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


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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  1. 1.

    Friedman, S.D., Hyttinen, T., and Kulikov, V.: Generalized descriptive set theory and classification theory. Mem. Am. Math. Soc. 230(1081), (American Mathematical Society, 2014)

  2. 2.

    Hyttinen, T., Kulikov, V.: Borel\(^*\) sets in the generalised Baire space. In: Sandu, G., van Ditmarsch, H. (eds.) Outstanding Contributions to Logic: Jaakko Hintikka. Springer (in print)

  3. 3.

    Hyttinen, T., Moreno, M.: On the reducibility of isomorphism relations. Math. Log. Q. (To appear)

  4. 4.

    Hyttinen, T., Tuuri, H.: Constructing strongly equivalent nonisomorphic models for unstable theories. Ann. Pure Appl. Logic. 52, 203–248 (1991)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Kunen, K.: Set theory, Studies in Logic: Mathematical Logic and Foundations 34. College Publications, London (2011)

    Google Scholar 

  6. 6.

    Mekler, A., Väänänen, J.: Trees and \(\varPi _1^1\) subsets of \(\omega _1^{\omega _1}\). J. Symb. Logic. 58(3), 1052–1070 (1993)

    Article  MATH  Google Scholar 

  7. 7.

    Shelah, S.: Classification Theory, Studies in Logic and the Foundations of Mathematics 92. North-Holland, Amsterdam (1990)

    Google Scholar 

  8. 8.

    Shelah, S.: Diamonds. Proc. Am. Math. Soc. 138(6), 2151–2161 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Vaught, R.: Invariant sets in topology and logic, Fund. Math. 82, 269–294, (1974/75)

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

Mathematics Subject Classification

  • 03E15
  • 03C45