A certain class of models of Peano arithmetic

Abstract

This paper is devoted to the study of recursively and short recursively saturated models of PA by means of so-called nonstandard satisfaction methods. The paper is intended to be self-contained. In particular, no knowledge of nonstandard satisfaction classes is assumed. In fact we shall not use this notion explicitly.We define a certain property of models of PA which we call the S-property and prove that properly short recursively saturated models (see Definition 2.1. below) are exactly short models with the S-property. The main result is that all properly short recursively saturated models are elementary cuts of recursively saturated models. This is a generalization to the uncountable case of the theorem of C. Smorynski [9] and is an easy application of some general results concerning cofinal extensions of models of PA which we discuss in §3.On the way we obtain another proof of the result of Smorynski and Stavi [10] which says that short recursive and recursive saturation is preserved under cofinal extensions.The author wants to thank H. Kotlarski and W. Marek for valuable suggestions concerning the subject of the paper.Special thanks must also go to J. Paris for the lemma used in the proof of Theorem 3.5.