On $M$-recursively saturated models of arithmetic

Abstract

Iiitroducaion. In [6], C. Smorynski investigated the properties of models of arithmetic using the notions of recursive saturation and short recursive saturation.In this paper, we shall generalize these notions and obtain new isomorphism criteria (Theorems A and B) and embeddability criteria(Theorems D and E) for countable models of arithmetic. Throughout, S'J.denotes Peano arithmetic with the induction schema for all formulas in some finitelanguage L2 {0,',+,･}. Ao denotes the set of all quantifierbounded formulas in L. Let M and N be countable models of SJ. with MQN. We say N is M-recursively saturated (Ms-recursively saturated)if N realizes every (short) type r which is Ax on MFM, where r may contain countably many parameters from M. It can be easilyshown that M-recursive saturation(Ms-recursive saturation)corresponds with(short)recursive saturation, if M=((d; 0,',+, ･>.For AQ＼N＼, Df(N, A) denotes the set of allelements in N which are definablein N using parameters from A. We put: