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 $mathrm{Lt}(mathcal{M}) cong gamma$.
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