{
  "aliases": [
    "zf",
    "zfc",
    "zermelo-fraenkel"
  ],
  "backlinks": null,
  "body": "\n## Summary\n\nZermelo–Fraenkel set theory, ZF, is an axiomatic theory of [[sets|0SJ3RX9TXA]] in [[first-order logic|086QKPF0A6]] with one binary relation $\\in$.\nIts axioms are extensionality, regularity, separation, pairing, union, replacement, infinity and power set. Separation and replacement are axiom schemas.\nZF with the [[axiom of choice|0HTGB13FE2]] is ZFC, the standard foundation of most of mathematics.\n\n## Remarks\n\nThe [[axiom of extensionality|0TB16T2C97]] says that a set is determined by its elements.\nSeparation forms only subsets $\\{x \\in A \\mid P(x)\\}$ of a given set $A$. This avoids [[Russell's paradox|0HESMMSYEG]].\nThe [[continuum hypothesis|0MN2MER9WN]] is independent of ZFC, if ZFC is consistent.\n\n## Explanations\n\n### L1 human\n\nZF is a short list of rules that says which sets exist.\nFor example, for two sets there is a set that contains both, and for each set there is the set of its subsets.\nFrom these rules, one can build the numbers, functions and most other objects of mathematics.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "zf",
      "zfc",
      "zermelo-fraenkel"
    ],
    "kind": "concept",
    "msc": [
      "03E30"
    ],
    "name": "Zermelo–Fraenkel set theory",
    "same_as": {
      "wikidata": "Q191849",
      "wikipedia": "Zermelo–Fraenkel set theory"
    },
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory"
      }
    ],
    "tag": "0HE5WDQ9FJ",
    "uses": [
      "0SJ3RX9TXA",
      "086QKPF0A6",
      "0TB16T2C97"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/0HE5WDQ9FJ/",
  "links": [
    {
      "field": "uses",
      "target": "0SJ3RX9TXA"
    },
    {
      "field": "uses",
      "target": "086QKPF0A6"
    },
    {
      "field": "uses",
      "target": "0TB16T2C97"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0HE5WDQ9FJ.md",
  "msc": [
    "03E30"
  ],
  "name": "Zermelo–Fraenkel set theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0HE/0HE5WDQ9FJ.md",
  "status": null,
  "tag": "0HE5WDQ9FJ",
  "uri": "m2://0HE5WDQ9FJ",
  "url": "https://math2.tamnd.com/wiki/zermelo-fraenkel-set-theory/",
  "writing": {
    "tag": "0HE5WDQ9FJ",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}