User blog comment:Lepton Adapter/Meta: What a growth rate is/@comment-35470197-20181127215527/@comment-30754445-20181128001347

> 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.

If I remember correctly, there's actually a theorem that guarantees this cannot happen as long as the fundamental sequences obey a few basic criteria (like being strictly increasing).