When I’ve tried to read up on proof-theory I’ve come across this point multiple times - that given a well-founded fast-growing hierarchy, the index of the fastest-growing function f that T can prove total corresponds to the proof-theoretic ordinal of T
There’s a vague intuitive sense in which these seem related to me - something along the lines of “T can’t actually compute f properly because it can’t recurse all the way down to the “base” function because it can’t prove such a chain is well-founded”
I’m not sure if that’s along the right track? But even if it is I’d appreciate if anyone could give a more formal/fleshed out explanation of the link
Thanks!