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

Abstract

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.

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Correspondence to Miguel Moreno.

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Hyttinen, T., Kulikov, V. & Moreno, M. A generalized Borel-reducibility counterpart of Shelah’s main gap theorem. Arch. Math. Logic 56, 175–185 (2017). https://doi.org/10.1007/s00153-017-0521-3

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Keywords

  • Generalized descriptive set theory
  • Classification theory
  • Main gap
  • Isomorphism

Mathematics Subject Classification

  • 03E15
  • 03C45