Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$?

Question

Is there a cardinal $\kappa$ with uncountable cofinality such that $\kappa<\kappa^{\aleph_0}$? The question is motivated by the observation that $\kappa< \kappa^{{\rm cf}\kappa}$ for any $\kappa$.

Answer 1

Assuming $\sf GCH$ then the answer is no, almost trivially. Since in that case $\kappa^{<\operatorname{cf}(\kappa)}=\kappa$ for every cardinal.

If $\sf CH$ is false, then $\aleph_1$ is an example; it is possible that $\sf GCH$ is false, but only $2^{\aleph_1}=\aleph_3$ is an example, in which case, despite the failure of $\sf GCH$, the statement is true.

And it is possible that the counterexamples are of this flavor, and they occur on a much later stage (e.g. $\aleph_{\omega_1+\omega}^{\aleph_0}=\aleph_{\omega_1+\omega+3}$ in which case $\aleph_{\omega_1+\omega+2}$ is an example as well).

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So from $\sf ZFC$ this is neither provable nor disprovable, and will depend a lot on additional assumptions.