Enayat Models – ASL 2016 | Athar Abdul-Quader

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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.