Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 4 / 16F / a / Solution

The Von Neumann hierarchy is defined by transfinite recursion:for a limit ordinal $\lambda$.

We prove by transfinite induction that every $V_{\alpha }$ is a transitive set. The claim is immediate for $V_0$. If $V_{\alpha }$ is transitive and $x\in y\in V_{\alpha +1}$, then $y\subseteq V_{\alpha }$, so $x\in V_{\alpha }$. Transitivity also gives $x\subseteq V_{\alpha }$, hence $x\in \mathcal{P}(V_{\alpha })=V_{\alpha +1}$. At a limit stage, if $x\in y\in V_{\lambda }$, then $y\in V_{\beta }$ for some $\beta <\lambda$, so $x\in V_{\beta }\subseteq V_{\lambda }$.

Transitivity implies $V_{\alpha }\subseteq \mathcal{P}(V_{\alpha })=V_{\alpha +1}$. Induction and unions at limit stages then give

By well-founded recursion on membership, define the rank of a setInduction on this rank gives $y\in V_{\mathrm{rank}(x)}$ for every $y\in x$. Thusand consequentlyEvery set therefore occurs at some level of the hierarchy.