Journal of Symbolic Logic

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

Harvey Friedman, Krzysztof Kurdyka, Chris Miller, and Patrick Speissegger

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 4 October 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

Export citation