{
  "aliases": [
    "fundamental-sequence-ordinals"
  ],
  "backlinks": null,
  "body": "\n## Summary\n\nIn set theory, a mathematical discipline, a **fundamental sequence** is a cofinal [[sequence|0G9EKM0E15]] of ordinals all below a given limit ordinal. Depending on author, fundamental sequences may be restricted to ω-sequences only or permit fundamental sequences of length $\\mathrm{\\omega}_1$. The $n^{\\text{th}}$ element of the fundamental sequence of $\\alpha$ is commonly denoted $\\alpha[n]$, although it may be denoted $\\alpha_n$ or $\\{\\alpha\\}(n)$. Additionally, some authors may allow fundamental sequences to be defined on successor ordinals. The term dates back to (at the latest) Veblen's construction of normal functions $\\varphi_\\alpha$, while the concept dates back to Hardy's 1904 attempt to construct a set of [[cardinality|0G0ASKNA6Z]] $\\aleph_1$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "aliases": [
      "fundamental-sequence-ordinals"
    ],
    "inventory": [
      "FM6Y2Q10F9"
    ],
    "kind": "concept",
    "name": "Fundamental sequence (set theory)",
    "same_as": {
      "wikidata": "Q118218208",
      "wikipedia": "Fundamental sequence (set theory)"
    },
    "sources": [
      {
        "locator": "Fundamental sequence (set theory)",
        "mention": "wikipedia://en/73752551",
        "note": "adopted",
        "sections": [
          "Summary"
        ],
        "wikipedia": "en:73752551@r1374629341"
      }
    ],
    "tag": "VB6YXF761M"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/VB6YXF761M/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/VB6YXF761M.md",
  "msc": null,
  "name": "Fundamental sequence (set theory)",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/VB6/VB6YXF761M.md",
  "status": null,
  "tag": "VB6YXF761M",
  "uri": "m2://VB6YXF761M",
  "url": "https://math2.tamnd.com/wiki/fundamental-sequence-set-theory/",
  "writing": {
    "tag": "VB6YXF761M",
    "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
      }
    ]
  }
}