If we assume ZFC to be consistent we have, by the Löwenheim-Skolem theorem, the existence of a countable model $\mathcal{U}_0$ of ZFC.
In $\mathcal{U}_0$ there is a infinite ordinal, that is a non-empty limit ordinal. Call the smallest one $\omega$. We can also construct the cardinal $2^\omega := \mathrm{card}(\wp (\omega))$, since the existence of the power set is given by the axioms.
However, the latter is uncountable, but it is a subset of $\mathcal{U}_0$, which is countable; this seems to be a contradiction.
I suspect that this "contradiction" can be resolved by distinguishing between infinity between models of ZFC, but I don't know how to do that.
So my question is: How can I resolve this?
Thanks!
Best Answer
The contradiction is inside the definition of "countable". A set is countable if there exists a surjection from $\mathbb{N}$ to our set. The function that would make our inside-the-model set countable doesn't exist inside of the model, so inside of the model, the set is uncountable.