---
tag: VB6YXF761M
kind: concept
name: Fundamental sequence (set theory)
aliases: [fundamental-sequence-ordinals]
sources:
  - wikipedia: en:73752551@r1374629341
    locator: Fundamental sequence (set theory)
    note: adopted
    mention: wikipedia://en/73752551
    sections: [Summary]
same_as:
  wikidata: Q118218208
  wikipedia: Fundamental sequence (set theory)
inventory: [FM6Y2Q10F9]
---

## Summary

In 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]] $\aleph_1$.