This is the last homework assignment of the term: Assume Evaluate the cardinal number
the size of the set of all functions
502 – Exponentiation
December 9, 2009502 – The constructible universe
December 9, 2009In 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.
1. Definability
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.