On Σ11-completeness of quasi-orders on

Tapani Hyttinen, Miguel Moreno, Vadim Kulikov

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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.

Original languageEnglish
Pages (from-to)245-268
Number of pages24
JournalFundamenta Mathematicae
Issue number3
StatePublished - 2020

Bibliographical note

Publisher Copyright:
© Instytut Matematyczny PAN, 2020


  • Completeness-completeness
  • Embeddability
  • Generalized Baire space
  • Generalized descriptive set theory
  • Quasi-orders
  • Reducibility


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