The Borel cardinality of Lascar strong types

TL;DR

This paper investigates the complexity of Lascar strong types in model theory, demonstrating that non-trivial restrictions of the Lascar equivalence relation are inherently non-smooth in the Borel hierarchy.

Contribution

It establishes a link between non-trivial Lascar types and their non-smooth Borel complexity, advancing understanding of their classification.

Findings

Non-trivial Lascar types are non-smooth as Borel equivalence relations.

The result applies to restrictions of Lascar equivalence to KP-strong types.

Provides a new perspective on the complexity of Lascar strong types in model theory.

Abstract

We show that if the restriction of the Lascar equivalence relation to a KP-strong type is non-trivial, then it is non-smooth (when viewed as a Borel equivalence relation on an appropriate space of types).