In this set of notes I want to sketch Gödel’s proof that is consistent with the other axioms of set theory. Gödel’s argument goes well beyond this result; his identification of the class of constructible sets eventually led to the development of inner model theory, one of the main areas of active research within set theory nowadays.
A good additional reference for the material in these notes is Constructibility by Keith Devlin.
The idea behind the constructible universe is to only allow those sets that one must necessarily include. In effect, we are trying to find the smallest possible transitive class model of set theory.
is defined as
where for limit, and where
The first question that comes to mind is whether this definition even makes sense. In order to formalize this, we need to begin by coding a bit of logic inside set theory. The recursive constructions that we did at the beginning of the term now prove useful.