Define the sets $V_{\alpha}, \alpha \in O N$. Show that each $V_{\alpha}$ is transitive, and explain why $V_{\alpha} \subseteq V_{\beta}$ whenever $\alpha \leqslant \beta$. Prove that every set $x$ is a member of some $V_{\alpha}$.

Which of the following are true and which are false? Give proofs or counterexamples as appropriate. You may assume standard properties of rank.

(a) If the rank of a set $x$ is a (non-zero) limit then $x$ is infinite.

(b) If the rank of a set $x$ is a successor then $x$ is finite.

(c) If the rank of a set $x$ is countable then $x$ is countable.