  To answer this second question, it's useful to begin by comparing the ordinary English interpretation of Ω(x)—the one introduced three paragraphs ago and the one which really does entail that x is uncountable—with the model-theoretic interpretation of Ω(x) that's given by M and  ⊨.  Clearly, it's this latter, model-theoretic interpretation which is most relevant to understanding the fact that M  ⊨  Ω[mˆ].  Further, it's only if this model-theoretic interpretation is pretty closely tied to the ordinary English interpretation—and so, derivatively, to the ordinary English expression “x is uncountable”—that we will have any real grounds for believing that the fact that M  ⊨ Ω[mˆ]  should entail that  mˆ is uncountable.