The elementary theory of Dedekind cuts in polynomially bounded structures

Let M be a polynomially bounded, o-minimal structure with archimedean prime model, for example if M is a real closed field. Let C be a convex and unbounded subset of M . We determine the first order theory of the structure M expanded by the set C. We do this also over any given set of parameters from M , which yields a description of all subsets of M, definable in the expanded structure. 1. Introdu tion This paper is a sequel to [Tr], where we began the model theoretic study of Dedekind cuts of o-minimal expansions of fields. Before explaining what we do here, we recall some terminology from [Tr]. If X is a totally ordered set, then a (Dedekind) cut p of X is a tuple p = (p, p) with X = pL∪pR and p < p. If Y ⊆ X then Y + denotes the cut p of X with p = {x ∈ X | x > Y }. Y + is called the upper edge of Y . Similarly the lower edge Y − of Y is defined. We fix an o-minimal expansion T of the theory of real closed fields in a language L . If M is a model of T and p is a cut of (the underlying set of) M , then by the model theoretic properties of p we understand the model theoretic properties of the structure M expanded by the set p in the language L (D) extending L , which has an additional unary predicate D interpreted as p. We write (M,p) for this expansion of M . Our aim here is to determine the full theory of the structure (M,p) in the language L (D) relative T and to give a description of the subsets ofM, definable in (M,p) relativeM . By “relative T ” and “relative M” we mean that the theory T and the subsets of M definable in M are assumed to be known. By basic model theory, this problem amounts to find the theory of the structure (M,p) in the language L (D) over a given set A of parameters. We can do this for all cuts in the case where T is polynomially bounded with archimedean prime model (c.f (3.4) below), in particular in the case of pure real closed fields. The main result is Theorem (4.4) below, which is of technical nature. For the moment, we describe what we get from this result by saying what the subsets of M, definable in (M,p), are. In order to speak about these sets, we first have to recall some invariants of a cut p from [Tr]. The o-minimal terminology is taken from [vdD1]. (1.1) Definition. Let p be a cut of an ordered abelian group K. The convex subgroup G(p) := {a ∈ K | a+ p = p} of K is called the invariance group of p (here a+ p := (a+ p, a+ p)). The cut G(p) is denoted by p̂. 2000 Mathematics Subject Classification: Primary 03C64; Secondary 03C10