Shelah's Main Gap and the generalized Borel-reducibility

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.