Define the von Neumann hierarchy of sets $V_{\alpha}$. Show that each $V_{\alpha}$ is transitive, and explain why $V_{\alpha} \subset V_{\beta}$ 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.]

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

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

(iii) If every finite subset of a set $x$ has rank at most $\alpha$ then $x$ has rank at most $\alpha$.

(iv) For every ordinal $\alpha$ there exists a set of $\operatorname{rank} \alpha$.