Is there a 'very fast growing' hierarchy that would capture System F? Particular ordinals in slow-growing and fast growing hierarchies can capture the expressiveness of many predicative type systems.

Is there a hierarchy of function that could possibly capture impredicative System F?
Typically, fast growing hierarchies are characterized by ordinal notations, which are really just ways to express fast-growing functions (but it's sometimes convenient to see them as ordinals in the mathematical sense).

There is a somewhat generic way of assigning an ordinal (notation) to a consistent theory, though it is very non-constructive. For various technical reasons, the ordinal for $\mathrm{PA}_2$ corresponds to the fast growing functions expressible in system $F$.

So the question really is:


  Is there an ordinal notation for $\mathrm{PA}_2$?


As far as I am aware, we do not know the answer to this question, and it is a very important open question in proof theory/ordinal analysis.

The following mathoverflow question explains this idea in more detail:

https://mathoverflow.net/questions/144041/proof-theoretic-ordinal-of-zfc-or-consistent-zfc-extensions