User blog comment:Simplicityaboveall/Extremely Large Numbers 2/@comment-1605058-20160801201054/@comment-5529393-20160803045518

It is possible in principle, however the Calculus of Constructions is so strong that it seems very difficult to describe an ordinal notation that is strong enough. The strongest ordinal notations in the literature only go up to a small fragment of second order arithmetic, whereas the Calculus of Constructions can express functions that are not provably recursive in nth order arithmetic for any n.