The hierarchy of sets constructed iteratively by repeatedly taking the power set of earlier stages. More formally, where $\wp$ is the power set operator, the hierarchy is defined recursively:

• • ${\text{V}}_0=\varnothing$

• • ${\text{V}}_{\alpha +1}=\wp \left({\text{V}}_{\alpha }\right)$ for ordinals $\alpha$

• • ${\text{V}}_{\beta }=\bigcup _{\alpha <\beta }\wp \left({\text{V}}_{\alpha }\right)$ for limit ordinals $\beta$

The union of all stages of the iterative hierarchy forms the von Neumann universe $\text{V}$. This method of constructing a universe is the most generous, distinguishing it from the constructible hierarchy and the constructible universe $\text{L}$, which is formed by iteratively taking subsets that are definable, i.e., correspond to first-order properties.