We study subsets of countable recursively saturated models of PA which can be defined using pathologies in satisfaction classes. More precisely, we characterize those subsets X such that there is a satisfaction class S where S behaves correctly on an idempotent disjunction of length c if and only if c∈X. We generalize this result to characterize several types of pathologies including double negations, blocks of extraneous quantifiers, and binary disjunctions and conjunctions. We find a surprising relationship between the cuts which can be defined in this way and arithmetic saturation: namely, a countable nonstandard model is arithmetically saturated if and only if every cut can be the “idempotent disjunctively correct cut” in some satisfaction class. We describe the relationship between types of pathologies and the closure properties of the cuts defined by these pathologies.

Abstract Simpson [6] showed that every countable model ${\cal M} \models PA$ has an expansion $\left( {{\cal M},X} \right) \models P{A^{\rm{*}}}$ that is pointwise definable. A natural question is whether, in general, one can obtain expansions of a nonprime model in which the definable elements coincide with those of the underlying model. Enayat [1] showed that this is impossible by proving that there is ${\cal M} \models PA$ such that for each undefinable class X of ${\cal M}$ , the expansion $\left( {{\cal M},X} \right)$ is pointwise definable. We call models with this property Enayat models. In this article, we study Enayat models and show that a model of $PA$ 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 γ, if there is a model ${\cal M}$ such that $Lt\left( {\cal M} \right) \cong \gamma$ , then there is an Enayat model ${\cal M}$ such that $Lt\left( {\cal M} \right) \cong \gamma$ .