User blog comment:Plain'N'Simple/A question for proof-theory experts/@comment-35470197-20191029224813/@comment-39541634-20191030210342

"If the growth rate of the proof length of the well-foundedness of ω↑↑n is significantly smaller than E, then we obtain a googological approximation between X and E."

That's what I thought.

So the googolplex dollar question is:

Given a typical theory T and a typical set of fundamental sequences for PTO(T), do we have any reason to believe that the proof length for the well-foundedness of PTO(T)[n] is - indeed - significantly smaller than E(n)? Or at least, not much larger?

(This weaker condition would still mean that X(n) is dominated by something like fPTO(T)+1(n) which would be good enough for me)