Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2009 / B / 4 / ii / c / Solution

Induct on $\alpha$. For every $\beta <\alpha$, the induction hypothesis says $\beta \in V_{\beta +1}$, and monotonicity gives $V_{\beta +1}\subseteq V_{\alpha }$. Thus every element of the ordinal $\alpha$ lies in $V_{\alpha }$, so $\alpha \subseteq V_{\alpha }$. Taking its power set givesFor $\alpha =0$ the inclusion is vacuous and $\varnothing \in \mathcal{P}(\varnothing )$ supplies the base case. This argument uses ordinal membership as strict order and the previously proved hierarchy monotonicity.