Can PA prove very fast growing functions to be total?

Question

The Goodstein-sequence is a total function, but PA cannot prove this.

Is this true for any other function with growth rate at least $f_{\epsilon_0}$ or are there functions growing at least as fast as the Goodstein-sequence that PA can prove to be total ?

I heard that the "power" of the PA is below the $f_{\epsilon_0}$-level, but I do not know if this answers my question.

Answer 1

Every function which eventually outgrows $f_{\varepsilon_0}$ cannot be proven to be total in Peano arithmetic. This is implied by more general result:

PA can prove a computable function $F$ to be total if and only if $F$ is primitively recursive in $f_{\omega\uparrow\uparrow n}$ for some finite $n$, where $\omega\uparrow\uparrow n=\omega^{...^\omega}$ with $\omega$ $n$ times.

In particular, if $F$ outgrows all of $f_{\omega\uparrow\uparrow n}$ for finite $n$, then PA cannot prove $F$ total.

Reference