Cohen’s Method

Abstract

In proving that the AC and the GCH are consistent with ZF Gödel used the so called method of internal models. From the assumption that the universe V is a model of ZF Gödel prescribed a method for producing a submodel L that is also a model of V = L, AC and GCH. This submodel is defined as the class of all sets having a certain property i.e.

$${{L}^{\mathcal{M}}} = \left\{ {a\left| {\left( {\exists \alpha \in \mathcal{M}} \right)\left[ {a = F{}^\backprime \alpha } \right]} \right.} \right\}$$

.