Submission date: 15 February 2010.

Abstract:

An open subset U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field (R,+,x) by U defines a Borel isomorph of (R,+,x,N) but does not define N, where N denotes the set of all natural numbers. It follows that (R,+,x,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,+,x). In particular, there is a Cantor set E such that (R,+,x,E) defines a Borel isomorph of (R,+,x,N) and, for every exponentially bounded o-minimal expansion M of (R,+,x), every subset of R definable in (M,E) either has interior or is Hausdorff null.

Mathematics Subject Classification: Primary 03C64; Secondary 03E15, 28E15

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

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