A Subfield of the Reals Defining a Non-Measurable Set

Abstract

In the area of Tame Geometry, different model-theoretic tameness conditions are established and their relationships are analyzed. We construct a subfield K of the real numbers that lacks several such tameness properties. As our main result, we present a first-order formula in the language of rings defining a set in K that is not Lebesgue measurable. Moreover, K has the independence property and admits both archimedean and non-archimedean orderings.