For theories with well known proof-theoretic-ordinals, (what) is there a correspondence between their proof-theoretic-ordinal and (ordinal indexed families of?) fast growing functions provable total in a given theory ?



Correspondence between proof-theoretic ordinals and fast growing functions?
Yes: for many theories, $\alpha$ is the proof-theoretic ordinal of T exactly when T proves that $f\_\beta$ is total for all $\beta<\alpha$, but does not prove $f\_\alpha$ is total. (Where $f\_\alpha$ is the $\alpha$-th function in the fast-growing hierarchy.)


[Avigad](http://www.andrew.cmu.edu/user/avigad/Papers/ordinals.pdf) has argued that the correct definition of proof-theoretic ordinals is in terms of provably total functions - specifically, that the proof theoretic ordinal of T is $\geq\alpha$ exactly when T proves the totality of all functions which are "$\prec\alpha$-recursive" functions, where $\prec\alpha$-recursive means that the function is given by a program together with a timer which uses ordinal notations $\prec\alpha$.


With some care about the encoding, there's a tight connection between proving fast-growing functions total and proving all $\prec\alpha$-recursive functions total, so in some sense one can take proving the totality of fast-growing functions to be the *definition* of the proof-theoretic ordinal.


As usual with proof-theoretic ordinals, all reasonable definitions are going to be equivalent for nice theories (which includes all strong enough theories which have appeared naturally elsewhere in logic), but there are artificial theories that make the various definitions no longer equivalent.


