Fundamental sequence

A fundamental sequence (FS) is an important concept in the study of ordinal hierarchies. Definitions of what constitutes a valid fundamental sequence vary, but here is a generally applicable one: if \(\alpha\) is a countable limit ordinal, an FS for \(\alpha\) is a sequence of length \(\omega\) consisting of ordinals, of which is equal to \(\alpha\).

Typically when we speak of FS's we refer to systems of FS's that systematically generate these sequences. If an FS system defines an FS for \(\alpha\), we use \(\alpha[n]\) to refer to the \(n\)th member of the FS.

In ZF, it is impossible to show that there exists an FS system that works for all countable limit ordinals, although with the axiom of choice we can nonconstructively prove that such a system exists. Unfortunately, there is no such constructive proof. Indeed, is necessary for this.

It is an open problem whether an ordinal hierarchy can be defined without using fundamental sequences. More explicitly, the problem concerns whether there is a model of ZF such that there exists an \(F : \omega_1 \rightarrow (\mathbb{N} \rightarrow \mathbb{N})\) where for all \(\alpha > \beta\), \(F(\alpha)\) eventually outgrows \(F(\beta)\), but there does not exist an \(S: \omega_1 \cap \text{Lim} \rightarrow (\mathbb{N} \rightarrow \omega_1)\) such that for all \(\alpha\), \(\sup(R) = \alpha\) where \(R\) is the range of \(S(\alpha)\).