Hanf number for Scott sentences of computable structures

Article
  • 5 Downloads

Abstract

The Hanf number for a set S of sentences in \(\mathcal {L}_{\omega _1,\omega }\) (or some other logic) is the least infinite cardinal \(\kappa \) such that for all \(\varphi \in S\), if \(\varphi \) has models in all infinite cardinalities less than \(\kappa \), then it has models of all infinite cardinalities. Friedman asked what is the Hanf number for Scott sentences of computable structures. We show that the value is \(\beth _{\omega _1^{CK}}\). The same argument proves that \(\beth _{\omega _1^{CK}}\) is the Hanf number for Scott sentences of hyperarithmetical structures.

Keywords

Computable structures Scott sentences Infinitary Logic Hanf Number Characterizing cardinals Hyperarithmetical structures 

Mathematics Subject Classification

Primary 03C57 03D45 Secondary 03C75 03C70 03C52 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Supplementary material

153_2018_615_MOESM1_ESM.pdf (370 kb)
Supplementary material 1 (pdf 370 KB)

References

  1. 1.
    Ash, C.J., Knight, J.F.: Computable Structures and the Hyperarithmetical Hierarchy. Elsevier, Amsterdam (2000)MATHGoogle Scholar
  2. 2.
    Csima, B., Harizanov, V., Miller, R., Montalbán, A.: Computability of Fraïssé limits. J. Symb. Logic 76, 66–93 (2011)CrossRefMATHGoogle Scholar
  3. 3.
    Goncharov, S.S.: Strong constructivizability of homogeneous models. Algebra i Logika 17, 363–388 (1978)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Hodges, W.: A Shorter Model Theory. Cambridge University Press, Cambridge (1997)MATHGoogle Scholar
  5. 5.
    Keisler, H.J.: Model Theory for Infinitary Logic. North-Holland, Amsterdam (1971)MATHGoogle Scholar
  6. 6.
    Nadel, M.: \(L_{\omega _1,\omega }\) and admissible fragments. In: Barwise, K.J., Feferman, S. (eds.) Model-Theoretic Logics, pp. 271–316. Springer, New York (1985)Google Scholar
  7. 7.
    Nadel, M.: Scott sentences and admissible sets. Ann. Math. Logic 7, 267–294 (1974)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Peretyat’kin, M.G.: A criterion of strong constructivizability of a homogeneous model. Algebra i Logika 17, 436–454 (1978)MathSciNetMATHGoogle Scholar
  9. 9.
    Ressayre, J.-P.: Boolean valued models and infinitary first order languages. Ann. Math. Logic 6, 41–92 (1973)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Ressayre, J.-P.: Models with compactness properties relative to an admissible language. Ann. Math. Logic 11, 31–55 (1977)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Rogers, H.: Theory of Recursive Functions and Effective Computability. McGraw Hill, New York (1967)MATHGoogle Scholar
  12. 12.
    Scott, D.: Logic with denumerably long formulas and finite strings of quantifiers. In: Addison, J., Henkin, L., Tarski, A. (eds.) The Theory of Models, pp. 329–341. North-Holland, Amsterdam (1965)Google Scholar
  13. 13.
    Souldatos, I.: Characterizing the power set by a complete Scott sentence. Fundam. Math. 6, 131–154 (2013)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.University of Notre DameNotre DameUSA
  3. 3.University of Detroit MercyDetroitUSA

Personalised recommendations