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

Abstract

AbstractAn openU⊆ 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 setE⊆ ℝ such that (ℝ, +, ·, ℕ) defines a Borel isomorph of (ℝ, +, ·, ℕ) and, for every exponentially bounded o-minimal expansionof (ℝ, +, ·), every subset of ℝ definable in (,E) either has interior or is Hausdorff null.