How to understand $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$ As a corollary of  König's theorem, we have $\operatorname{cf}(2^{\aleph_0}) > \aleph_0$
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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 $\sup ({2^n:n<\omega}) = 2^{\aleph_0}$? 

Note that in ordinal arithmetic $2^\omega$ is the supremum of $2^n$ for all $n<\omega$, so it is $\omega$ and thus has cofinality $\aleph_0$.
$\omega$ and $\aleph_0$ are the same set but they nevertheless behave differently in practice -- because tradition is to use the notation $\omega$ for operations where a limiting process is involved and $\aleph_0$ for operations where the "countably infinite" is used all at once in a single step. Cardinal exponentatiation $2^{\aleph_0}$ is such an all-at-once process.
