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On $\Sigma _1^1$-completeness of quasi-orders on $\kappa ^\kappa $

Volume 251 / 2020

Tapani Hyttinen, Vadim Kulikov, Miguel Moreno Fundamenta Mathematicae 251 (2020), 245-268 MSC: Primary 03E15; Secondary 03C55, 03C45, 54H05. DOI: 10.4064/fm679-1-2020 Published online: 13 May 2020

Abstract

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 $.

Authors

  • Tapani HyttinenDepartment of Mathematics and Statistics
    University of Helsinki
    Pietari Kalmin katu 5
    00014 University of Helsinki, Finland
    e-mail
  • Vadim KulikovDepartment of Mathematics and Statistics
    University of Helsinki
    Pietari Kalmin katu 5
    00014 University of Helsinki, Finland
    and
    Department of Mathematics
    and Systems Analysis
    Aalto University
    Otakaari 1
    02150 Espoo, Finland
    e-mail
  • Miguel MorenoDepartment of Mathematics and Statistics
    University of Helsinki
    Pietari Kalmin katu 5
    00014 University of Helsinki, Finland
    and
    Department of Mathematics
    Bar-Ilan University
    Ramat-Gan, 5290002 Israel
    e-mail

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