Good evening to all. I have two exercises I tried to resolve without a rigorous success:
Is it true or false that if $\kappa$ is a non-numerable cardinal number then $\omega^\kappa = \kappa$, where the exponentiation is the ordinal exponentiation?
I found out that the smallest solution for this Cantor's Equation is $\epsilon_0$ but it is numerable. I was thinking about $\omega_1$ but i don't know how to calculate something like $\omega^{\omega_1}$ and see if it is equal or not to $\omega_1$.
Exists an ordinal number $\alpha > \omega$ that verify $\alpha \times \alpha \subseteq V_\alpha$ , where $V_\alpha$ is the von Neumann hierarchy?
Thanks in advance