Define the sets $V_{\alpha}$ for ordinals $\alpha$. Show that each $V_{\alpha}$ is transitive. Show also that $V_{\alpha} \subseteq V_{\beta}$ whenever $\alpha \leqslant \beta$. Prove that every set $x$ is a member of some $V_{\alpha}$.

For which ordinals $\alpha$ does there exist a set $x$ such that the power-set of $x$ has rank $\alpha$ ? [You may assume standard properties of rank.]