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Valuation theoretic content of the Marker-Steinhorn theorem

Published online by Cambridge University Press:  12 March 2014

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

Extract

The Marker-Steinhorn Theorem (cf. [2] and [3]), says the following. If T is an o-minimal theory and M ≺ N is an elementary extension of models of T such that M is Dedekind complete in N, then for every N-definable subset X of Nk, the trace XMk is M-definable. The original proof in [2] gives an explicit method how to construct a defining formula of XMk out of a defining formula of X. A geometric reformulation of the Marker-Steinhorn Theorem is the definability of Hausdorff limits of families of definable sets. An explicit construction of these Hausdorff limits for expansions of the real field has recently been achieved in [1]. Both proofs and also the treatment [3] are technically involved.

Here we give a short algebraic, but not constructive proof, if T is an expansion of real closed fields. In fact we'll identify the statement of the Theorem with a valuation theoretic property of models of T (namely condition (†) below). Therefore our proof might be applicable to other elementary classes which expand fields, if a notion of dimension and a reasonable valuation theory are available.

From now on, let T be an o-minimal expansion of real closed fields. We have to show the following (cf. [2], Th. 2.1. for this formulation). If M is a model of T and p is a tame n-type over M (i.e., M is Dedekind complete in M ⟨ᾱ⟩ := dcl(Mᾱ) for some realization ᾱ of p), then p is a definable type (cf. [4], 11 .b).

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2004

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References

[1]Lion, J.-M. and Speissegger, P., A geometric proof of the definability of Hausdorff limits, preprint, 02 2003.Google Scholar
[2]Marker, D. and Steinhorn, C., Definable types in o-minimal theories, this Journal, vol. 59 (1994). pp. 185198.Google Scholar
[3]Pillay, A., Definability of types and paires of o-minimal structures, this Journal, vol. 59 (1994), no. 4. pp. 14001409.Google Scholar
[4]Poizat, B., Cours de théorie des modèles, 1985.Google Scholar
[5]van den Dries, L. and Lewenberg, A. H., T-convexity and tame extensions, this Journal, vol. 60 (1995), no. 1, pp. 74101.Google Scholar

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