User blog comment:Hyp cos/Fundamental Sequences in Taranovsky's Notation/@comment-40154718-20190916083849/@comment-35470197-20190916092714

Given an ordinal notation, there are several meanings of "fundamental sequences". For example, consider Buchholz's ordinal notation. The fundamental sequence of D_1 0, which corresponds to Ω_1, is defined as (D_1 0)[n] = n, where n runs through all ordinal terms in the notation. On the other hand, we can define another fundamental sequence as a strictly increasing sequence D_0 D_ω … D_ω 0.

The former one is used to define fundamental sequences for "countable ordinal terms", i.e. ordinal terms smaller than D_1 0 so that they corresponds to fundamental sequences of the corresponding countable ordinals. The second one is a portion of the whole system of fundamental sequences such that they corresponds to fundamental sequence of the corresponding ordinal types.

Hyp cos's definition is exactly the latter one. It does not give a system of fundamental sequences of the corresponding ordinals with respect to certain conversion whose codomain includes uncountable ordinals, but gives a system of fundamental sequences of the corresponding ordinal types.