{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nConsider first-order logic without equality over a [[signature|WDVVEYD6AT]] $\\sigma$ with at least one constant symbol.\nLet $\\varphi = \\forall x_1 \\cdots \\forall x_n \\, \\psi(x_1, \\dots, x_n)$ be a sentence in [[Skolem normal form|GCPQGMJYEV]], with $\\psi$ quantifier-free.\nThen $\\varphi$ is unsatisfiable if and only if there are finitely many tuples $\\bar t_1, \\dots, \\bar t_k$ of closed [[terms|6D3WA7N2W4]] with the following property.\nThe conjunction $\\psi(\\bar t_1) \\land \\cdots \\land \\psi(\\bar t_k)$ is unsatisfiable as a propositional formula, with its atomic sentences read as propositional variables.\n\n## Remarks\n\nThe instances $\\psi(\\bar t)$ with closed terms $\\bar t$ are the ground instances of $\\varphi$.\nFor the proof, suppose every finite set of ground instances is propositionally [[satisfiable|QN7M1JCGT9]].\nBy propositional compactness, all ground instances are satisfiable at once by some assignment.\nThis assignment defines a Herbrand structure, whose elements are the closed terms, and this structure is a model of $\\varphi$.\nThe other direction holds because $\\varphi$ implies each of its ground instances.\nExample: let $\\varphi = \\forall x \\, (P(x) \\land \\neg P(f(x)))$ with a constant $c$.\nThe instances for $x = c$ and $x = f(c)$ contain $\\neg P(f(c))$ and $P(f(c))$, so their conjunction is unsatisfiable, and so is $\\varphi$.\nIf $\\sigma$ has no constant symbol, one adds one; without closed terms there are no ground instances.\n\n## Explanations\n\n### L1 human\n\nThe theorem reduces a question about all possible structures to a question about finitely many concrete cases.\nTo show that a universal sentence is contradictory, it is enough to plug in finitely many concrete terms and find a contradiction by truth tables.\nThis is the basis of many automatic theorem provers.\n\n### L2 human\n\nCombined with Skolemization, the theorem gives a semi-decision procedure for validity in first-order logic.\nTo test whether $\\chi$ is valid, put $\\neg \\chi$ in Skolem normal form and search through larger and larger sets of ground instances.\nIf $\\chi$ is valid, the search ends; by [[Church's theorem|VA6BNNY2YC]] it cannot always end when $\\chi$ is not valid.\nThe theorem is closely related to the [[compactness theorem|1NR3HAWG9D]] and to [[Gödel's completeness theorem|WQ1ZE255SF]].\nWith equality, the same holds after adding the equality axioms for $\\sigma$ to $\\varphi$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "about": [
      "QN7M1JCGT9"
    ],
    "kind": "statement",
    "msc": [
      "03B10",
      "03F05",
      "03B35"
    ],
    "name": "Herbrand's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q1930577",
      "wikipedia": "Herbrand's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Herbrand's_theorem"
      }
    ],
    "tag": "VZ21F10WRS",
    "uses": [
      "GCPQGMJYEV",
      "6D3WA7N2W4",
      "QN7M1JCGT9",
      "WDVVEYD6AT"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/VZ21F10WRS/",
  "links": [
    {
      "field": "about",
      "target": "QN7M1JCGT9"
    },
    {
      "field": "uses",
      "target": "GCPQGMJYEV"
    },
    {
      "field": "uses",
      "target": "6D3WA7N2W4"
    },
    {
      "field": "uses",
      "target": "QN7M1JCGT9"
    },
    {
      "field": "uses",
      "target": "WDVVEYD6AT"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/VZ21F10WRS.md",
  "msc": [
    "03B10",
    "03F05",
    "03B35"
  ],
  "name": "Herbrand's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VZ2/VZ21F10WRS.md",
  "status": null,
  "tag": "VZ21F10WRS",
  "uri": "m2://VZ21F10WRS",
  "url": "https://math2.tamnd.com/wiki/herbrands-theorem/",
  "writing": {
    "tag": "VZ21F10WRS",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}