Enayat Models of Peano Arithmetic

Simpson showed that every countable model $\mathcal{M} \models \mathsf{PA}$ has an expansion $(\mathcal{M}, X) \models \mathsf{PA}^*$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a non-prime model in which the definable elements coincide with those of the underlying model. Enayat showed that this is impossible by proving that there is $\mathcal{M} \models \mathsf{PA}$ such that for each undefinable class $X$ of $\mathcal{M}$, the expansion $(\mathcal{M}, X)$ is pointwise definable. We call models with this property Enayat models. In this paper, we study Enayat models and show that a model of $\mathsf{PA}$ is Enayat if it is countable, has no proper cofinal submodels and is a conservative extension of all of its elementary cuts. We then show that, for any countable linear order $\gamma$, if there is a model $\mathcal{M}$ such that $\mathrm{Lt}(\mathcal{M}) \cong \gamma$, then there is an Enayat model $\mathcal{M}$ such that $\mathrm{Lt}(\mathcal{M}) \cong \gamma$.