I will review some recent results in the Borel reducibility on uncountable cardinals of the Helsinki logic group. Borel reducibility on the generalised Baire space \kappa^\kappa for uncountable \kappa is defined analogously to that for \kappa=\omega. One of the corollaries of this work is that under some mild cardinality assumptions on kappa, if T1 is classifiable and T2 is unstable or superstable with OTOP, then the ISOM(T1) is continuously reducible ISOM(T2) and ISOM(T2) is not Borel reducible to ISOM(T1). Another result, on which we will focus more in the talk is that under certain cardinality assumptions on kappa the inclusion modulo the non-stationary ideal is a Sigma^1_1-complete quasiorder which has the corollary that if V=L, then the embeddability of dense linear orders of cardinality kappa is Sigma^1_1-complete.
Wed, 24/01/2018 - 11:00 to 13:00