ABSTRACT
This chapter is devoted to prove Cohen's theorems: the continuum hypothesis CH is independent from the ZFC-axioms, and the axiom of choice AC is independent from the ZF-axioms. For both, Weaver's ZFC+-theory is used. In this theory, the universe V contains a denumerable transitive set M which is a ZFC-universe too, and ZFC+ is consistent if ZFC is consistent. With that assumption, a generic extension M[G] is constructed where CH does not hold, thus proving that if the ZFC-theory is consistent, then there is no proof of CH.
In order to deal with the second theorem, a construction parallel to that of generic extensions is needed. Thus, given the same data of that construction, plus a group of automorphisms of the complete Boolean algebra A and a filter of subgroups, another intermediate extension M(G,F) is constructed, and it is shown to be a ZF-universe. Then a particular such extension is chosen having the property that AC does not hold in M(G,F). Again, this shows that AC cannot be proved from the ZF-axioms if these are consistent.
