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A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

  • Ehud Hrushovski (a1) and Ya'acov Peterzil (a2)

Abstract

We use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

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[1]Berarducci, A. and Servi, T., An effective version of Wilkie's theorem of the complement and some effective o-minimality results, Annals of Pure and Applied Logic, vol. 125 (2004), no. 1–3, pp. 43–74.
[2]van den Dries, Lou, o-minimal structures, Logic: From Foundations to Applications (Staffordshire, 1993), Oxford Scientific Publications, Oxford University Press, New York, 1996, pp. 137–185.
[3]Edmundo, M. J. and Otero, M., Definably compact abelian groups, Journal of Mathematical Logic, vol. 4 (2004), no. 2, pp. 163–180.
[4]Lipshitz, L. and Robinson, Z., Overconvergent real closed quantifier elimination, Bulletin of the London Mathematical Society, to appear.
[5]Peterzil, Y. and Starchenko, S., A trichotomy theorem for o-minimal structures, Proceedings of the London Mathematical Society (3), vol. 77 (1998), no. 3, pp. 481–523.
[6]— Peterzil, Y. and Starchenko, S., Expansions of algebraically closed fields in o-minimal structures, Selecta Mathematica (New Series), vol. 7 (2001), no. 3, pp. 409–445.
[7]Woerheide, A., O-minimal homology, Ph .D.thesis, University of Illinois at Urbana-Champaign, 1996.

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