Axiom of constructibility

The axiom of constructibility is a possible axiom for set theory in mathematics that asserts that every set is constructible. The axiom is usually written as V = L, where V represents the von Neumann universe of all well-founded sets, and L represents the constructible sets. In Zermelo–Fraenkel set theory (ZF), the property of being constructible is expressible as a single formula ${\displaystyle \mathrm {Constructible} (x)}$, and every set is in V, so the axiom can be written in the language of ZF in the form ${\displaystyle \forall x\mathrm {Constructible} (x)}$.

The axiom of constructibility, first investigated by Kurt Gödel, is inconsistent with the proposition that zero sharp exists and stronger large cardinal axioms (see list of large cardinal properties). Generalizations of this axiom are explored in inner model theory.