User blog comment:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals/@comment-25601061-20180729214112/@comment-25601061-20180802163700

1. H is not actually the set-theoretic huge cardinal, but rather a planned extension of UNOCF that was later dropped in favor of another extension, stellar cardinals, which might be added to the page once Username comes back (he said he was going to be on a hiatus about two or three weeks ago) and approves the extension.

2. Again, stage cardinals are not actually cardinals, but rather a placeholder for cardinals that can extend weakly compacts, so as long as we can prove that stage cardinals can express an ordinal that works like a weakly compact, we can work under any set of axioms (or even none at all) that can (1) prove the existence of a set of ordinals large enough to replace stage cardinals and the ordinals produced by them and keep the behaviour of the notation the same, and (2) prove that the resulting notation is well-founded. (Oh, and stage cardinals are already being used in UNOCF, extending UNOCF's collapse of weakly compact cardinals)