How to understand $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$ As a corollary of  König's theorem, we have $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$
.
On the other hand, we have $\operatorname{cf}(\aleph_\omega) = \aleph_0$.
Why the logic in the latter equation can't apply to the former one?
To be precise, why we can't have $\sup ({2^n:n<\omega}) = 2^{\aleph_0}$? 

We have $2^n < \omega$ for all $n < \omega$, so the supremum of the $2^n$ is the supremum of a set of ordinals cofinal in $\omega$, and hence is just $\omega$.
