{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $T$ be a [[complete theory|655K8Y7VS8]] in a countable signature, with infinite models, and let $\\lambda$ be an infinite [[cardinal|0EBACQFAZW]].\nThe theory $T$ is *$\\lambda$-stable* if the following holds for every [[model|MVCH6N9SNW]] $M$ of $T$ and every set $A$ of elements of $M$ with $\\abs{A} \\le \\lambda$.\nThe set $S_1^M(A)$ of complete $1$-[[types|9V71Q3SD1G]] over $A$ has [[cardinality|0G0ASKNA6Z]] at most $\\lambda$.\nThe theory $T$ is *stable* if it is $\\lambda$-stable for some infinite cardinal $\\lambda$.\n\n## Remarks\n\nAn $\\aleph_0$-stable theory is called $\\omega$-stable.\nShelah proved that $T$ is stable if and only if no formula $\\varphi(\\bar{x}, \\bar{y})$ has the order property.\nThe order property means that some model of $T$ has tuples $\\bar{a}_i, \\bar{b}_i$ for $i \\in \\N$ with $\\varphi(\\bar{a}_i, \\bar{b}_j)$ true exactly when $i \u003c j$.\n\nExample: the theory $\\mathrm{ACF}_0$ of [[algebraically closed fields|MT8TDVPBXB]] of characteristic $0$ is $\\omega$-stable.\nFor a countable set $A$ in a model $K$, let $k$ be the subfield generated by $A$.\nBy quantifier elimination, a complete $1$-type over $A$ is determined by the ideal of polynomials in $k[x]$ that vanish at $x$.\nThis ideal is $0$ or is generated by an irreducible polynomial, and $k[x]$ has countably many polynomials, so there are countably many types.\n\nNon-example: the theory of [[dense linear orders without endpoints|2W4S2H4VYA]] is not stable.\nThe formula $x \u003c y$ has the order property: in $(\\Q, \u003c)$ take $a_i = b_i = i$, so $a_i \u003c b_j$ exactly when $i \u003c j$.\nFor $\\lambda = \\aleph_0$ one sees the failure directly, with $A = \\Q$ inside the model $\\R$.\nEach irrational number $r$ realizes its own complete type over $\\Q$, determined by the cut $\\set{q \\in \\Q : q \u003c r}$.\nSo $S_1^{\\R}(\\Q)$ has size at least $2^{\\aleph_0}$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "T7JW5QY6KR",
    "kind": "definition",
    "msc": [
      "03C45"
    ],
    "name": "Stable theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Stable_theory"
      }
    ],
    "tag": "C44JJKGCBF",
    "uses": [
      "655K8Y7VS8",
      "9V71Q3SD1G",
      "MVCH6N9SNW",
      "0EBACQFAZW",
      "0G0ASKNA6Z",
      "MT8TDVPBXB",
      "2W4S2H4VYA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/C44JJKGCBF/",
  "links": [
    {
      "field": "defines",
      "target": "T7JW5QY6KR"
    },
    {
      "field": "uses",
      "target": "655K8Y7VS8"
    },
    {
      "field": "uses",
      "target": "9V71Q3SD1G"
    },
    {
      "field": "uses",
      "target": "MVCH6N9SNW"
    },
    {
      "field": "uses",
      "target": "0EBACQFAZW"
    },
    {
      "field": "uses",
      "target": "0G0ASKNA6Z"
    },
    {
      "field": "uses",
      "target": "MT8TDVPBXB"
    },
    {
      "field": "uses",
      "target": "2W4S2H4VYA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/C44JJKGCBF.md",
  "msc": [
    "03C45"
  ],
  "name": "Stable theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/C44/C44JJKGCBF.md",
  "status": null,
  "tag": "C44JJKGCBF",
  "uri": "m2://C44JJKGCBF",
  "url": "https://math2.tamnd.com/wiki/stable-theory-definition/",
  "writing": {
    "tag": "C44JJKGCBF",
    "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
      }
    ]
  }
}