A generalized Borel-reducibility counterpart of Shelah’s main gap theorem

Abstract

We study the $\kappa$-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and $T^{\prime }$, if T is classifiable and $T^{\prime }$ is not, then the isomorphism of models of $T^{\prime }$ is strictly above the isomorphism of models of T with respect to $\kappa$-Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions.