User blog comment:P進大好きbot/Full References of Arguments on Ordinal Notations with Large Cardinals/@comment-25601061-20180729214112/@comment-35470197-20180803001045

The consistency of large cardinal axioms is not provable, even if we assume the consistency of the usual mathematics, i.e. \(\textrm{ZFC}\) set theory. So you are using stronger axioms, which may imply the contradition.

Also, the existence of sufficiently large ordinals which makes stage cardinals work as you desire may contain contradiction.

For example, if one says that "My large number is greater than anything, and the existence is provable under the contradiction \(0 = 1\)", then you accept the definition, right?