Most(?) theories have Borel complete reducts

TL;DR

The paper demonstrates that many simple theories possess Borel complete reducts, especially when they have uncountably many types or are not small or $ ext{omega}$-stable, revealing complex classification properties.

Contribution

It establishes new conditions under which theories have Borel complete reducts, linking model-theoretic properties to Borel complexity.

Findings

Countable theories with uncountably many 1-types have Borel complete reducts.

Non-small theories have Borel complete reducts in their $Th(M)^{eq}$.

Non-$ ext{omega}$-stable theories have models with Borel complete reducts.

Abstract

We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has a Borel complete reduct, and if a theory $T$ is not $\omega$-stable, then the elementary diagram of some countable model of $T$ has a Borel complete reduct.