{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "AGYBSTD7BG"
    },
    {
      "field": "uses",
      "target": "AGYBSTD7BG"
    },
    {
      "field": "uses",
      "target": "TK6R2C9KRE"
    },
    {
      "field": "defines",
      "target": "VFPGADHJYA"
    }
  ],
  "body": "\n## Summary\n\nAn ultraproduct combines a family of structures of the same language into one new structure.\nAn [[ultrafilter|J64Y3JC0RE]] on the index set decides which properties hold in it.\nA property holds in the ultraproduct when it holds in a \"large\" set of factors, where \"large\" means \"in the ultrafilter\".\nBy [[Łoś's theorem|AGYBSTD7BG]], this rule works for every first-order formula.\nUltraproducts give a short proof of the [[compactness theorem|1NR3HAWG9D]] and build nonstandard models such as the [[hyperreal numbers|421QMTFG9T]].\n\n## Explanations\n\n### L1 human\n\nTake infinitely many copies of a structure, say the real numbers, one for each natural number.\nAn element of the new structure is a sequence, one entry from each copy.\nTwo sequences count as equal when they agree on a large set of indices, and an ultrafilter says which sets are large.\nFor example, with a non-principal ultrafilter on $\\N$, the sequence $(1, 2, 3, \\dots)$ becomes a number larger than every ordinary real number.\nThis is how infinitely large and infinitely small numbers can be built rigorously.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "kind": "concept",
    "msc": [
      "03C20"
    ],
    "name": "Ultraproduct",
    "same_as": {
      "wikidata": "Q1535225",
      "wikipedia": "Ultraproduct"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Ultraproduct"
      }
    ],
    "tag": "AX9Q86GB8M"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/AX9Q86GB8M/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/AX9Q86GB8M.md",
  "msc": [
    "03C20"
  ],
  "name": "Ultraproduct",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AX9/AX9Q86GB8M.md",
  "status": null,
  "tag": "AX9Q86GB8M",
  "uri": "m2://AX9Q86GB8M",
  "url": "https://math2.tamnd.com/wiki/ultraproduct/",
  "writing": {
    "tag": "AX9Q86GB8M",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}