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

Abstract

We study the $$\kappa $$ź-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 $$T^{\prime }$$Tź, if T is classifiable and $$T^{\prime }$$Tź is not, then the isomorphism of models of $$T^{\prime }$$Tź is strictly above the isomorphism of models of T with respect to $$\kappa $$ź-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.