A Borel-reducibility Counterpart of Shelah's Main Gap Theorem

TL;DR

This paper explores the Borel-reducibility hierarchy of isomorphism relations of first order theories, establishing a structured comparison aligned with Shelah's Main Gap, and demonstrating the existence of intermediate equivalence relations.

Contribution

It introduces a Borel-reducibility counterpart to Shelah's Main Gap, showing the relative complexity of isomorphism relations for classifiable versus non-classifiable theories.

Findings

Classifiable theories have strictly lower isomorphism complexity.

Non-classifiable theories' isomorphism relations are above those of classifiable theories.

Intermediate equivalence relations can be positioned between these isomorphism complexities.

Abstract

We study the Borel-reducibility of isomorphism relations of complete first order theories and show the consistency of the following: For all such theories T and T', if T is classifiable and T' is not, then the isomorphism of models of T' is strictly above the isomorphism of models of T with respect to Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions.