Proving unprovability

Abstract

We have seen in Chap. 15 how we can extend models of ZFC to models in which for example CH fails—supposed we have suitable generic filters at hand. On the other hand, we have also seen that there is no way to prove that generic filters exist. However, in order to show that for example CH is independent of ZFC we have to show that ZFC +CH as well as ZFC + CH has a model. In other words we are not interested in the generic filters themselves, but rather in the sentences which are true in the corresponding generic models. On the other hand, if there are no generic filters, then there are also no generic models. The trick used to avoid generic filters (over models of ZFC) is to carry out the whole forcing construction within a given model V of ZFC. How this can be done will be shown in this chapter.