User blog comment:Deedlit11/Ordinal Notations VI: Up to a weakly compact cardinal/@comment-1605058-20130718061544

I see some repeating scheme here - to extend previous notation you add larger and larger cardinals just to use them as diagonalization operator. Even though it looks a bit ad hoc, it seems to turn out perfect to define larger and larger countable ordinals. Is there an "absolute" limit for this method of diagonalization?