ω1-like recursively saturated models of Presburger's arithmetic

Abstract

Let Pr be Presburger's arithmetic, i.e., the complete theory of the structure (ω, +). Lipshitz and Nadel showed in [4] that a countable model of Pr is recursively saturated iff it can be expanded to a model of Peano arithmetic, PA. As a starting point for an introductory discussion, let us mention one more fact about countable recursively saturated models of Pr. If is such a model then the following is readily seen (as explained also in §1):Any two countable recursively saturated elementary endextensions of are isomorphic.If we drop “countable” from the assumption of this statement, we can still say that the two models are ∞ω-equivalent. Must they be isomorphic if both have cardinality ℵ1? Certainly not, since one of the models can be ω1-like while the other is not. Once we realized this much, let us concentrate on ω1-like structures. We prove in §3:Theorem. Any countable recursively saturated modelof Pr hasisomorphism types of ω1-like recursively saturated elementary endextensions. Only one of these is the isomorphism type of a structure that can be expanded to a model of PA.The key technical result is proven in §2. It says that an as above has precisely two countable recursively saturated elementary endextensions which are nonisomorphic over .