  These, then, are two ways of thinking about the relationship between Ω(x) and “x is uncountable.” Together they explain why it's so natural for mathematicians to use “x is uncountable” as an abbreviation for Ω(x) and (so) why someone might be inclined to think that the fact that M  ⊨ Ω[mˆ]  should entail that mˆ is  uncountable. They also bring us back to our second question: why doesn't the fact that M  ⊨ Ω[mˆ]  really entail that  mˆ is uncountable.