Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2007 / B / 5 / i

Here $Z{F}^{\ast }$ denotes ZF without foundation. Transfinite recursion on ordinals defines the cumulative hierarchy byThe associated class is $V=\{x:\exists \alpha \in \mathrm{Ord}(x\in V_{\alpha })\}$. Each level is a set, while $V$ is a definable class. Without Foundation it need not contain every ambient set; it consists of the sets built by this well-founded hierarchy.