User blog:Plain'N'Simple/A question for proof-theory experts

1. Let K(n) = { f :N→N | f is computable and can be proven to be a total function in ACA0 in less then 10n symbols } (*)

2. Let X(n) = max (F(n) | F∈K(n))

3. Let E(n) = fε 0 (n) in the fast growing hierarchy (using the Wainer fundamental sequences).

How does X(n) compare to E(n)? And how does the answer relate to the fact that PTO(ACA0)=ε0?

Thanks in advance.

(*) I'm fully aware of the fact that a proper definition of K(n) would require a specific symbolic formulation of ACA0. Unfortunately I am not knowledgable enough to write one myself. So feel free to pick a syntax you're comfortable with, when you answer my question (your specific choice shouldn't change the final result by much).