User blog comment:Alemagno12/Huge Ordinal Analysis/@comment-1605058-20180203102941/@comment-1605058-20180203144517

Axiom of choice has nothing to do with expressiveness of the language - both ZF and ZFC are in the same language FOST. You probably mean something like proof-theoretic strength - in this case, the axiom of choice doesn't make any difference to ZF and the small large cardinals (like inaccessibles, Mahlos, indescribables), but it might make a difference for large large cardinals (measurable and up).