User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-25418284-20130718062620/@comment-5529393-20130718165106

Actually, I think that the "higher order logic" that Loader refers to is just higher order arithmetic. I believe I have heard the calculus of constructions associated with higher order arithmetic, and I have heard system F associated with second order arithmetic, so I think that they are using nth order logic and nth order arithmetic interchangeably.

But yes, we are nowhere near Loader's ordinal. There is doubt that Taranovsky's notation actually reaches second-order arithmetic.