Cumulative Hierarchy

• Built via transfinite recursion: $V_0=\varnothing$ $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$ $V_{\lambda }={\bigcup }_{\beta <\lambda }{V}_{\beta }$

• $V={\bigcup }_{\alpha \in \text{ON}}{V}_{\alpha }$

• Each stage is transitive.