Bounds on Scott rank for various nonelementary classes

Abstract

Ideally we would like to answer the following question. Suppose T is a first order theory and there is a bound ~<0~1 on the Scott rank of countable models of T. What can we say about 7? In I-S] Sacks points out that ~ < 6~(T) and asks if this can be improved to ~ < co~. If T has only countably many countable models, then Martin 's conjecture would imply ~ < ~o. 2. While this question seems very difficult, it is possible to obtain exact bounds on Scott rank in some more abstract settings. For example in [ K M S ] we showed that i fE is S~ equivalence relation where every class is Borel and there is a bound on the Borel rank of the equivalence classes, then ~ < 7,1 (we define 7z 1 below) and 7~ is the optimal bound. In this paper we will consider bounds on Scott rank for PCL~I -sentences. Basically we will show that 62~(~o) is an optimal bound on the Scott ranks of countable models a PCL~ -sentence ~o and 7~0P) is an optimal bound on the well founded models of an L,ol`o-sentence ~p. These results are proved in Sect. 2 and Sect. 3. In Sect. 4 we examine bounds on Scott rank for these classes when we also consider uncountable models. In this case the bounds coincide. The results of Sect. 4 were obtained earlier in unpublished work by Nadel and Stavi.