Hanf number for Scott sentences of computable structures

  • S. S. Goncharov
  • J. F. Knight
  • I. Souldatos


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.


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

Mathematics Subject Classification

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


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Supplementary material

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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

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