Fundamental sequence (set theory)

Summary

In set theory, a mathematical discipline, a fundamental sequence is a cofinal sequence 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 $\aleph_1$.