Shelah's Main Gap and the generalized Borel-reducibility

TL;DR

This paper establishes a connection between Shelah's Main Gap and generalized Borel reducibility, demonstrating that the complexity of model isomorphism relations aligns with the classification of theories, and shows how to force certain complexity levels.

Contribution

It links Shelah's Main Gap to generalized Borel reducibility and demonstrates the possible complexity levels of model isomorphism relations via forcing.

Findings

Classifiable theories have lower Borel complexity in isomorphism.

Non-classifiable theories have strictly higher Borel complexity.

Isomorphism of models can be forced to be analytic co-analytic or complete.

Abstract

We answer one of the main questions in generalized descriptive set theory, the Friedman-Hyttinen-Kulikov conjecture on the Borel reducibility of the Main Gap. We show a correlation between Shelah's Main Gap and generalized Borel reducibility notions of complexity. For any $\kappa$ satisfying $\kappa=\lambda^+=2^\lambda$ and $2^{\mathfrak{c}}\leq\lambda=\lambda^{\omega_1}$, we show that if $T$ is a classifiable theory and $T'$ is a non-classifiable theory, then the isomorphism of models of $T'$ is strictly above the isomorphism of models of $T$ with respect to Borel-reducibility. We also show that the following can be forced: for any countable first-order theory in a countable vocabulary, $T$, the isomorphism of models of $T$ is either analytic co-analytic, or analytically-complete.