Archive for Mathematical Logic

, Volume 56, Issue 3–4, pp 175–185 | Cite as

A generalized Borel-reducibility counterpart of Shelah’s main gap theorem

  • Tapani Hyttinen
  • Vadim Kulikov
  • Miguel MorenoEmail author


We study the \(\kappa \)-Borel-reducibility of isomorphism relations of complete first order theories in a countable language and show the consistency of the following: For all such theories T and \(T^{\prime }\), if T is classifiable and \(T^{\prime }\) is not, then the isomorphism of models of \(T^{\prime }\) is strictly above the isomorphism of models of T with respect to \(\kappa \)-Borel-reducibility. In fact, we can also ensure that a range of equivalence relations modulo various non-stationary ideals are strictly between those isomorphism relations. The isomorphism relations are considered on models of some fixed uncountable cardinality obeying certain restrictions.


Generalized descriptive set theory Classification theory Main gap Isomorphism 

Mathematics Subject Classification

03E15 03C45 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of HelsinkiHelsinkiFinland

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