Prime models of theories of computable linear orderings

Abstract

We answer a long-standing question of Rosenstein by exhibiting a complete theory of linear orderings with both a computable model and a prime model, but no computable prime model. The proof uses the relativized version of the concept of limitwise monotonic function. A linear ordering is computable if both its domain and its order relation are computable; it is computably presentable if it is isomorphic to a computable linear ordering. (There are natural generalizations of these notions to other kinds of structures; for instance, see [10] for details.) There is a large body of research on computable linear orderings ([4] gives an extensive overview). Much of this work has been focused on the relationship between classical and effective order types, but it is also interesting to take an approach inspired by classical model theory and study the relationship between effective order types and theories of linear orderings, asking, for instance, what kinds of computable linear orderings exist within the models of a given theory of linear orderings. Taking this approach, Rosenstein ended his book Linear Orderings [12] by asking whether a complete theory of linear orderings with a computable model and a prime model must have a computable prime model. This question was repeated in the Problem Sessions section of [11] (Problem 7.20), and it has recently been included in [3] (Question 3.18). In this paper, we give a negative answer to Rosenstein's question. We assume familiarity with basic notions and results from computability theory and model theory (standard references are [13] and [5], respectively). For structures A and B in the same language, we will write A ok B to mean that player 3 has a winning strategy in the Ehrenfeucht-Fraisse game EFk [A, B] of length k. Recall that if A -k B for all k e w then A B. See Section 3.3 of [5] for details. We will use the following relativized version of a notion due originally to Khisamiev [6]. 1. Definition. Let a be a Turing degree. A function f is a-limitwise monotonic if there exists an a-computable binary function g such that, for all n, s E w, g(n, s) ? Received by the editors December 29, 1999 and, in revised form, February 25, 2000. 2000 Mathematics Subject Classification. Primary 03D45; Secondary 06A05, 03C57, 03C50, 03C65. The author's research was supported by the Marsden Fund of New Zealand. The author thanks the anonymous referee for valuable historical comments. ?)2001 American Mathematical Society