Define the von Neumann hierarchy of sets $V_{\alpha}$, and show that each $V_{\alpha}$ is a transitive set. Explain what is meant by saying that a binary relation on a set is well-founded and extensional. State Mostowski's Theorem.

Let $r$ be the binary relation on $\omega$ defined by: $\langle m, n\rangle \in r$ if and only if $2^{m}$ appears in the base-2 expansion of $n$ (i.e., the unique expression for $n$ as a sum of distinct powers of 2 ). Show that $r$ is well-founded and extensional. To which transitive set is $(\omega, r)$ isomorphic? Justify your answer.