I am reading Kunen's book on set theory and am puzzled by Exercise IV.10, which is:
Find a sentence $\phi$ such that for any $\beta$, if $\phi$ is absolute for $R(\beta)$, then $\beta=\omega_\beta$. Then, find a formula $\psi(x)$, such that for any non-0 transitive $M$, if $\psi(x)$ is absolute for $M$, then $M=R(\beta)$ for some $\beta$ such that $\beta=\omega_\beta$. Hint. $\phi$ will be enough of ZF to guarantee that $\forall\alpha(\omega_\alpha\hbox{ exists})$. $\psi(x)$ can be "$\phi \wedge (x\hbox{ is an }R(\alpha))$".
(Here, $R(\alpha)$ is the set of sets of rank $<\alpha$, which is also called $V_{\alpha}$.)
Finding $\phi$ is no problem, but I don't see how the provided hint for $\psi(x)$ can work. If we use it, then given absoluteness, we have $R(\beta)\in M \Rightarrow R(\beta)=R(\alpha)^M$ for some $\alpha\in M$, but this says nothing about the possibility $R(\beta)\notin M$.