User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35392788-20191029194318

We know from the Kirby-Paris theorem that an hydra game restricted to all ordinals \(<\varepsilon_0\) (modelled in that case by Goodstein's function) cannot be proven to terminate in both PA and ACA0 (ACA0 is conservative over PA) because such a proof requires transfinite induction along \(\varepsilon_0\), known to prove PA's (and ACA0's) consistency.

Now, it should be quite straightfoward to equate the computation of \(f_\alpha\) for \(\alpha<\epsilon_0\) (using the Wainer hierarchy) to an hydra game restricted to all ordinals \(<\alpha\). But then the proof of such a game terminating only requires transfinite induction along \(\alpha\), which is provable in PA and ACA0.

Therefore, the fact that X(n) cannot be proven to terminate in ACA0 would imply that it outgrows all \(f_{\alpha}\) for \(\alpha<\varepsilon_0\). QED.

This proof is not completely rigorous I think ; it might be hastily making bridges between hydra games, FGH and transfinite induction. However, I still believe the proof strategy to be mostly correct. It also doesn't answer your question completely, since it doesn't give a very precise relation between X(n) and E(n). I think it could be proven that E(n) outgrows Goodstein's function G(n) because G(n) is an hydra game starting at some ordinal determined by n, and that ordinal sequence is outgrown by \(\varepsilon_0[n]\) using an appropriate fundamental sequence. However, I don't think such a strategy can be applied to X(n) because I don't think it can be nicely mapped to an hydra game.

To answer your second question in a bit more detail, I conjecture that a function F not provably terminating in some theory T outgrows \(f_\alpha\) for all \(\alpha<\text{PTO}(T)\), where PTO(T) is of course its proof-theoretic ordinal. It might even be provable using the same technique I just used, but I'm less certain about that.