On ‐complete equivalence relations on the generalized Baire space

Abstract

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if , then many of them are ‐complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is ‐complete (it is, if , but can be forced not to be).