On \Sigma^1_1-complete Equivalence Relations on the Generalized Baire Space

TL;DR

This paper explores the complexity of equivalence relations on uncountable structures, demonstrating their -completeness under certain set-theoretic assumptions and showing the independence of this property in ZFC.

Contribution

It establishes -completeness of equivalence relations on uncountable structures under V=L and shows the undecidability of their complexity in ZFC for certain theories.

Findings

Many equivalence relations are -complete if V=L

The -completeness of isomorphism relations can be forced to fail in ZFC

The complexity status of some isomorphism relations is independent of ZFC

Abstract

Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V = L, then many of them are \Sigma^1_1-complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in ZFC whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is \Sigma^1_1-complete (it is, if V = L, but can be forced not to be).