Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 16H / c / Solution

The collection of hereditarily finite sets iswhere $V_0=\varnothing$ and $V_{n+1}=\mathcal{P}(V_n)$. Every $V_n$ is finite, and the axiom of infinity together with replacement and union makes $V_{\omega }$ a set. Hence this collection is not a proper class.

It is transitive: if $x\in V_n$, then $x\subset V_{n-1}$, so every $y\in x$ belongs to $V_{n-1}\subset V_{\omega }$. It is also $\phi$-closed. If $x\in V_n$ andthen $s\subset x\subset V_{n-1}$, so $s\in V_n\subset V_{\omega }$.