Is there a computable function that grows faster than any function in a fast growing hierarchy with index less than the Church–Kleene ordinal, where computable fundamental sequences are used?
If the answer is negative, is there any proof that can be referenced or provided?
Fast growing heiarchies are defined on their wiki page, https://en.wikipedia.org/wiki/Fast-growing_hierarchy.
As discussed in the comments, I do not think the particular fundamental sequences chosen will affect this question. If I am wrong on that point, please let me know.