Journal of Mathematical Sciences

, Volume 202, Issue 1, pp 40–49 | Cite as

Hyperarithmetical Categoricity of Boolean Algebras of Type \( \mathfrak{B} \)(ωα× η)

Article

We study the Δβ0 -categoricity of Boolean algebras. We prove that if δ is a limit ordinal or 0, n ∈ ω, and δ+n ≥ 1, then the Boolean algebra \( \mathfrak{B} \)δ+n×η) is Δδ + 2n + 10 -categorical, but not Δδ + 2n0 -categorical.

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References

  1. 1.
    S. S. Goncharov, Countable Boolean Algebras and Decidability [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1996); English transl.: Plenum, New York (1997).Google Scholar
  2. 2.
    Yu. L. Ershov and S. S. Goncharov, Constructive Models, [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (1999); English transl.: Consultants Bureau (2000).Google Scholar
  3. 3.
    C. J. Ash and J. F. Khight, Computable Structures and the Hyperarithmetical Hierarchy, Elsevier, Amsterdam (2000).MATHGoogle Scholar
  4. 4.
    C. J. Ash, J. F. Khight, M. Manasse, and T. Slaman, “Generic copies of countable structures,” Ann. Pure Appl. Logic 42, No. 3, 195–205 (1989).CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    J. Chisholm, “Effective model theory vs. recursive model theory,” J. Symb. Log. 55, No. 3, 1168–1191 (1990).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    S. S. Goncharov, “On the number of nonequivalent constructivizations” [in Russian], Algebra Logika 16, No. 3, 257–282 (1977).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    S. Goncharov, V. Harizanov, J. Knight, Ch. McCoy, R. Miller, and R. Solomon, “Enumerations in computable structure theory,” Ann. Pure Appl. Logic 136, No. 3, 219–246 (2005).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    J. Chisholm, E. B. Fokina, S. S. Goncharov, V. S. Harizanov, J. F. Knight, and S. Quinn, “Intrinsic bounds on complexity and definability at limit levels,” J. Symb. Log 74, No. 3, 1047–1060 (2009).CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    S. S. Goncharov, “Autostability and computable families of constructivizations” [in Russian], Algebra Logika 14, No. 6, 647–680 (1975).CrossRefMATHGoogle Scholar
  10. 10.
    S. S. Goncharov and V. D. Dzgoev, “Autostability of models” [in Russian], Algebra Logika 19, No. 1, 45–58 (1980); English transl.: Algebra Logic 19, 28–37 (1980).CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    J. B. Remmel, “Recursive isomorphism types of recursive Boolean algebras,” J. Symb. Log. 46, No. 3, 572–594 (1981).CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ch. McCoy, Δ 20 -Categoricity in Boolean algebras and linear orderings,” Ann. Pure Appl. Logic 119, No. 1–3, 85–120 (2003).CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Ch. F. McCoy, “Partial results in Δ 30 -categoricity in linear orderings and Boolean algebras” [in Russian], Algebra Logika 41, No. 5, 531–552 (2002); English transl.: Algebra Logic 41, No. 5, 295–305 (2002).CrossRefMathSciNetGoogle Scholar
  14. 14.
    C. J. Ash, “Categoricity in hyperarithmetical degrees,” Ann. Pure Appl. Logic 34, No. 1, 1–14 (1987).CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussia

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