Journal of Symbolic Logic

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

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,+,·,E) defines a Borel isomorph of (ℝ,+,·,ℕ) and, for every exponentially bounded o-minimal expansion ℜ of (ℝ,+,·), every subset of ℝ definable in (ℜ,E) either has interior or is Hausdorff null.

Article information

Source
J. Symbolic Logic Volume 75, Issue 4 (2010), 1311-1325.

Dates
First available: 4 October 2010

http://projecteuclid.org/euclid.jsl/1286198148

Digital Object Identifier
doi:10.2178/jsl/1286198148

Zentralblatt MATH identifier
05835167

Mathematical Reviews number (MathSciNet)
MR2767970

Citation

Friedman, Harvey; Kurdyka, Krzysztof; Miller, Chris; Speissegger, Patrick. Expansions of the real field by open sets: definability versus interpretability. Journal of Symbolic Logic 75 (2010), no. 4, 1311--1325. doi:10.2178/jsl/1286198148. http://projecteuclid.org/euclid.jsl/1286198148.