  This brings us to a final point. The above discussion explains why those of us who are willing take a naively realistic attitude towards the language of set theory—e.g., those of us who have no qualms about expressions like “the ordinary English interpretation of Ω(x)”—should remain untroubled by Skolem's Paradox. It's important to emphasize that this analysis also explains why Skolem's Paradox doesn't introduce contradictions into various forms of axiomatized set theory, even when these axiomatizations are themselves understood formalistically or model-theoretically. From a proof-theoretic standpoint, for example, there is a difference between unrelativized quantification and quantification which has been explicitly relativized to some formula in our language (where this formula is one that, from an intuitive perspective, serves to “pick out” the domain of countable model of ZFC). So, there's no a priori reason to think that a sentence with unrelativized quantifiers will conflict with that sentence's fully-relativised  counterpart.[30]  Similarly, from a model-theoretic perspective, there is a difference between quantifiers which range over the whole domain of a model and quantifiers which only range over the “elements” of some particular member of the model (where, once again, this member is one which the larger model “thinks” is a model of ZFC). So, although the naive realism of sections 2.1–2.4 is useful for expository purposes, it's not essential to the underlying analysis of Skolem's Paradox.