December 2010 Expansions of the real field by open sets: definability versus interpretability
Harvey Friedman, Krzysztof Kurdyka, Chris Miller, Patrick Speissegger
J. Symbolic Logic 75(4): 1311-1325 (December 2010). DOI: 10.2178/jsl/1286198148


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.


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Harvey Friedman. Krzysztof Kurdyka. Chris Miller. Patrick Speissegger. "Expansions of the real field by open sets: definability versus interpretability." J. Symbolic Logic 75 (4) 1311 - 1325, December 2010.


Published: December 2010
First available in Project Euclid: 4 October 2010

zbMATH: 1220.03030
MathSciNet: MR2767970
Digital Object Identifier: 10.2178/jsl/1286198148

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

Rights: Copyright © 2010 Association for Symbolic Logic


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Vol.75 • No. 4 • December 2010
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