User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190916083849/@comment-35470197-20190916111232

For a well-founded set (S,<), e.g. an ordinal notation, the ordinal type of (S,<) is the ordinal α admitting an order preserving bijective map S→α. For an s∈S, the initial segment {t∈S,t<s} is also a well-founded set with respect to the restriction of <, and hence its ordinal type makes sense.

Consider the case where (S,<) is Buchholz's ordinal notation. Then the canonical order-preserving map f:S→ε_{Ω_ω+1} assigns Ω_1 to D_1 0. On the other hand, the ordinal type of the initial segment below D_1 0 is ψ_0(ε_{ω+1}), which is countable. The map ot which assigns to each s∈S the ordinal type of the initial segment below s coincides with f when it is restricted to the subset of countable ordinal terms, i.e. the initial segment of D_1 0, but differs from f when we consider the whole system.

The standard system of fundamental sequences respects fundamental sequences of the images of f, while another fundamental sequence (D_1 0)[n] = D_0 D_ω … D_ω 0 respects fundamental sequences of the images of ot. The length-based fundamental sequence of an ordinal notation, e.g. Hyp cos's one, is the latter one.