The isomorphism relation of theories with S-DOP in generalized Baire spaces

TL;DR

This paper investigates the complexity of classifying models of certain theories in generalized Baire spaces, showing that for inaccessible cardinals, the isomorphism relation of models of superstable theories with S-DOP can be as complex as the most complicated analytic sets.

Contribution

It establishes the Borel reducibility of isomorphism relations between classifiable and superstable theories with S-DOP in generalized Baire spaces, and demonstrates the $oldsymbol{ ext{Σ}}_1^1$-completeness under certain conditions.

Findings

Isomorphism of models of classifiable theories reduces to that of superstable theories with S-DOP.

For inaccessible $oldsymbol{ ext{kappa}}$, the isomorphism relation of superstable theories with S-DOP is $oldsymbol{ ext{Σ}}_1^1$-complete.

The results hold under the assumption of the inaccessibility of $oldsymbol{ ext{kappa}}$.

Abstract

We study the Borel-reducibility of isomorphism relations in the generalized Baire space $\kappa^\kappa$. In the main result we show for inaccessible $\kappa$, that if $T$ is a classifiable theory and $T'$ is superstable with the strong dimensional order property (S-DOP), then the isomorphism of models of $T$ is Borel reducible to the isomorphism of models of $T'$. In fact we show the consistency of the following: If $\kappa$ is inaccessible and $T$ is a superstable theory with S-DOP, then the isomorphism of models of $T$ is $\Sigma_1^1$-complete.