Constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted L, is a particular class of sets that can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this, he proved that the constructible universe is an inner model of ZF 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, 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.
L can be thought of as being built in "stages" resembling the von Neumann universe, V. The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes Vα+1 to be the set of all subsets of the previous stage, Vα. By contrast, in Gödel's constructible universe L, one uses only those subsets of the previous stage that are:
By limiting oneself to sets defined only in terms of what has already been constructed, one ensures that the resulting sets will be constructed in a way that is independent of the peculiarities of the surrounding model of set theory and contained in any such model.
Define
L is defined by transfinite recursion as follows:
If z is an element of Lα, then z = {y | y ∈ Lα and y ∈ z} ∈ Def (Lα) = Lα+1. So Lα is a subset of Lα+1, which is a subset of the power set of Lα. Consequently, this is a tower of nested transitive sets. But L itself is a proper class.
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