model $\\overline{\\Q}$.\n\nIf $L$ is countable, a prime model of $T$ is countable by the downward Löwenheim–Skolem theorem.\nThen a model is prime if and only if it is countable and atomic, that is, every tuple of it realizes an isolated type.\nTwo prime models of $T$ are isomorphic.\nAnother example is $(\\Q, \u003c)$, the prime model of the theory of dense linear orders without endpoints.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "S8W40PYNHX",
    "kind": "definition",
    "msc": [
      "03C50",
      "03C15"
    ],
    "name": "Prime model",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Prime_model"
      }
    ],
    "tag": "FJ79HREXG4",
    "uses": [
      "655K8Y7VS8",
      "MVCH6N9SNW",
      "CGH8C8EA9N",
      "MT8TDVPBXB"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/FJ79HREXG4/",
  "links": [
    {
      "field": "defines",
      "target": "S8W40PYNHX"
    },
    {
      "field": "uses",
      "target": "655K8Y7VS8"
    },
    {
      "field": "uses",
      "target": "MVCH6N9SNW"
    },
    {
      "field": "uses",
      "target": "CGH8C8EA9N"
    },
    {
      "field": "uses",
      "target": "MT8TDVPBXB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FJ79HREXG4.md",
  "msc": [
    "03C50",
    "03C15"
  ],
  "name": "Prime model",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FJ7/FJ79HREXG4.md",
  "status": null,
  "tag": "FJ79HREXG4",
  "uri": "m2://FJ79HREXG4",
  "url": "https://math2.tamnd.com/wiki/prime-model-definition/",
  "writing": {
    "tag": "FJ79HREXG4",
    "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
      }
    ]
  }
}