  There are two ways to approach the first question. On the one hand, we could start with the formula Ω(x), and give this formula what we might call its “ordinary English” interpretation. This is the interpretation which lets “∈” refer to the real set-theoretic membership relation, which lets “∀” and “∃” range over the whole (real) set-theoretic universe, and which interprets “=” and the propositional connectives in the usual  manner.[14] Then for any set m, Ω(m) will come out true if and only if m is  uncountable.[15]  This shows that there is at least one interpretation of Ω(x) under which this formula really does capture—at least from an extensional perspective—the ordinary mathematical notion of uncountability. So, there's at least one interpretation under which  Ω(mˆ)  really does say that mˆ is uncountable.