Finite set of rank $\alpha$

Question

Defining the rank function $\in$-recursively as $\text{rank}(x)=\bigcup\{\text{rank}(y)^{+}\mid y\in x\}$ for a set $x$, for which ordinals $\alpha$ does there exist a finite set of rank $\alpha$? And what happens if we require the set to be transitive?

Answer 1

In general, it works for any successor ordinal $\alpha^+$, since you can consider $x=\{\alpha\}$.

If you require $x$ to be transitive and finite, then it is clear by induction that it has finite rank, so only the finite ordinals $\alpha$ work.