ENAYAT MODELS OF PEANO ARITHMETIC

摘要

Simpson [6] showed that every countable model M satisfies PA has an expansions (M, X) satisfies PA* 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 M satisfies PA such that for each undefinable class X of M, the expansion (M, X) 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 gamma, if there is a model M such that Lt(M) congruent to gamma, then there is an Enayat model M such that Lt(M) congruent to gamma.