Constructible universe

In set theory, the constructible universe (or Gödel's constructible universe), denoted by

L

,

{\displaystyle L,}

is a particular class of sets that can be described entirely in terms of simpler sets.

L

{\displaystyle L}

is the union of the constructible hierarchy

L

α

{\displaystyle L_{\alpha }}

. It was introduced by Kurt Gödel in 1938, who 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.

Definition

L

{\displaystyle L}

can be thought of as being built in "stages" resembling the construction of the von Neumann universe,

V

{\displaystyle V}

. The stages are indexed by ordinals. In von Neumann's universe, at a successor stage, one takes

V

α

+

1

{\displaystyle V_{\alpha +1}}

to be the set of all subsets of the previous stage,

V

α

{\displaystyle V_{\alpha }}

. By contrast, in Gödel's constructible universe

L

{\displaystyle L}

, one uses only those subsets of the previous stage that are:

definable by a formula in the formal language of set theory,

with parameters from the previous stage, and

with the quantifiers interpreted to range over the previous stage.

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.