V=L
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 and L denote the von Neumann universe and the constructible universe, respectively. The axiom, 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.
The axiom of constructibility implies the axiom of choice over Zermelo–Fraenkel set theory. It also settles many natural mathematical questions independent of Zermelo–Fraenkel set theory with the axiom of choice (ZFC); for example, the axiom of constructibility implies the generalized continuum hypothesis, the negation of Suslin's hypothesis, and the existence of an analytical (in fact, ) non-measurable set of real numbers, all of which are independent of ZFC.
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