Advertisement

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
Article
  • 56 Downloads

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.

Keywords

Generalized descriptive set theory Classification theory Main gap Isomorphism 

Mathematics Subject Classification

03E15 03C45 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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)Google Scholar
  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)Google Scholar
  3. 3.
    Hyttinen, T., Moreno, M.: On the reducibility of isomorphism relations. Math. Log. Q. (To appear)Google Scholar
  4. 4.
    Hyttinen, T., Tuuri, H.: Constructing strongly equivalent nonisomorphic models for unstable theories. Ann. Pure Appl. Logic. 52, 203–248 (1991)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Vaught, R.: Invariant sets in topology and logic, Fund. Math. 82, 269–294, (1974/75)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.University of HelsinkiHelsinkiFinland

Personalised recommendations