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 ∼ = κ T on its models of size κ, for κ any cardinal satisfying κ <κ = κ > 2 ℵ 0 . This is achieved by establishing a link between said rank and the L∞κ-Scott height of the κ-sized models of T , and yields to the following descriptive set-theoretical analogue of Shelah's Main Gap Theorem: Given a countable complete firstorder theory T , either ∼ = κ T is Borel with a countable Borel rank (i.e. very simple, given that the length of the relevant Borel hierarchy is κ + > ℵ 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 ∼ = κ T , and provide a characterization of categoricity of T in terms of the descriptive set-theoretical complexity of ∼ = κ T .