This is a talk at CUNY’s MoPA seminar on 19 November 2014.

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.