User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-1605058-20171029134223

Two things aren't clear for me:

1. that for any \(\beta\), you will find a function satisfying 3a-c, e.g. it might be that, for the sake of a simple example, \(f_{\omega k+l}\) functions take values growing so rapidly that there isn't a computable function which would work as \(f_{\omega^2}\).

2. that this hierarchy isn't "bounded", i.e. there is a computable function which outgrows all of the \(f_\beta\) in this hierarchy.

I don't claim I have an example where such things happen, but I don't see why they can't happen.