von Neumann Universe (Cumulative Hierarchy)

Defined by transfinite recursion:

• $V_0=\varnothing$

• $V_{\alpha +1}=\mathcal{P}(V_{\alpha })$

• $V_{\lambda }={\bigcup }_{\beta <\lambda }{V}_{\beta }$

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

• Foundation for ZFC semantics.