{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "16B7TWCEX9"
    }
  ],
  "body": "\n## Summary\n\nA strongly minimal theory is a complete first-order [[theory|92SWXST1ZM]] with infinite models.\nIn each of its models, every definable subset of the line is finite or cofinite.\nThese theories are the simplest infinite structures from the point of view of definability.\nIn a strongly minimal theory, algebraic closure behaves like linear span, so each model has a well-defined dimension.\nThey are basic building blocks of [[stable theories|T7JW5QY6KR]] and are central to the study of [[categorical theories|1JJCGNX240]].\n\n## Explanations\n\n### L1 human\n\nThink of a structure where a formula in one variable can only cut out a few points, or everything except a few points.\nNothing in between, like a half-line or an infinite set with an infinite complement, can be described.\nThe theory of an infinite set with no structure at all is the simplest case: a formula can only pick out finitely many named elements or their complement.\nAnother case is the theory of [[algebraically closed fields|MT8TDVPBXB]]: a nonzero polynomial in one variable has finitely many roots.\nSo in $\\C$, every definable set of numbers is finite or cofinite.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "03C45"
    ],
    "name": "Strongly minimal theory",
    "same_as": {
      "wikidata": "Q7624689",
      "wikipedia": "Strongly minimal theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Strongly_minimal_theory"
      }
    ],
    "specializes": [
      "T7JW5QY6KR"
    ],
    "tag": "PNGY60EBDJ"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/PNGY60EBDJ/",
  "links": [
    {
      "field": "specializes",
      "target": "T7JW5QY6KR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PNGY60EBDJ.md",
  "msc": [
    "03C45"
  ],
  "name": "Strongly minimal theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PNG/PNGY60EBDJ.md",
  "status": null,
  "tag": "PNGY60EBDJ",
  "uri": "m2://PNGY60EBDJ",
  "url": "https://math2.tamnd.com/wiki/strongly-minimal-theory/",
  "writing": {
    "tag": "PNGY60EBDJ",
    "kind": "concept",
    "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
      }
    ]
  }
}