A descriptive Main Gap Theorem

Abstract: Answering one of the main questions of [FHK14, Chapter 7], we show that there is a tight connection between the depth of a classifiable shallow theory $T$ and the Borel rank of the isomorphism relation $\cong^\kappa_T$ on its models of size $\kappa$, for $\kappa$ any cardinal satisfying $\kappa^{< \kappa} = \kappa > 2^{\aleph_0}$. This is achieved by establishing a link between said rank and the $\mathcal{L}_{\infty \kappa}$-Scott height of the $\kappa$-sized models of $T$, and yields to the following descriptive set-theoretical analogue of Shelah's Main Gap Theorem: Given a countable complete first-order theory $T$, either $\cong^\kappa_T$ is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is $\kappa^+ > \aleph_1$), or it is not Borel at all. The dividing line between the two situations is the same as in Shelah's theorem, namely that of classifiable shallow theories. We also provide a Borel reducibility version of the above theorem, discuss some limitations to the possible (Borel) complexities of $\cong^\kappa_T$, and provide a characterization of categoricity of $T$ in terms of the descriptive set-theoretical complexity of $\cong^\kappa_T$.