User blog comment:Lepton Adapter/Meta: What a growth rate is/@comment-30754445-20181128000924

Technically you are correct.

But practically, your choice of fundamental sequences doesn't matter much, unless you deliberately try to create pathological results (like setting ω[n]=BB[n]). That's one of the advantages of the FGH over (say) the SGH.

For example, if I write fΓ 0 (100), the general ballpark of the size of that number is quite clear even though there are several possible reasonable fundamental sequences for Γ0. This is also why different array notations yield very similarly-sized numbers at the same ordinal. For example, consider:

1. Bowers' Goppatoth

2. Saibian's Tetheratoth

3. Q10 in my letter-notation

4. (0,0)(1,1)10 in PSS (any version you wish)

5. fε 0 (10) in the FGH with the Wainer Hierarchy.

All these numbers are the output of functions that do ε0-level recursion, and they are all roughly the same size (#4 will be the largest, assuming the f(n)=n2 variant, by not by much).

So I don't really see any problem in using ordinals as a yardstick for function growth-rates. I do share your dislike of the phrasing "this function has a growth rate of [ordinal]", though. It's just sloppy. Personally, I usually phrase it as "this function is on level-[ordinal] of the FGH" or something similar.