Classifying Enayat Models of Peano Arithmetic – ASL Winter Meeting (JMM)

Update (1/15): See slides here.

I will be contributing a session at the 2018 ASL Winter Meeting at the Joint Mathematics Meetings on Saturday, January 13. This talk is based on a recent paper of mine which can be found on the arxiv here.

Abstract: Simpson used arithmetic forcing to show that every countable model $\mathcal{M} \models \mathrm{PA}$ has an undefinable, inductive subset $X \subseteq M$ such that the expansion $(\mathcal{M}, X)$ is pointwise definable. Enayat later showed that there are many models with the property that every expansion upon adding a predicate for an undefinable class is pointwise definable. We refer to models with this property as Enayat models. That is, a model $\mathcal{M} \models \mathrm{PA}$ is Enayat if for each undefinable class $X \subseteq M$, the expansion $(\mathcal{M}, X)$ is pointwise definable. In this talk we show that a model is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of each of its elementary cuts.