Assume that $2^{\aleph_0}=\aleph_1$ and $(\aleph_{\omega_1})^{\aleph_0}=\aleph_{\omega_1+5}$.
Prove that there exist $\alpha<\omega_1$ such that:
$\alpha$ is a limit ordinal
$(\aleph_\alpha)^{\aleph_0}\geq\aleph_{\omega_1}$
for all $\gamma<\alpha:(\aleph_\gamma)^{\aleph_0}<\aleph_{\omega_1}$
I suspect some smart usage of Tarski, Bukovsky or Hausdorff is needed here but I do not see how to use their formulas.