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Archive for Mathematical Logic

, Volume 48, Issue 1, pp 25–38 | Cite as

Categoricity of computable infinitary theories

  • W. Calvert
  • S. S. Goncharov
  • J. F. Knight
  • Jessica Millar
Article

Abstract

Computable structures of Scott rank \({\omega_1^{CK}}\) are an important boundary case for structural complexity. While every countable structure is determined, up to isomorphism, by a sentence of \({\mathcal{L}_{\omega_1 \omega}}\), this sentence may not be computable. We give examples, in several familiar classes of structures, of computable structures with Scott rank \({\omega_1^{CK}}\) whose computable infinitary theories are each \({\aleph_0}\)-categorical. General conditions are given, covering many known methods for constructing computable structures with Scott rank \({\omega_1^{CK}}\), which guarantee that the resulting structure is a model of an \({\aleph_0}\)-categorical computable infinitary theory.

Mathematics Subject Classification (2000)

03C57 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  • W. Calvert
    • 1
  • S. S. Goncharov
    • 2
  • J. F. Knight
    • 3
  • Jessica Millar
    • 4
  1. 1.Department of Mathematics and StatisticsMurray State UniversityMurrayUSA
  2. 2.Institute of MathematicsAcademy of Sciences, Siberian BranchNovosibirskRussia
  3. 3.Department of MathematicsUniversity of Notre DameNotre DameUSA
  4. 4.Department of MathematicsBrown UniversityProvidenceUSA

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