In 1938 Gödel published “The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis”. This paper introduces the constructible universe, a so-called inner model of ZFC. This is a class L that satisfies the ZFC axioms, plus GCH, provided that V satisfies the ZF axioms. So if ZF is consistent, then so is ZF+AC+GCH.
The 1938 paper did not employ classes in the formal sense, as the context was ZF set theory. A couple of years later Gödel elaborated his proof in a monograph, introducing the NBG axioms and treating classes formally. We’ll stick with the ZF version.
Cohen’s book gives a more readable account of all this. I will quote from it from time to time.
The proof rests on five foundation stones:
- First-order logic:
- Concepts such as sentence, satisfaction, model, and definability all play key roles.
- Constructible Sets:
- Gödel’s universe of constructible sets is the centerpiece of the whole argument; it revolves around the notion of definability. The next post introduces the constructible universe.
- Absoluteness:
- Certain concepts of set theory are absolute, in that they do not depend on the surrounding model. Absoluteness is covered in a later post.
- Reflection Principles:
- At two pivotal points, Gödel made use of a version of the Löwenheim-Skolem Theorem. The theorem says that we can always find a “small” submodel that “reflects” certain aspects of a larger model.
- Mostowski Collapsing Lemma:
- At a key point in the argument, Gödel needed to remove “superfluous” sets from a model; he applied the Mostowski collapsing lemma. (Cohen calls it “the trivial result… concerning ∈-isomorphisms”.)
The following sections give the argument in broad strokes. For the logic background, I’ll refer to my notes Basics of First-order Logic (henceforth Logic Notes).
Diagonal Argument 