Past exam of the analytic topology and axiomatic set theory course of the University of Oxford / 2007 / B / 5 / ii

Use transfinite induction. The inclusion at zero is immediate. If $V_{\beta }\subseteq V_{\beta +1}$, monotonicity of the power set givesIf $\alpha$ is a limit and $x\in V_{\alpha }$, choose $\beta <\alpha$ with $x\in V_{\beta }$. The inductive inclusion $V_{\beta }\subseteq \mathcal{P}(V_{\beta })$ means that $x\subseteq V_{\beta }\subseteq V_{\alpha }$. Thus $x\in \mathcal{P}(V_{\alpha })=V_{\alpha +1}$. This proves $V_{\alpha }\subseteq V_{\alpha +1}$ for every ordinal. These successive inclusions and the union definition also give $V_{\beta }\subseteq V_{\alpha }$ whenever $\beta \le \alpha$.

The preceding inclusion is exactly $V_{\alpha }\subseteq \mathcal{P}(V_{\alpha })$. Therefore every $x\in V_{\alpha }$ is a subset of $V_{\alpha }$. Equivalently,which says that $V_{\alpha }$ is a transitive set. This uses the power set definition and the proved inclusion, without any ambient use of Foundation.

We prove the stronger ordinal identityby transfinite induction. It holds at zero. At a successor stage an ordinal $\gamma$ belongs to $\mathcal{P}(V_{\beta })$ exactly when every $\delta <\gamma$ belongs to $V_{\beta }$. By induction this means every $\delta <\gamma$ is less than $\beta$, equivalently $\gamma \le \beta$. Thus the ordinals in $V_{\beta +1}$ are precisely those less than $\beta +1$. At a limit the identity follows by taking unions.

Every $\beta <\alpha$ consequently belongs to $V_{\alpha }$, so $\alpha \subseteq V_{\alpha }$ and $\alpha \in \mathcal{P}(V_{\alpha })=V_{\alpha +1}$. But $\alpha \notin V_{\alpha }$, since an ordinal is not less than itself. HenceThe only well-foundedness used is that of the ordinal order, available independently of Foundation for arbitrary ambient sets.

Every set is a subset of itself, so $V_{\alpha }\subseteq V_{\alpha }$. By the successor definition,Membership and inclusion are different statements here; this argument establishes membership of the whole level in its successor.