The Journal of Symbolic Logic

Research Article

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

Harvey Friedmana1, Krzysztof Kurdykaa2, Chris Millera3 and Patrick Speisseggera4

a1 Department of Mathematics, The Ohio State University, 231 West 18th Avenue Columbus. Ohio 43210., USA. E-mail: friedman@math.ohio-state.edu

a2 Laboratoire de Mathématiques, Université de Savoie, UMR 5127 CNRS, 73376 Le Bourget-Du-Lac, France. E-mail: Krzysztof.Kurdyka@univ-savoie.fr

a3 Department of Mathematics, The Ohio State University, 231 West 18th Avenue Columbus, Ohio 43210, USA. E-mail: miller@math.ohio-state.edu

a4 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.

(Received November 29 2008)

Key words and phrases

  • expansion of the real field;
  • o-minimal;
  • projective hierarchy;
  • Cantor set;
  • Hausdorff dimension;
  • Minkowski dimension