Skip to main content
SpringerLink
Log in

Degrees of autostability relative to strong constructivizations

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The spectra of the Turing degrees of autostability of computable models are studied. For almost prime decidable models, it is shown that the autostability spectrum relative to strong constructivizations of such models always contains a certain recursively enumerable Turing degree; moreover, it is shown that for any recursively enumerable Turing degree, there exist prime models in which this degree is the least one in the autostability spectrum relative to strong constructivizations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. I. Mal’tsev, “On Recursive Abelian Groups,” Dokl. Akad. Nauk SSSR 146(5), 1009–1012 (1962) [Sov. Math., Dokl. 32, 1431–1434 (1963)].

    MathSciNet  Google Scholar 

  2. A. I. Mal’tsev, “Constructive Algebras. I,” Usp. Mat. Nauk 16(3), 3–60 (1961) [Russ. Math. Surv. 16 (3), 77–129 (1961)].

    Google Scholar 

  3. A. Fröhlich and J. C. Shepherdson, “Effective Procedures in Field Theory,” Philos. Trans. R. Soc. London, Ser. A 248, 407–432 (1956).

    Article  MATH  Google Scholar 

  4. S. I. Adyan, “Unsolvability of Some Algorithmic Problems in the Theory of Groups,” Tr. Mosk. Mat. Obshch. 6, 231–298 (1957).

    MathSciNet  MATH  Google Scholar 

  5. S. I. Adyan, “The Problem of Identity in Associative Systems of a Special Form,” Dokl. Akad. Nauk SSSR 135(6), 1297–1300 (1960) [Sov. Math., Dokl. 1, 1360–1363 (1960)].

    Google Scholar 

  6. P. S. Novikov and S. I. Adyan, “Das Wortproblem für Halbgruppen mit einseitiger Kürzungsregel,” Z. Math. Log. Grundlagen Math. 4, 66–88 (1958).

    Article  MathSciNet  MATH  Google Scholar 

  7. A. A. Markov, “Impossibility of Certain Algorithms in the Theory of Associative Systems. II,” Dokl. Akad. Nauk SSSR 58(3), 353–356 (1947).

    MATH  Google Scholar 

  8. B. L. van der Waerden, “Eine Bemerkung über die Unzerlegbarkeit von Polynomen,” Math. Ann. 102, 738–739 (1930).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. S. Goncharov, “Constructivizability of Superatomic Boolean Algebras,” Algebra Logika 12(1), 31–40 (1973) [Algebra Logic 12, 17–22 (1973)].

    MATH  Google Scholar 

  10. S. Wehner, “Enumerations, Countable Structures and Turing Degrees,” Proc. Am. Math. Soc. 126, 2131–2139 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  11. T. A. Slaman, “Relative to Any Nonrecursive Set,” Proc. Am. Math. Soc. 126, 2117–2122 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  12. S. S. Goncharov, “Limit Equivalent Constructivizations,” in Mathematical Logic and the Theory of Algorithms (Nauka, Novosibirsk, 1982), Tr. Inst. Mat., Sib. Otd. Akad. Nauk SSSR 2, pp. 4–12.

    Google Scholar 

  13. S. Goncharov, V. Harizanov, J. Knight, C. McCoy, R. Miller, and R. Solomon, “Enumerations in Computable Structure Theory,” Ann. Pure Appl. Logic 136(3), 219–246 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  14. S. S. Goncharov, “Computability and Computable Models,” in Mathematical Problems from Applied Logic. II, Ed. by D. Gabbay, S. S. Goncharov, and M. Zakharyashev (Springer, New York, 2007), Int. Math. Ser. 5, pp. 99–216.

    Chapter  Google Scholar 

  15. C. J. Ash, “Categoricity in Hyperarithmetical Degrees,” Ann. Pure Appl. Logic 34(1), 1–14 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  16. C. J. Ash and J. F. Khight, Computable Structures and the Hyperarithmetical Hierarchy (Elsevier, Amsterdam, 2000).

    MATH  Google Scholar 

  17. C. Ash, J. Knight, M. Manasse, and T. Slaman, “Generic Copies of Countable Structures,” Ann. Pure Appl. Logic 42(3), 195–205 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  18. 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. Logic 74(3), 1047–1060 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  19. E. B. Fokina, I. Kalimullin, and R. Miller, “Degrees of Categoricity of Computable Structures,” Arch. Math. Logic 49, 51–67 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  20. Yu. L. Ershov, “Constructive Models,” in Selected Topics of Algebra and Logic (Nauka, Novosibirsk, 1973), pp. 111–130 [in Russian].

