{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $\\mathcal{L}$ be a regular abstract logic that extends [[first-order logic|086QKPF0A6]].\nRegular means that $\\mathcal{L}$ is closed under the Boolean connectives, existential quantification, renaming of symbols and relativization.\nSuppose that $\\mathcal{L}$ has the following two properties.\n\n1. Compactness: a set of $\\mathcal{L}$-sentences has a model whenever each of its finite subsets has a model.\n2. Downward Löwenheim–Skolem property: each $\\mathcal{L}$-sentence with an infinite model has a countable model.\n\nThen each $\\mathcal{L}$-sentence is equivalent to a first-order [[sentence|DKFGQTRYZE]], that is, it has the same models.\n\n## Remarks\n\nThe proof argues by contradiction with Ehrenfeucht–Fraïssé games.\nIf an $\\mathcal{L}$-sentence $\\varphi$ is not first-order, then for each $n$ there are [[structures|983GZBQYT4]] $\\mathcal{A}_n \\models \\varphi$ and $\\mathcal{B}_n \\models \\neg\\varphi$ that satisfy the same first-order sentences of quantifier depth $n$.\nCompactness and the Löwenheim–Skolem property, applied to an $\\mathcal{L}$-sentence coding these pairs and partial isomorphisms between them, give countable $\\mathcal{A} \\models \\varphi$ and $\\mathcal{B} \\models \\neg\\varphi$ with a back-and-forth system.\nThen $\\mathcal{A} \\cong \\mathcal{B}$, which contradicts $\\mathcal{A} \\models \\varphi$ and $\\mathcal{B} \\models \\neg\\varphi$.\n\nExample: second-order logic is a regular extension of first-order logic that is not equivalent to it.\nBy the theorem, it must lack the [[compactness theorem|1NR3HAWG9D]] or the [[Löwenheim–Skolem theorem|484M0TZ1S7]]; in fact it lacks both.\nThe infinitary logic $L_{\\omega_1\\omega}$ has the Löwenheim–Skolem property, so it must fail compactness, and it does.\n\n## Explanations\n\n### L1 human\n\nFirst-order logic has two famous properties: compactness and the Löwenheim–Skolem property.\nLindström's theorem says that no stronger logic keeps both.\nSo these two properties characterise first-order logic among all reasonable logics.\n\n### L2 human\n\nThe theorem, proved by Per Lindström in 1969, founded abstract model theory, which compares logics by their properties.\nIt explains why first-order logic is the default logic of model theory: any gain in expressive power costs compactness or the Löwenheim–Skolem theorem.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "086QKPF0A6"
    ],
    "kind": "statement",
    "msc": [
      "03C95",
      "03B10"
    ],
    "name": "Lindström's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2379128",
      "wikipedia": "Lindström's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lindström's_theorem"
      }
    ],
    "tag": "FYZ3CJGZYV",
    "uses": [
      "086QKPF0A6",
      "1NR3HAWG9D",
      "484M0TZ1S7",
      "983GZBQYT4",
      "DKFGQTRYZE"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/FYZ3CJGZYV/",
  "links": [
    {
      "field": "about",
      "target": "086QKPF0A6"
    },
    {
      "field": "uses",
      "target": "086QKPF0A6"
    },
    {
      "field": "uses",
      "target": "1NR3HAWG9D"
    },
    {
      "field": "uses",
      "target": "484M0TZ1S7"
    },
    {
      "field": "uses",
      "target": "983GZBQYT4"
    },
    {
      "field": "uses",
      "target": "DKFGQTRYZE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/FYZ3CJGZYV.md",
  "msc": [
    "03C95",
    "03B10"
  ],
  "name": "Lindström's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/FYZ/FYZ3CJGZYV.md",
  "status": null,
  "tag": "FYZ3CJGZYV",
  "uri": "m2://FYZ3CJGZYV",
  "url": "https://math2.tamnd.com/wiki/lindstroms-theorem/",
  "writing": {
    "tag": "FYZ3CJGZYV",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}