Gödel’s constructible universe is a subclass$L$ of the von Neumann universe $V$ of well-founded pure sets, defined by transfinite induction as $L = \bigcup_\alpha L_\alpha$ where

The elements of the constructible universe are called constructible sets; the idea is similar to the constructible sets in topology and algebraic geometry.

Properties

$L$ is a transitivebig class containing all the ordinals. In fact, it is the smallest transitive model of the set theory containing all the ordinals. On the other hand, the sets in this class can be effectively enumerated by von Neumann ordinals. The question weather $L\neq V$ can not be decided in ZF. If $L\neq V$ then still we do not know how, without an axiom of choice, to produce specific sets which are not constructible.