Full Idea
Gödel proved the classical relative consistency of the axiom V = L (which implies the axiom of choice and the generalized continuum hypothesis). This established the full independence of the continuum hypothesis from the other axioms.
Gist of Idea
Gödel proved the classical relative consistency of the axiom V = L
Source
report of Kurt Gödel (What is Cantor's Continuum Problem? [1964]) by Hilary Putnam - Mathematics without Foundations
Book Reference
'Philosophy of Mathematics: readings (2nd)', ed/tr. Benacerraf/Putnam [CUP 1983], p.304
A Reaction
Gödel initially wanted to make V = L an axiom, but the changed his mind. Maddy has lots to say on the subject.