User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-5029411-20130811125215/@comment-5150073-20140623162000

Well, in that case, cof(alpha) function becomes partial. It becomes computable not for all ordinals, but only for ones which are needed is $$\psi$$. That's not problematic, though.