{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $M$ be an $L$-structure that is $\\aleph_0$-[[saturated|RMTZ5J64FX]], and let $\\varphi(\\bar{x})$ be an $L$-[[formula|HYJ10AF6RT]] with parameters from $M$.\nWrite $\\varphi(M)$ for the [[definable set|CY1TXHF6KQ]] of tuples of $M$ that satisfy $\\varphi$.\nFor each [[ordinal|0YSCSX5YZC]] $\\alpha$, the relation $\\mathrm{RM}(\\varphi) \\ge \\alpha$ is defined by induction on $\\alpha$:\n\n1. $\\mathrm{RM}(\\varphi) \\ge 0$ if $\\varphi(M)$ is nonempty;\n2. $\\mathrm{RM}(\\varphi) \\ge \\alpha + 1$ if there are formulas $\\psi_0, \\psi_1, \\dots$ with parameters from $M$ whose sets $\\psi_i(M)$ are pairwise disjoint subsets of $\\varphi(M)$ with $\\mathrm{RM}(\\psi_i) \\ge \\alpha$;\n3. for a limit ordinal $\\alpha$, $\\mathrm{RM}(\\varphi) \\ge \\alpha$ if $\\mathrm{RM}(\\varphi) \\ge \\beta$ for every $\\beta \u003c \\alpha$.\n\nThe *Morley rank* $\\mathrm{RM}(\\varphi)$ is $-1$ if $\\varphi(M)$ is empty.\nOtherwise it is the largest ordinal $\\alpha$ with $\\mathrm{RM}(\\varphi) \\ge \\alpha$ if one exists, and $\\infty$ if $\\mathrm{RM}(\\varphi) \\ge \\alpha$ for every ordinal $\\alpha$.\nFor an arbitrary $L$-structure $N$ and a formula with parameters from $N$, the Morley rank is computed in an $\\aleph_0$-saturated [[elementary extension|CGH8C8EA9N]] of $N$.\nIt does not depend on the choice of the extension.\n\n## Remarks\n\nExample: in an [[algebraically closed field|MT8TDVPBXB]] $K$, the formula $x = x$ in one variable has Morley rank $1$.\nThe singletons $\\set{a}$ for distinct $a \\in K$ are infinitely many disjoint sets of rank $0$, so the rank is at least $1$.\nBy quantifier elimination every definable subset of $K$ is finite or cofinite, so no two disjoint definable subsets are both infinite.\nA set of rank at least $1$ is infinite, so the rank of $x = x$ is not at least $2$.\nMore generally, in $K^n$ the Morley rank of an irreducible variety is its Krull dimension.\n\nNon-example: in the theory of dense linear orders without endpoints, the formula $x = x$ has Morley rank $\\infty$.\nThe rank of an open interval is at least $\\alpha$ for every $\\alpha$, by induction, since an interval contains infinitely many pairwise disjoint open intervals.\n\nA complete theory $T$ in a countable signature is $\\omega$-stable, a strong form of [[stability|T7JW5QY6KR]], if and only if every formula has Morley rank less than $\\infty$.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "WN43DRPNBX",
    "kind": "definition",
    "msc": [
      "03C45"
    ],
    "name": "Morley rank",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Morley_rank"
      }
    ],
    "tag": "PXX4P2BHJW",
    "uses": [
      "RMTZ5J64FX",
      "CY1TXHF6KQ",
      "HYJ10AF6RT",
      "0YSCSX5YZC",
      "CGH8C8EA9N",
      "MT8TDVPBXB",
      "T7JW5QY6KR"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/PXX4P2BHJW/",
  "links": [
    {
      "field": "defines",
      "target": "WN43DRPNBX"
    },
    {
      "field": "uses",
      "target": "RMTZ5J64FX"
    },
    {
      "field": "uses",
      "target": "CY1TXHF6KQ"
    },
    {
      "field": "uses",
      "target": "HYJ10AF6RT"
    },
    {
      "field": "uses",
      "target": "0YSCSX5YZC"
    },
    {
      "field": "uses",
      "target": "CGH8C8EA9N"
    },
    {
      "field": "uses",
      "target": "MT8TDVPBXB"
    },
    {
      "field": "uses",
      "target": "T7JW5QY6KR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PXX4P2BHJW.md",
  "msc": [
    "03C45"
  ],
  "name": "Morley rank",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PXX/PXX4P2BHJW.md",
  "status": null,
  "tag": "PXX4P2BHJW",
  "uri": "m2://PXX4P2BHJW",
  "url": "https://math2.tamnd.com/wiki/morley-rank-definition/",
  "writing": {
    "tag": "PXX4P2BHJW",
    "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
      }
    ]
  }
}