  Second, the model-theoretic interpretation understands “∃x” and “∀x” to range only over the domain of M, while the ordinary English interpretation understands these quantifiers to range over the whole set-theoretic universe. Clearly, these two understandings are quite different. Further, the differences in question are pretty closely related to the sorts of sets that are involved in Skolem's Paradox. Suppose, for instance, that M  ⊨  “mˆ is the set of real numbers.” Then a simple cardinality argument shows that there are 2ℵ0 real numbers which do not live in the domain of M (and so, in particular, which do not live in {m  |  M  ⊨  m ∈  mˆ }).  Hence, there's a real difference between the genuinely uncountable set ℜ and the merely countable set {m  |  M ⊨  m ∈  mˆ }—between,  that is, the real set of real numbers and the set of things which M merely thinks are real numbers. On the model-theoretic interpretation of Ω(x), the quantifiers only range over the latter, smaller set, while on the ordinary English interpretation, they range over the entire, larger set. Similarly, suppose that M  ⊨ “m is infinite.” Then we can show that there are exactly 2ℵ0 bijections f : ω → {m′ ∈ M  | M  ⊨  m′ ∈  m }.[20]  However, at most countably many of these bijections live in the domain of M. Hence, only countably many of them are “seen” by ∃x and ∀x under the model-theoretic interpretation of Ω(x), although all 2ℵ0 of them are “seen” under the ordinary English interpretation.