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

TL;DR

This paper constructs an open set in the real numbers that expands the real field to define complex sets without defining all projective sets, revealing nuanced relationships between definability and interpretability.

Contribution

It introduces a specific open set expansion of the real field that differentiates between definability and interpretability of complex sets, with implications for geometric measure theory and o-minimal structures.

Findings

Defines a Borel isomorph of (R,+,x,N) without defining N

Shows the expansion defines sets at every projective hierarchy level

Identifies a Cantor set with specific definability properties in o-minimal expansions

Abstract

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.