If it's the idea of completed infinity that's the objection, then it's the first step, constructing the powerset, that would be problematic. Various forms of finitism would not accept that one can 'take all the subsets, finite or infinite, and quantify over them' and obtain a meaningful result past the formal level.
They should still then accept that there is no surjection from a set to a set that is a set of all subsets of that set (because of there being no such set).
