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autostability of prime models under strong constructivizations

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Algebra and Logic Aims and scope

We furnish an example of an Ehrenfeucht theory whose prime model is autostable under strong constructivizations and there exists a prime model in a finite expansion by constants that is nonautostable under strong constructivizations of the theory constructed.

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References

  1. A. Fröhlich and J. Shepherdson, “Effective procedures in field theory,” Philos. Trans. Roy. Soc. London, Ser. A, 248, 407-432 (1956).

    Article  MATH  MathSciNet  Google Scholar 

  2. A. I. Mal’tsev, “On recursive Abelian groups,” Dokl. Akad. Nauk SSSR, 146, No. 5, 1009-1012 (1962).

    MathSciNet  Google Scholar 

  3. A. I. Mal’tsev, “Constructive models. I,” Usp. Mat. Nauk, 16, No. 3, 3-60 (1961).

    Google Scholar 

  4. E. B. Fokina, “Arithmetical Turing degrees and categorical theories of computable models,” in Mathematical Logic in Asia, Proc. 9th Asian Logic Conf. (2006), pp. 58-69.

  5. S. S. Goncharov and Yu. L. Ershov, Constructive Models, Sib. School Alg. Log. [in Russian], Nauch. Kniga, Novosibirsk (1999).

  6. E. B. Fokina, “Spectra of computable models,” Vestnik NGU, Mat., Mekh., Inf., 6, No. 6, 69-73 (2006).

    Google Scholar 

  7. J. H. Schmerl, “A decidable ℵ0-categorical theory with a non-recursive Ryll-Nardzewski function,” Fund. Math., 98, No. 2, 121-125 (1978).

    MATH  MathSciNet  Google Scholar 

  8. A. Gavryushkin, “On complexity of Ehrenfeucht theories with computable model,” in Logical Approaches to Computational Barriers, Second Conf. Comput. Europe, Rep. Ser., Swansea Univ. (2006), pp. 105-108.

  9. A. N. Gavryushkin, “Complexity of Ehrenfeucht models,” Algebra Logika, 46, No. 5, 507-519 (2006).

    MathSciNet  Google Scholar 

  10. M. G. Peretyatkin, “Complete theories with finite number of countable models,” Algebra Logika, 12, No. 5, 550-576 (1973).

    Google Scholar 

  11. S. Goncharov and B. Khoussainov, “Open problems in the theory of constructive algebraic systems,” Cont. Math., 257, Am. Math. Soc., Providence, RI (2000), pp. 145-170.

  12. T. S. Millar, “Foundations of recursive model theory,” Ann. Math. Log., 13, No. 1, 45-72 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Millar, “Persistently finite theories with hyperarithmetic models,” Trans. Am. Math. Soc., 278, No. 1, 91-99 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  14. T. Millar, “Decidability and the number of countable models,” Ann. Pure Appl. Log., 27, No. 2, 137-153 (1984).

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Millar, “Decidable Ehrenfeucht theories,” Proc. Symp. Pure Math., 42 (1985), pp. 311-321.

    MathSciNet  Google Scholar 

  16. J. Knight, “Nonarithmetical ℵ0-categorical theories with recursive models,” J. Symb. Log., 59, No. 1, 106-112 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Reed, “A decidable Ehrenfeucht theory with exactly two hyperarithmetic models,” Ann. Pure Appl. Log., 53, No. 2, 135-168 (1991).

    Article  MATH  Google Scholar 

  18. C. Ash and T. Millar, “Persistently finite, persistently arithmetic theories,” Proc. Am. Math. Soc., 89, 487-492 (1983).

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Lerman and J. Scmerl, “Theories with recursive models,” J. Symb. Log., 44, No. 1, 59-76 (1979).

    Article  MATH  Google Scholar 

  20. S. Lempp and T. A. Slaman, “The complexity of the index sets of ℵ0-categorical theories and of Ehrenfeucht theories,” Cont. Math., 425, Am. Math. Soc., Providence, RI (2007), pp. 43-47.

  21. M. Morley, “Decidable models,” Israel J. Math., 25, 233-240 (1976).

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Vaught, “Denumerable models of complete theories,” in Infinitistic Methods, Proc. Symp. Found. Math., Pergamon, London (1961), pp. 303-321.

    Google Scholar 

  23. S. S. Goncharov, “Strong constructivizability of homogeneous models,” Algebra Logika, 17, No. 4, 363-388 (1978).

    MathSciNet  Google Scholar 

  24. M. G. Peretyatkin, “A strong constructivizability criterion for homogeneous models,” Algebra Logika, 17, No. 4, 436-454 (1978).

    MathSciNet  Google Scholar 

  25. A. N. Gavryushkin, “Spectra of computable models for Ehrenfeucht theories,” Algebra Logika, 46, No. 3, 275-289 (2007).

    MATH  MathSciNet  Google Scholar 

  26. A. N. Gavryushkin, “Constructive models of theories with linear Rudin–Keisler ordering” Vestnik NGU, Mat., Mekh., Inf., 9, No. 2, 30-37 (2009).

    Google Scholar 

  27. A. Gavryushkin, “Computable models spectra of Ehrenfeucht theories,” in Logic Coll. 2007, Uniw. Wroclawski (2007), pp. 46-47.

  28. A. Gavryushkin, “Computable models spectra of Ehrenfeucht theories,” in Logic and Theory of Algorithms, Fourth Conf. Comput. Europe, Univ. Athens (2008), pp. 50-51.

  29. H. Rogers, Theory of Recursive Functions and Effective Computability, McGraw-Hill, New York (1967).

    MATH  Google Scholar 

  30. C. C. Chang and H. J. Keisler, Model Theory, North-Holland, Amsterdam (1973).

    MATH  Google Scholar 

  31. A. T. Nurtazin, “Strong and weak constructivizations and computable families,” Algebra Logika, 13, No. 3, 311-323 (1974).

    MATH  MathSciNet  Google Scholar 

  32. E. A. Palyutin, “Algebras of formulas for countably categorical theories,” Coll. Math., 31, 157-159 (1974).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    S. S. Goncharov

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    S. S. Goncharov

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  1. S. S. Goncharov
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Correspondence to S. S. Goncharov.

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Supported by RFBR (project No. 08-01-00336) and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-335.2008.1).

Translated from Algebra i Logika, Vol. 48, No. 6, pp. 729-740, November-December, 2009.

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Goncharov, S.S. autostability of prime models under strong constructivizations. Algebra Logic 48, 410–417 (2009). https://doi.org/10.1007/s10469-009-9072-y

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  • DOI: https://doi.org/10.1007/s10469-009-9072-y

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