Past exam of the axiomatic set theory course of the University of Oxford / 2011 / 1 / a

The cumulative hierarchy is the transfinite recursionHere $\mathcal{P}$ is the power set. The class $V={\bigcup }_{\alpha \in \mathrm{Ord}}{V}_{\alpha }$ consists of all sets in ZF, by the Axiom of foundation and Axiom schema of replacement.

Two facts about a transitive set $A$ drive the transfinite induction. First, $A\subseteq \mathcal{P}(A)$: every $x\in A$ is a subset of $A$. Second, $\mathcal{P}(A)$ is a transitive set: if $x\in y\in \mathcal{P}(A)$, then $x\in A$, hence $x\subseteq A$, hence $x\in \mathcal{P}(A)$.

Starting with $V_0=\varnothing$, these facts prove the successor step. At a limit ordinal $\lambda$, the union of the earlier increasing transitive sets is transitive: if $x\in y\in V_{\lambda }$, some $\alpha <\lambda$ contains $y$ in $V_{\alpha }$, and transitivity puts $x$ there too. The union contains every earlier stage. Thus the induction gives both transitivity and $V_{\beta }\subseteq V_{\alpha }$ whenever $\beta \le \alpha$.

Finally $V_{\alpha }\subseteq V_{\alpha }$, so $V_{\alpha }\in \mathcal{P}(V_{\alpha })=V_{\alpha +1}$. The Axiom of foundation gives $V_{\alpha }\notin V_{\alpha }$. This witnesses strictness:This is the transitivity and strict growth of the cumulative hierarchy.