This chapter provides a brief introduction to forcing in models of PA, and includes some applications. It is proved that in general the MacDowell-Specker Theorem does not hold for uncountable languages. However, in some situations it does producing models with many generic classes. Perfect forcing is used to show that countable languages can, in some sense, be replaced by finite languages.
Keywords: generics, forcing, MacDowell-Specker Theorem, Cohen forcing, perfect generics