Constructible universe

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by ${\displaystyle L}$, is a particular class of sets that can be described entirely in terms of simpler sets. ${\displaystyle L}$ is the union of the constructible hierarchy ${\displaystyle L_{\alpha }}$. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis".^[1] In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.