User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-11227630-20180208013330/@comment-11227630-20180209111614

What about expressing axioms of KP in SOST and adding the second-order formula "there exists a correct cardinal"? Does it have a larger, equal, or smaller PTO than ZFC?

And what about KP + "for all positive integer n, there exists a $$\Sigma_n$$-correct cardinal" (expressed in SOST) vs KP + "there exists a correct cardinal"?