User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412/@comment-30754445-20171029130815

@Littlepeng

"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"

I wasn't trying to give a "definition" of such a sequence. Just a simple description of one.

At any rate, this could be turned into an actual workable definition by using a variation on the Kleene's O idea:

(1) We remember that any computable function can be represented by a Turing Machine.

(2) We enumerate them (there are plenty of standard orderings to choose from).

(3) We use the usual framework of the FGH. For a limit ordinal β, fβ(n) is defined as the first function in our enumeration that obeys the usual rules:

(3a) For any n, there exists βn < β such that fβ(n) = fβ n (n).

(3b) The sequence β1, β2, β3... is monotonically increasing,

(3c) β is the limit of β1, β2, β3...

And that's it! Unless I've missed something, this defines fundamental sequences for all ordinals below ω₁ck.

For the fundamental sequence of ω₁ck itself, we also need to define the "ordinal growth rate" of  any unbounded computable function g(n): "the smallest ordinal α such that g(n) does not outgrow f α (n)".

And now, finally, my original statement becomes a well-defined one:

"Look at the fastest growing function which can be programmed into a Turing Machine with n states. The ordinal growth rate of this function can be set as the nth member of the fundamental sequence of ω₁ck"

(If there's anything wrong with what I've did here, I'll be happy to hear about it)