User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-30754445-20171029162208

I'm fairly certain that #2 is impossible:

Say such a situation arises, and we artificially add this bounding computable function to our hierarchy. What would be it's ordinal position in the hierarchy? Obviously, it would be the first nonrecursive ordinal, which is ω₁ck.

This would mean that we found a set of fundamental sequences such that f_ω₁ck(n) is a computable function. Surely there's a theorem that states that this is can never happen?

As for #1, that is indeed a problem. I think it can be fixed, though, by adding more conditions. I'll think about it.