Constructible universe

In mathematics, the constructible universe (or Gödel's constructible universe) is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1940 paper Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory. In this, he proved that the constructible universe is a model of set theory, 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. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

The constructible universe is defined by transfinite recursion as follows:

Define

(the empty set)

For α an ordinal, define

(where P(x) is the power set of x for any set x).

For β a limit ordinal, define

(that is, the union of all the V-sets so far).

Note that in the case of β = ω (ω being the set of natural numbers), V_ω is the set of hereditarily finite sets. The constructible universe itself is not a set, but a class. It is defined as the union of all the V-sets:

The constructible universe, L, is the class of sets x such that x is an element of V_α for some ordinal α.