State the Axiom of Foundation and the Principle of $\in$-Induction, and show that they are equivalent (in the presence of the other axioms of ZF). [You may assume the existence of transitive closures.]

Explain briefly how the Principle of $\in$-Induction implies that every set is a member of some $V_{\alpha}$.

For each natural number $n$, find the cardinality of $V_{n}$. For which ordinals $\alpha$ is the cardinality of $V_{\alpha}$ equal to that of the reals?