Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2010 / B / 6 / d / i / Solution

The cumulative hierarchy is defined by transfinite recursion:This recursion on ordinals is available without assuming the Axiom of foundation for all sets. The stages are increasing and transitive, by transfinite induction using their definitions.

Work internally in the specified model of ZF without the Axiom of foundation, and assume the membership induction theorem. Apply it toSuppose every member $y$ of $x$ lies in some stage. Assign to each such $y$ the least ordinal ${\alpha }_y$ for which $y\in V_{{\alpha }_y}$. The Axiom schema of replacement gives a set of these ordinals, and let $\beta$ be their supremum, taking $\beta =0$ for the empty collection. Monotonicity gives $x\subseteq V_{\beta }$, and therefore $x\in V_{\beta +1}$.

This proves progressivity of $\phi$, so membership induction gives it for every set of the model. Every stage already consists of sets in that model. ConsequentlyThe equality is internal to the given model; its ordinals and power sets are computed there.