  The interesting thing about these large cardinal hypotheses is that for n &gt; 0 they are inconsistent with AC (by a result of Kunen). Moreover, there is now reason to believe that for n &gt; 0 they are outright inconsistent (in ZF) (see Woodin (2011)). Should this turn out to be the case it would be rather interesting since it would be a case where one derived structure theory from an inconsistency without awareness of the inconsistency. 