We prove under V = L that the inclusion modulo the non-stationary ideal is a Σ11-complete quasi-order in the generalized Borel-reducibility hierarchy (κ > ω). This improvement to known results in L has many new consequences concerning the Σ11completeness 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 ∆11, then it is Σ11-complete. We also study the case V 6= L and prove Σ11-completeness results for weakly ineffable and weakly compact κ.
Bibliographical notePublisher Copyright:
© Instytut Matematyczny PAN, 2020
- Generalized Baire space
- Generalized descriptive set theory