# 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.

# Enayat Models – ASL 2016

I will be contributing a session at the ASL 2016 North American Annual Meeting, on Wednesday, May 25 at 4:45 PM.

Abstract: Simpson used arithmetic forcing to show that every countable model $\mathcal{M} \models \textsf{PA}$ has an expansion $(\mathcal{M}, X) \models \textsf{PA}^*$ that is pointwise definable. The natural question then is whether this method can be used to obtain expansions of (countable models) with the property that the definable elements of the expansion coincide with the definable elements of the original model. Enayat later showed that this is impossible. He proved that there are models with the property that every expansion upon adding a predicate for an undefinable class is pointwise definable. We call models with this property Enayat models. It is easy to iterate Enayat’s construction and obtain other models with this property. Elementary submodels of any Enayat model formed in this way are well-ordered by inclusion. I will present a construction of an Enayat model whose elementary substructures form an infinite descending chain.