User blog comment:Vel!/FGH Gripe/@comment-30754445-20170220134751/@comment-30754445-20170220155011

To say that g(x) grows as ordinal X in the FGH means that:

(1) for all ag(n) for all n>M

(2) for all a>X, there exists M such that fa(n)<g(n) for all n<M

Assuming reasonable fundamental sequences.

As for the dependence of the FGH on the fundamental sequences, I know that's the situation for the SGH. But it's a situation the arises pretty naturally there. It's not like it is difficult to detect, once you look for it.

The FGH, on the other hand, seems to behave nicely. Sure, there's a risk that we've missed something, but there's nothing wrong with provisionally assuming we didn't. If this assumption turns out to be wrong, we can always correct the articles later.

At any rate, if we are really serious about doing things right, the dependence of FS is hardly the biggest problem we need to fix. We don't even have an agreed upon standard of how to write large countable ordinals (as there are several different definitions of ψ and θ), so worrying about something which might not even be a problem seems a bit pointless to me.