  And in the case of Skolem's Paradox, they actually do come apart. Since M is countable, the set  mˆ = {m | M  ⊨ m ∈  mˆ }  must also be countable. So, there really does exist a bijection (indeed, 2ℵ0 bijections), f : ω →  mˆ. On the ordinary English interpretation of  Ω(mˆ), the quantifiers “see” these bijections, and so  Ω(mˆ)  comes out false. What Skolem's Paradox shows is that M itself does not contain any such bijections. Hence, the quantifiers in the model-theoretic interpretation of  Ω(mˆ)  don't see any bijections between ω and  mˆ,  and so  Ω(mˆ) comes out true. In this case, then, differences in the ways the model-theoretic and ordinary English interpretations of Ω(x) handle the quantifiers provide a perfectly natural explanation of what's going on in Skolem's Paradox.