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

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.