Define the sets $V_{\alpha}, \alpha \in O N$. What is meant by the rank of a set?

Explain briefly why, for every $\alpha$, there exists a set of rank $\alpha$.

Let $x$ be a transitive set of rank $\alpha$. Show that $x$ has an element of rank $\beta$ for every $\beta<\alpha$.

For which $\alpha$ does there exist a finite set of rank $\alpha$ ? For which $\alpha$ does there exist a finite transitive set of rank $\alpha$ ? Justify your answers.

[Standard properties of rank may be assumed.]