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

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

2008-12-04

arxiv.org/abs/0812.0836v2

Mathematics

Logic

14 pages

Scientific paper

An open set U of the real numbers R is produced such that the expansion (R,+,x,U) of the real field by U defines a Borel isomorph of (R,+,x,N) but does not define N. 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 subset K of R such that for every exponentially bounded o-minimal expansion M of (R,+,x), every subset of R definable in (M,K) either has interior or is Hausdorff null.

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