Is there an uncountable cardinal number of cofinality $\aleph_0$?

Question

The first uncountable cardinal number $\aleph_1$ is a regular cardinal which means that its cofinality is $\aleph_1$. Is it provable in ZFC that there are uncountable cardinal numbers (bigger than $\aleph_1$) of cofinality $\aleph_0$? Or is it consistent with ZFC at least?

Answer 1

Yes, it is provable in ZF, indeed much less: consider $\aleph_\omega$. (Using Replacement, ZF proves that for each ordinal $\alpha$ there is an $\alpha$th cardinal, $\aleph_\alpha$; if $\alpha$ has countable cofinality, then so does $\aleph_\alpha$.)