User blog comment:Username5243/Hedrondude just updated his large number pages/@comment-1605058-20160202140040/@comment-5529393-20160203103429

I don't know that f_w_1^CK is unacceptable;  you just have to define it. Of course, it may be simpler just to refer to Turing degrees once you get to nonrecursive functions. (However Turing degrees aren't directly related to growth rate.)

Googology Noob, the term you probably want is "cofinality". Countable limit ordinals have cofinality w, so there exists sequences of length w with those ordinals as their supremum, but w_1 has cofinality w_1, so a set of smaller ordinals has to have cardinality w_1 to have w_1 as their supremum.