Dedekind cuts in polynomially bounded , O - minimal expansions of real closed fields Dissertation

O-minimal expansions of real closed fields Dissertation Zur Erlangung des Doktorgrades der Naturwissenschaften aus Regensburg 1996 Introduction We are concerned with o-minimal theories. The main result (the Box Theorem 17.3) is about polynomially bounded, o-minimal expansions T of real closed fields with archimedean prime model. In this Introduction, T always denotes such a theory-the reader may think of T as the theory of real closed fields. A Dedekind cut p ('cut' for short) of a real closed field M is a pair (p L , p R) of subsets p L , p R (the Left, Right options of p) of M such that p L < p R and M = p L ∪ p R. We examine the structure (M, D), where D is a symbol for the set p L. Sometimes p cuts out an algebraic substructure of M. For example A := {a ∈ M | |a| ∈ p L } can be a convex subgroup or a convex valuation ring of M. In the latter case, (M, D) is the same as the real valued field (M, A). This case is treated in real valuation theory which is therefore a main tool here. In fact we are far under results like the examination of the real valuation spectra in general (we have no analysis of the type space S n (M, D)). Rather our first goals are model completeness results for all structures (M, D) in small, definable expansions of (M, D) (§16). There are six sorts of such expansions according to certain properties of the cut p. I describe these properties and explain the word " sort " : If a is an element of a model M of T , then the cut a + := ((−∞, a], (a, +∞)) has defin-able options (for real closed fields, definable sets/maps are semi algebraic sets/maps), that is (M, D) is a definable expansion of M. If s is an M-definable map, then s moves a + to another cut with M-definable options. This means, that lim xa s(x) exists in M ∪ {±∞}. In general if p and q are cuts over M , we say that p and q are equivalent, if there is an M-definable map, which moves p to q (this is an equivalence relation). If p is a cut of M we write W 0 (p) for the convex subgroup {a …