The cumulative hierarchy

In ZF without Foundation define the cumulative hierarchy by

$V_0=\mathrm{\varnothing },V_{\alpha +1}=\mathcal{P}(V_{\alpha }),V_{\lambda }={\bigcup }_{\beta <\lambda }{V}_{\beta }(\lambda \text{ a nonzero limit ordinal})\mathrm{.}$

For each ordinal $\theta$, use the set well-order recursion schema on $\theta +1$. On histories of domain $0$ return $\mathrm{\varnothing }$; on domain $\beta +1$ return the power set of the last value; on nonzero limit domains return the union of the range. Each is a unique set. Recursions on different ordinal intervals agree on overlaps by the uniqueness clause applied to the smaller interval. Hence the definition of $V_{\alpha }$ as the value at $\alpha$ is uniform and independent of the chosen interval. Power Set is used at successors and Replacement and Union at limits. The notation $⟨V_{\alpha }:\alpha \in \mathrm{Ord}⟩$ denotes a definable class function, not a set sequence.

Conventions and prerequisites: Transfinite recursion, Basic closure properties of ordinals, Successor and limit ordinals.