Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 16I / a / Solution

The rank is defined recursively byFor a nonzero limit ordinal $\alpha$, every function $f:\alpha \to \alpha$, viewed as a set of ordered pairs, has rank $\alpha$, and the set of all such functions consequently has rank $\alpha +1$.

The von Neumann hierarchy isfor limit $\lambda$. Foundation permits induction down the membership relation and shows that the ranks of all elements of a set form a set of ordinals. If $\rho =\mathrm{rank}(x)$, then every element of $x$ lies in $V_{\rho }$, so $x\subseteq V_{\rho }$ and hence $x\in V_{\rho +1}$. Thus every set occurs in the hierarchy.