I will talk about forcing over models of arithmetic. Our primary application will be the following theorem, due to Simpson: if a model M of PA is countable, then M has a subset U such that that (M,U) is a pointwise definable model of PA*. Time permitting, we will see that the MacDowell–Specker theorem fails for uncountable languages: for M countable and nonstandard, there are U_alpha for alpha < omega_1 such that (M, U_alpha)_{alpha < omega_1} is a model of PA* and has no elementary end extensions.