User blog comment:Triakula/FGH and SGH at large ordinals/@comment-35470197-20191214094756

Which one stands for FGH? Anyway, I think that FGH_α(x) > SGH_α(x) holds for any countable ordinal α and x > 1, and hence FGH_α eventually dominates SGH_α for any countable ordinal α with respect to any system of fundamental sequences. Sorry if I am incorrect.

Also, the comparison between FGH_α and SGH_α up to a certain comparability is partially known in the study of Catching function by hyp cos. Unfortunately, the original Catching function is known to be ill-defined, because of the heavy dependency of those hierarchies on fundamental sequences.

> Generally, is it true that initially faster hierarchies imply that they are always faster at all ordinals regardless of fundamental sequences?

I could not understand the precise meaning. You mean that if SGH_β is eventually dominated by FGH_β for any non-zero β smaller than a given ordinal α equipped with a system of fundamental sequences, then SGH_α is eventually dominated by FGH_α with respect to any fundamental sequence of α, right? Then it is true by the reason above. If you consider other hierachies, then I do not think so.