The Borel Complexity of Isomorphism for O-Minimal Theories

TL;DR

This paper classifies the Borel complexity of isomorphism relations for countable models of o-minimal theories, showing they are either very simple or maximally complex depending on the presence of nonsimple types.

Contribution

It provides a complete characterization of the Borel complexity for models of o-minimal theories based on their model-theoretic properties, especially the existence of nonsimple types.

Findings

If nonsimple types exist, isomorphism is Borel complete.

If no nonsimple types, isomorphism is either smooth or Borel equivalent to F_2.

The complexity depends on the theory's smallness and the structure of invariants.

Abstract

Given a countable o-minimal theory T, we characterize the Borel complexity of isomorphism for countable models of T up to two model-theoretic invariants. If T admits a nonsimple type, then it is shown to be Borel complete by embedding the isomorphism problem for linear orders into the isomorphism problem for models of T. This is done by constructing models with specific linear orders in the tail of the Archimedean ladder of a suitable nonsimple type. If the theory admits no nonsimple types, then we use Mayer's characterization of isomorphism for such theories to compute invariants for countable models. If the theory is small, then the invariant is real-valued, and therefore its isomorphism relation is smooth. If not, the invariant corresponds to a countable set of reals, and therefore the isomorphism relation is Borel equivalent to $F_2$. Combining these two results, we conclude that (Mod(T),$\cong$) is either maximally complicated or maximally uncomplicated (subject to completely general model-theoretic lower bounds based on the number of types and the number of countable models).