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  "body": "\n## Statement\n\nLet $T$ be a [[complete theory|655K8Y7VS8]] in a countable first-order signature.\nLet $I(T, \\aleph_0)$ be the number of [[countable|01TQB88H4C]] [[models|MVCH6N9SNW]] of $T$ up to isomorphism.\nThen $I(T, \\aleph_0)$ is finite, or equal to [[$\\aleph_0$|0V2H5PGJ9B]], or equal to $2^{\\aleph_0}$.\n\n## Remarks\n\nThe conjecture is open.\nMorley proved in 1970 that $I(T, \\aleph_0)$ is at most $\\aleph_1$ or equal to $2^{\\aleph_0}$.\nSo the conjecture holds under the continuum hypothesis, and the open question is whether a theory can have exactly $\\aleph_1$ countable models when $\\aleph_1 \u003c 2^{\\aleph_0}$.\nShelah, Harrington and Makkai proved the conjecture for $\\omega$-stable theories.\nVaught himself proved that $I(T, \\aleph_0)$ is never equal to $2$.\n\nAll the allowed values occur.\nThe theory of dense linear orders without endpoints has exactly one countable model.\nThe theory of algebraically closed fields of characteristic $0$ has $\\aleph_0$ countable models, one for each transcendence degree $0, 1, 2, \\dots, \\aleph_0$.\nThe complete theory of $(\\Z, \u003c)$ has $2^{\\aleph_0}$ countable models.\n\n## Explanations\n\n### L1 human\n\nA complete theory can have many non-isomorphic countable models.\nThe conjecture says that their number is never strictly between \"countably many\" and \"as many as there are real numbers\".\nWithout the continuum hypothesis, there could be a size in between, and the question is whether some theory has exactly that many countable models.\n",
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      "655K8Y7VS8"
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    "history": {
      "first_stated": "1961",
      "posed_by": "Robert Vaught"
    },
    "kind": "conjecture",
    "msc": [
      "03C15",
      "03C45"
    ],
    "name": "Vaught conjecture",
    "same_as": {
      "wikidata": "Q7917378",
      "wikipedia": "Vaught conjecture"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Vaught_conjecture"
      }
    ],
    "tag": "YVBHE2SEE6",
    "uses": [
      "655K8Y7VS8",
      "MVCH6N9SNW",
      "01TQB88H4C",
      "0V2H5PGJ9B"
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  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/YVBHE2SEE6/",
  "links": [
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      "field": "about",
      "target": "655K8Y7VS8"
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  "markdown": "https://math2.tamnd.com/node/YVBHE2SEE6.md",
  "msc": [
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    "03C45"
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  "name": "Vaught conjecture",
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  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YVB/YVBHE2SEE6.md",
  "status": null,
  "tag": "YVBHE2SEE6",
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  "url": "https://math2.tamnd.com/wiki/vaught-conjecture/",
  "writing": {
    "tag": "YVBHE2SEE6",
    "kind": "conjecture",
    "level": "W0",
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