User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813/@comment-39541634-20191030065255

Wait... are you using "N" here as "the set of natural numbers" or as a set of ordinals? It seems like you're doing both, and I can't always tell which is which. In my question, "N" was the set of natural numbers.

At any rate, my question has nothing to do with finite promse games. I've asked it because I want to know the general connection between fPTO(T)(n) and X(n) (when X is defined similarly to the X in my question, but for theory T). I know the commonly given answer (that X ~ E with E growing a teeny bit faster) but I've gathered from your previous posts here that this common answer is not true.

Hence my question.

Now, I know that without specification of fundamental sequences for PTO(T), there can be no meaningful answer (you can just set f(n)=BB(n) and then E automatically "wins"). I suppose it might also be possible to dream up crazy syntaxes for the specific purpose of inflating the values of X.

But I'm not intersted in such pathological cases. I'm interested at what typically happens when you use an accepted standard formulation of a theory T, and an accepted standard set of fundamental sequences for E.

And T doesn't have to be ACA0. If you can answer an analogous question to a theory you are more familiar with, that would be just as good. The only limitation is that PTO(T) must be small enough so talking about "standard FS system of PTO(T)" actually makes sense (so T=Z2, for example, won't be a good choice because there's no such thing as "a standard FS system of PTO(Z2)").