The hierarchy of constructible sets that together constitute the constructible universe $\text{L}$, used by Kurt Gödel to show that if Zermelo-Frankel set theory ($\text{ZF}$) is consistent, then $\text{L}$ models both the axiom of choice and the generalized continuum hypothesis, entailing that $\text{ZFC}$ ($\text{ZF}$ with the axiom of choice) and $\text{ZF}+\text{GCH}$ must be consistent as well. The recursive definition of $\text{L}$ mirrors that of the iterative hierarchy $\text{V}$: where each stage in the definition of $\text{V}$ follows from taking all subsets of elements from preceding stages, the construction of $\text{L}$ proceeds by taking definable (or constructible) subsets of the preceding stages. The definable subset operation $𝒟$ corresponds to the sets of elements defined by first-order sentences in the language of set theory with constants for each element in preceding stages. Formally, the iterative hierarchy is defined as follows:

• • ${\text{L}}_0=\varnothing$

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

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

$\text{L}$ is then the union of all ${\text{L}}_{\alpha }$ for all ordinals $\alpha$, i.e., all constructible sets. Assuming that $\text{ZF}$ is consistent, $\text{L}$ is a model of $\text{ZF}$ and the axiom of choice, which demonstrates that consistency of $\text{ZF}$ entails the consistency of $\text{ZFC}$.