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Expansions of the real field by open sets: definability versus interpretability

Published online by Cambridge University Press:  12 March 2014

Harvey Friedman
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue Columbus. Ohio 43210., USA. E-mail: friedman@math.ohio-state.edu
Krzysztof Kurdyka
Affiliation:
Laboratoire de Mathématiques, Université de Savoie, UMR 5127 CNRS, 73376 Le Bourget-Du-Lac, France. E-mail: Krzysztof.Kurdyka@univ-savoie.fr
Chris Miller
Affiliation:
Department of Mathematics, The Ohio State University, 231 West 18th Avenue Columbus, Ohio 43210, USA. E-mail: miller@math.ohio-state.edu
Patrick Speissegger
Affiliation:
Department of Mathematics & Statistics, McMaster University, 1280 Main Street West Hamilton, Ontario L8S 4K1, Canada. E-mail: speisseg@math.mcmaster.ca

Abstract

An open U ⊆ ℝ is produced such that (ℝ, +, ·, U) defines a Borel isomorph of (ℝ, +, ·, ℕ) but does not define ℕ. It follows that (ℝ, +, ·, U) defines sets in every level of the projective hierarchy but does not define all projective sets. This result is elaborated in various ways that involve geometric measure theory and working over o-minimal expansions of (ℝ, +, ·). In particular, there is a Cantor set E ⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansion of (ℝ, +, ·), every subset of ℝ definable in (, E) either has interior or is Hausdorff null.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2010

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