Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 4 / i

The cumulative hierarchy is defined by transfinite recursion on the ordinals:The recursion theorem of ZF supplies each set stage. The whole universe is $V={\bigcup }_{\alpha \in \mathrm{Ord}}{V}_{\alpha }$, a proper-class union. Equivalently, $x\in V_{\alpha }$ exactly when the rank of $x$ is less than $\alpha$, where $\mathrm{rank}(x)=\sup \{\mathrm{rank}(z)+1:z\in x\}$.