  Second, Skolem's Paradox does not depend on the specific axiomatization of set theory that we happen to be working with. Any first-order axiomatization of set theory can have the Löwenheim-Skolem theorems applied to it, so every such axiomatization is subject to Skolem's Paradox. This means, in particular, that we can't resolve the paradox by simply choosing a new axiomatization of set theory (or adding some new axioms to the axiomatization that we're already using). The fact that Skolem's paradox is, in this way, intrinsic to the first-order context—that it is an inescapable fact about first-order axiomatizations of set theory—is another reason why Skolem's Paradox may seem so initially puzzling.