    Google Scholar 

  21. M. D. Morley, “Decidable Models,” Isr. J. Math. 25, 233–240 (1976).

    Article  MathSciNet  MATH  Google Scholar 

  22. S. S. Goncharov and A. T. Nurtazin, “Constructive Models of Complete Solvable Theories,” Algebra Logika 12(2), 125–142 (1973) [Algebra Logic 12, 67–77 (1973)].

    Article  MATH  Google Scholar 

  23. Yu. Ershov and S. Goncharov, “Elementary Theories and Their Constructive Models,” in Handbook of Recursive Mathematics, Vol. 1: Recursive Model Theory (Elsevier, Amsterdam, 1998), Stud. Logic Found. Math. 138, pp. 115–165.

    Google Scholar 

  24. A. T. Nurtazin, “Strong andWeak Constructivization and Computable Families,” Algebra Logika 13(3), 311–323 (1974) [Algebra Logic 13, 177–184 (1974)].

    MathSciNet  MATH  Google Scholar 

  25. L. Harrington, “Recursively Presentable Prime Models,” J. Symb. Logic 39, 305–309 (1974).

    Article  MathSciNet  MATH  Google Scholar 

  26. T. S. Millar, “Foundations of Recursive Model Theory,” Ann. Math. Logic 13(1), 45–72 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Chisholm, “Effective Model Theory vs. Recursive Model Theory,” J. Symb. Logic 55(3), 1168–1191 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  28. S. S. Goncharov, Countable Boolean Algebras and Decidability (Nauchnaya Kniga, Novosibirsk, 1996; Plenum, New York, 1997).

    MATH  Google Scholar 

  29. S. S. Goncharov and Yu. L. Ershov, Constructive Models (Nauchnaya Kniga, Novosibirsk, 1999); Engl. transl.: Yu. L. Ershov and S. S. Goncharov, Constructive Models (Consultants Bureau, New York, 2000).

    MATH  Google Scholar 

  30. S. S. Goncharov, “Autostability of Prime Models under Strong Constructivizations,” Algebra Logika 48(6), 729–740 (2009) [Algebra Logic 48, 410–417 (2009)].

    MathSciNet  Google Scholar 

  31. S. S. Goncharov, “On Autostability of Almost Prime Models Relative to Strong Constructivizations,” Usp. Mat. Nauk 65(5), 107–142 (2010) [Russ. Math. Surv. 65, 901–935 (2010)].

    MathSciNet  Google Scholar 

  32. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability (McGraw-Hill, Maidenhead, Berksh., 1967; Mir, Moscow, 1972).

    MATH  Google Scholar 

  33. C. C. Chang and H. J. Keisler, Model Theory (North-Holland, Amsterdam, 1973; Mir, Moscow, 1977).

    MATH  Google Scholar 

  34. A. I. Mal’cev, Algorithms and Recursive Functions (Nauka, Moscow, 1965; Wolters-Noordhoff, Groningen, 1970).

    Google Scholar 

  35. E. A. Palyutin, “The Algebras of Formulae of Countably Categorical Theories,” Colloq. Math. 31(2), 157–159 (1974).

    MathSciNet  MATH  Google Scholar 

  36. S. S. Goncharov and J. F. Knight, “Computable Structure and Non-structure Theorems,” Algebra Logika 41(6), 639–681 (2002) [Algebra Logic 41, 351–373 (2002)].

    MathSciNet  MATH  Google Scholar 

  37. Yu. L. Ershov, Decision Problems and Constructive Models (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  38. C. F. D. McCoy, “Δ02 -Categoricity in Boolean Algebras and Linear Orderings,” Ann. Pure Appl. Logic 119(1–3), 85–120 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  39. C. F. D. McCoy, “Partial Results in Δ 03 -Categoricity in Linear Orderings and Boolean Algebras,” Algebra Logika 41(5), 531–552 (2002) [Algebra Logic 41, 295–305 (2002)].

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State University and Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, pr. akademika Koptyuga 4, Novosibirsk, 630090, Russia

    S. S. Goncharov

Authors
  1. S. S. Goncharov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. S. Goncharov.

Additional information

Dedicated to the anniversary of Academician S.I. Adian

Original Russian Text © S.S. Goncharov, 2011, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2011, Vol. 274, pp. 119–129.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goncharov, S.S. Degrees of autostability relative to strong constructivizations. Proc. Steklov Inst. Math. 274, 105–115 (2011). https://doi.org/10.1134/S0081543811060071

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543811060071

Keywords

Search

Navigation