{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $L$ be a [[signature|WDVVEYD6AT]] and let $T$ be an $L$-[[theory|92SWXST1ZM]].\nThe theory $T$ *has quantifier elimination* if, for every $L$-[[formula|HYJ10AF6RT]] $\\varphi(x_1, \\dots, x_n)$, there is a quantifier-free $L$-formula $\\psi(x_1, \\dots, x_n)$ such that\n$$T \\models \\forall x_1 \\cdots \\forall x_n \\, \\bigl(\\varphi(x_1, \\dots, x_n) \\leftrightarrow \\psi(x_1, \\dots, x_n)\\bigr).$$\n\n## Remarks\n\nFor sentences ($n = 0$) one allows the quantifier-free sentences $\\top$ and $\\bot$ when $L$ has no constant symbols.\nIt suffices to treat formulas of the form $\\exists y \\, \\chi(x_1, \\dots, x_n, y)$ with $\\chi$ a conjunction of atomic and negated atomic formulas.\n\nExample: the theory of dense linear orders without endpoints, in the signature $\\set{\u003c}$, has quantifier elimination.\nFor instance, $\\exists y \\, (x_1 \u003c y \\wedge y \u003c x_2)$ is equivalent to $x_1 \u003c x_2$ by density, and $\\exists y \\, (x_1 \u003c y)$ is equivalent to $x_1 = x_1$ since there is no greatest element.\n\nNon-example: the theory of $(\\R, +, \\cdot, 0, 1)$, without the order symbol, does not have quantifier elimination.\nThe formula $\\exists y \\, (x = y \\cdot y)$ defines $[0, \\infty)$, which is infinite and co-infinite.\nA quantifier-free formula in $x$ is a Boolean combination of polynomial equations $p(x) = 0$, so it defines a finite or cofinite subset of $\\R$.\nWith the order symbol added, the theory has quantifier elimination; this is [[Tarski's theorem|FSJ9NGRRPA]].\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "7FDDXV3JK7",
    "kind": "definition",
    "msc": [
      "03C10"
    ],
    "name": "Quantifier elimination",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Quantifier_elimination"
      }
    ],
    "tag": "RE5TFTDP4D",
    "uses": [
      "92SWXST1ZM",
      "HYJ10AF6RT",
      "WDVVEYD6AT",
      "FSJ9NGRRPA"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/RE5TFTDP4D/",
  "links": [
    {
      "field": "defines",
      "target": "7FDDXV3JK7"
    },
    {
      "field": "uses",
      "target": "92SWXST1ZM"
    },
    {
      "field": "uses",
      "target": "HYJ10AF6RT"
    },
    {
      "field": "uses",
      "target": "WDVVEYD6AT"
    },
    {
      "field": "uses",
      "target": "FSJ9NGRRPA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/RE5TFTDP4D.md",
  "msc": [
    "03C10"
  ],
  "name": "Quantifier elimination",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/RE5/RE5TFTDP4D.md",
  "status": null,
  "tag": "RE5TFTDP4D",
  "uri": "m2://RE5TFTDP4D",
  "url": "https://math2.tamnd.com/wiki/quantifier-elimination-definition/",
  "writing": {
    "tag": "RE5TFTDP4D",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}