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Model completeness of o-minimal structures expanded by Dedekind cuts

Published online by Cambridge University Press:  12 March 2014

Marcus Tressl*
Affiliation:
Universität Regensburg, NWF I - Mathematik, D-93040 Regensburg, Germany, E-mail: marcus.tressl@mathematik.uni-regensburg.de

Extract

§1. Introduction. Let M be a totally ordered set. A (Dedekind) cut p of M is a couple (pL, pR) of subsets pL, pR of M such that pLpR = M and pL < pR, i.e., a < b for all apL, bpR. In this article we are looking for model completeness results of o-minimal structures M expanded by a set pL for a cut p of M. This means the following. Let M be an o-minimal structure in the language L and suppose M is model complete. Let D be a new unary predicate and let p be a cut of (the underlying ordered set of) M. Then we are looking for a natural, definable expansion of the L(D)-structure (M, pL) which is model complete.

The first result in this direction is a theorem of Cherlin and Dickmann (cf. [Ch-Dic]) which says that a real closed field expanded by a convex valuation ring has a model complete theory. This statement translates into the cuts language as follows. If Z is a subset of an ordered set M we write Z+ for the cut p with pR = {aMa > Z} and Z for the cut q with qL = {aMa < Z}.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2005

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