User blog comment:Deedlit11/Ordinal Notations V: Up to a weakly Mahlo cardinal/@comment-11227630-20140818113703/@comment-11227630-20140818140626

Even if a is limit, we need an a-weakly inaccessible is regular. So "the first ω-weakly inaccessible cardinal" is not the limit of $$I,I(1,0),I(2,0),I(3,0),\cdots$$ simply. (Well, this is another thing)