Enayat Models of Peano Arithmetic

TL;DR

This paper investigates Enayat models of Peano Arithmetic, characterizing their properties and demonstrating their existence for any countable linear order, thus advancing understanding of models with specific definability features.

Contribution

It provides a characterization of Enayat models of PA and proves their existence for all countable linear orders, expanding the class of known models with unique definability properties.

Findings

Enayat models are countable, have no proper cofinal submodels, and are conservative extensions of all their elementary cuts.

Such models exist for any countable linear order.

The paper establishes conditions under which Enayat models can be constructed.

Abstract

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