User blog comment:Lepton Adapter/Meta: What a growth rate is/@comment-35470197-20181127215527/@comment-35470197-20181127222020

Another serious problem is that diagonalisation in FGH construction does not necessarily give a right hierarchy of growth rate if the choice of fundamental sequences is "bad". For example, there is a choice of fundamental sequences which gives \(f_{\omega} = f_{\omega^{\omega}}\) in FGH. Namely, the inequality \(\alpha < \beta\) of ordinals does not necessarily ensure the inequality \(f_{\alpha} < f_{\beta}\) of growth rate.

We do not intend to choose such a system of fundamental sequences when we deal with small ordinals, but this problem (unwillingly) more possibly occurs when we deal with large ordinals described by large cardinals, because the corresponding FGH cotains monstrously many diagonalisation steps.

So avoiding such a "bad" choice is one of my interest when I create a new large function using FGH-like recursion.