Source: J. Symbolic Logic
Volume 75, Issue 4
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
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--------, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998.
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