Constructible universe

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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 an inner 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.

Gödel's universe can be thought of as being built in "stages" resembling von Neumann's universe. The stages are indexed by ordinals; unlike von Neumann's construction, where one takes at a successor stage n+1 the full power set of the previous stage n (i.e., the set of all subsets of the previous stage) in Gödel's construction one uses only the definable subsets of the previous stage. (Some care is necessary to precisely define what "definable" means in this context.)

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