User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5150073-20130716084312/@comment-5529393-20130717212604

I'm not sure what you're asking here;  in both cases, we define the ordinal from the funciton f as the smallest ordinal alpha such that F_a dominates f, where F is the fast-growing hierarchy. Loader's ordinal will be greater than the proof-theoretic ordinal for nth order logic for all n.