Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 75, Issue 4 (2010), 1311-1325.
Expansions of the real field by open sets: definability versus interpretability
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.
J. Symbolic Logic Volume 75, Issue 4 (2010), 1311-1325.
First available in Project Euclid: 4 October 2010
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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. http://projecteuclid.org/euclid.jsl/1286198148.