Journal of Symbolic Logic 75 (4):1311-1325 (2010)

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 $\germ{R}$ of (ℝ, +, ·), every subset of ℝ definable in ( $\germ{R}$ , E) either has interior or is Hausdorff null
Keywords expansion of the real field   o-minimal   projective hierarchy   Cantor set   Hausdorff dimension   Minkowski dimension
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DOI 10.2178/jsl/1286198148
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Expansions of o-Minimal Structures by Iteration Sequences.Chris Miller & James Tyne - 2006 - Notre Dame Journal of Formal Logic 47 (1):93-99.

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