On Tameness, Measurability and the Independence Property

TL;DR

This paper explores the boundaries of tameness in model theory by constructing a real subfield with complex properties, including the independence property and non-Borel definable sets, challenging existing notions of geometric tameness.

Contribution

It introduces a specific subfield of real numbers that exhibits both tameness violations and the independence property, providing new insights into model-theoretic and geometric complexity.

Findings

Constructed a subfield of real numbers lacking several tameness properties.

Defined a non-Borel set using a first-order formula in the ring language.

Showed the subfield has the independence property and supports multiple orderings.

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 of such tameness properties. As our main result, we present a first-order formula in the language of rings that defines a non-Borel set in $K$. Moreover, $K$ has the independence property and admits both archimedean and non-archimedean orderings.