User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-1605058-20171029104431

Using growth rates to define fundamental sequences doesn't seem like too sensible of an idea - in order to define growth rate (regardless of how we define it) we need to have some sort of fast-growing hierarchy of functions to compare to, but to construct it we need to have fundamental sequences beforehand.

What one can do instead is use Kleene's O again. The rough idea is as follows: given a number \(n\), we look at the ordinals which are "tagged" in Kleene's O by numbers up to \(n\), and then we take the largest one of them and declare it to be \(\omega_1^\mathrm{CK}[n]\).

(another note which might be of interest: in Kleene's O, not every number represents an ordinal, and the same ordinal can be represented in multiple ways, but importantly every ordinal is represented at least one, which suffices for showing countability of \(\omega_1^\mathrm{CK}\))