The isomorphism relation of theories with S-DOP in the generalised Baire spaces (Q2067635)

The paper under review lies at the intersection of so-called generalized descriptive set theory and model-theoretic classification. Generalized descriptive set theory studies definable subsets of generalized Baire spaces, which are spaces of the form \(\kappa^\kappa\) for \(\kappa\) an uncountable cardinal equipped with the so-called bounded topology. Given a first-order complete theory \(T\) in a countable language, one can view the isomorphism relation \(\cong_T^\kappa\) on models of \(T\) of cardinality \(\kappa\) as a relation on the generalized Baire space \(\kappa^\kappa\) and the line of research that this paper is contributing to attempts to see if there is a connection between the descriptive set-theoretic complexity of \(\cong_T^\kappa\) and the model-theoretic complexity of the theory \(T\). In particular, the driving question is whether or not Borel reducibility is a refinement of the stability-theoretic notion of complexity. The main result of this paper is that if \(\kappa\) is an inaccessible cardinal, \(T\) is a superstable theory with S-DOP, then the relation \(\cong_T^\kappa\) is \(\Sigma_1^1\)-complete. Another result along these lines is that if \(\kappa\) is inaccessible and \(T_1\) is classifiable while \(T_2\) is a superstable theory with S-DOP, then \(\cong_{T_1}^\kappa\) continuously reduces to \(\cong_{T_2}^\kappa\). Both of these results follow from the result that if \(\kappa\) is inaccessible and \(T\) is a theory with S-DOP, then a particular equivalence relation \(E^\kappa_{\lambda\text{-club}}\) continuously reduces to \(\cong_T^\kappa\), where \(\lambda=(2^{\aleph_0})^+\) and \(f,g\in \kappa^\kappa\) are \(E^\kappa_{\lambda\text{-club}}\) related if the set of \(\alpha<\kappa\) for which \(f(\alpha)=g(\alpha)\) contains a \(\lambda\)-club.