On -completeness of quasi-orders on \kappa ^\kappa
Tom 251 / 2020
Streszczenie
We prove under V=L that the inclusion modulo the non-stationary ideal is a \Sigma_1^1 -complete quasi-order in the generalized Borel-reducibility hierarchy (\kappa \gt \omega ). This improvement to known results in L has many new consequences concerning the \Sigma_1^1 -completeness of quasi-orders and equivalence relations such as the embeddability of dense linear orders as well as the equivalence modulo various versions of the non-stationary ideal. This serves as a partial or complete answer to several open problems stated in the literature. Additionally the theorem is applied to prove a dichotomy in L: If the isomorphism of a countable first-order theory (not necessarily complete) is not \Delta_1^1 , then it is \Sigma_1^1 -complete.
We also study the case V\ne L and prove \Sigma_1^1 -completeness results for weakly ineffable and weakly compact \kappa .
