User blog comment:Deedlit11/Ordinal Notations IV: Up to a weakly inaccessible cardinal/@comment-5150073-20130811164029/@comment-1605058-20130811171452

But Omega is larger than CK ordinal, so is larger than any recursive extension anyway. We can easily work with ordinal collapsing using admissible and recursively inaccessible ordinals instead of cardinals. Choice of cardinals is fairly arbitrary.