There is a similar question in this site but I am not satisfied with the answer, which is basically the same as the proof in the mentioned textbook.
The book(Karel Hrbacek&Thomas Jech, Introduction to Set Theory 3e, p165) states a lemma: For every $alpha$, $text{cf}(2^{aleph_alpha})>aleph_alpha$. Then it asserts that $2^{aleph_0}$ cannot be $aleph_omega$, since $text{cf}(2^{aleph_omega})=aleph_0$. But I can’t see the connection. According to the lemma, $text{cf}(2^{aleph_omega})$ should be larger than $aleph_omega>aleph_0$, how can it equal $aleph_0$?
On the other hand, I can’t see why $text{cf}(2^{aleph_omega})=aleph_0$ is false either. Since $2^{aleph_omega}=limlimits_{nrightarrowomega}2^{aleph_n}$, it is the limit of an increasing sequence of ordinals of length $omega$, so its cofinality should not be greater than $aleph_0$. Is there something wrong within this reasoning?
