  Before turning to examine some of the more-purely philosophical issues concerning Skolem's Paradox, a few further points about the paradox's mathematics are in order. First, in order to get a better feel for how the differences between the model-theoretic and the ordinary English interpretations of Ω(x) actually give rise to Skolem's Paradox, it's worth tracking these differences through a slightly more refined version of the paradox. We say that a set X is transitive if every member of X is a set, and every member of a member of X is also a member of X (so, y ∈ x ∈ X  ⇒ y ∈ X). We say that a model for the language of set theory is transitive if the model's domain is a transitive set and the model's “membership” relation is just the real membership relation restricted to the model's domain (so, for any m1, m2 ∈ M,   m1 ∈  m2 ⇔ M  ⊨ m1 ∈ m2). Then, as noted in section  1,  the Transitive Submodel Theorem says that if we start with any transitive model of ZFC, then we can find a transitive model whose domain is countable (indeed, we may assume that this countable model is a submodel of the model with which we started).