Models with the ω-property

Abstract

In [KP] we have studied the problem of determining when a subset of a (countable) model M of PA can be coded in an elementary end extension of M. Sets with this property are called elementary extensional. In particular we can ask whether there are elementary extensional subsets of a model which have order type ω. It turns out that having elementary extensional subsets of order type ω is an interesting property connected with other structural properties of models of PA. We will call this property the ω-property. In [KP] the problem of characterizing models with the ω-property was left open. It is still open, and the aim of this paper is to present a collection of results pertaining to it. It should be mentioned that the same notion was studied by Kaufmann and Schmerl in [KS2] in connection with some weak notions of saturation which they discuss there. Our notion of a model with the ω-property corresponds to the notion of an upward monotonically ω-lofty cut.It is fairly easy to see that countable recursively saturated models (or in fact all recursively saturated models with cofinality ω) and all short recursively saturated models have the ω-property (Proposition 1.2 below). On the other hand, if we had asked the question about the existence of models with the ω-property before 1975 (when recursively saturated models were introduced) the answer would probably not have been that easy and we would have to come to notions close to recursive saturation.