On Sigma(1)(1)-completeness of quasi-orders on kappa(kappa)

Tapani Hyttinen*, Vadim Kulikov, Miguel Moreno

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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 > 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 not equal L and prove Sigma(1)(1)-completeness results for weakly ineffable and weakly compact kappa.

Original languageEnglish
Pages (from-to)245-268
Number of pages24
JournalFUNDAMENTA MATHEMATICAE
Volume251
Issue number3
DOIs
Publication statusPublished - 2020
MoE publication typeA1 Journal article-refereed

Keywords

  • generalized Baire space
  • generalized descriptive set theory
  • reducibility
  • quasi-orders
  • embeddability
  • Sigma(1)(1)-completeness
  • EQUIVALENT NONISOMORPHIC MODELS
  • BOREL-REDUCIBILITY

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