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A Computable Structure with Non-Standard Computability

Abstract

We find an example of a computable admissible set whose level of computability is higher than that of the standard model of Peano arithmetic. As a byproduct, we construct a 1-decidable model of an undecidable submodel complete theory.

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References

  1. 1.

    R. R. Avdeev, “On the admissible sets of type HYP(M) over recursively saturated models,” Sib. Matem. Zh. 52, 1199 (2011) [Siberian Math. J.52, 951 (2011)].

    Google Scholar 

  2. 2.

    J. Barwise, Admissible Sets and Structures (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1975).

    Book  MATH  Google Scholar 

  3. 3.

    C. C. Chang and H. J. Keisler, Model Theory (North-Holland, Amsterdam, 1990).

    MATH  Google Scholar 

  4. 4.

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

    MATH  Google Scholar 

  5. 5.

    Yu. L. Ershov, “Σ-definability in admissible sets,” Dokl. Akad. Nauk SSSR 285, 792 (1985) [Soviet Math., Dokl. 32, 767 (1985)].

    MathSciNet  Google Scholar 

  6. 6.

    Yu. L. Ershov, Definability and Computability (Nauchnaya Kniga, Novosibirsk, 1996) [Definability and Computability (Consultants Bureau, New York, 1996)].

    Google Scholar 

  7. 7.

    Yu. L. Ershov and S. S. Goncharov, Constructive Models (Nauchnaya Kniga, Novosibirsk, 1999) [Constructive Models (Consultants Bureau, New York, 2000)].

    Book  MATH  Google Scholar 

  8. 8.

    Yu. L. Ershov, V. G. Puzarenko, and A. I. Stukachev, “HF-computability,” in Computability in Context. Computation and Logic in the Real World, 169 (Imperial College Press, London, 2011).

    MATH  Google Scholar 

  9. 9.

    S. S. Goncharov, Countable Boolean Algebras and Decidability (Nauchnaya Kniga, Novosibirsk, 1996) [Countable Boolean Algebras and Decidability (Consultants Bureau, New York, 1997)].

    Google Scholar 

  10. 10.

    M. Harrison-Trainor, A. Melnikov, R. Miller, and A. Montalb'an, “Computable functors and effective inter-pretability,” J. Symbolic Logic 82, 77 (2017).

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    A. N. Khisamiev, “On the Ershov upper semilattice ♠E,” Sib. Matem. Zh. 45, 211 (2004) [Siberian Math. J. 45, 173 (2004)].

    MathSciNet  MATH  Google Scholar 

  12. 12.

    O. Melnikov, R. I. Tyshkevich, V. A. Yemelichev, and V. I. Sarvanov, Lectures on Graph Theory (Nauka, Moscow, 1990) [Lectures on Graph Theory (B. I. Wissenschaftsverlag, Mannheim, 1994)].

    MATH  Google Scholar 

  13. 13.

    V. G. Puzarenko, “Computability overmodels of decidable theories,” Algebra ilogika 39, 170 (2000) [Algebra and Logic 39, 98 (2000)].

    MathSciNet  Google Scholar 

  14. 14.

    V. G. Puzarenko, “Generalized numerations and definability of the field R in admissible sets,” Vestnik Novosibirsk. Gos. Univ., Ser. Mat. Mekh. Inform. 3,no. 2, 107 (2003) [in Russian].

    MATH  Google Scholar 

  15. 15.

    V. G. Puzarenko, “Computability in special models,” Sib. Matem. Zh. 46, 185 (2005) [Siberian Math. J. 46, 148 (2005)].

    MathSciNet  MATH  Google Scholar 

  16. 16.

    V. G. Puzarenko, “A certain reducibility on admissible sets,” Sib. Matem. Zh. 50, 414 (2009) [Siberian Math. J. 50, 330 (2009)].

    MathSciNet  MATH  Google Scholar 

  17. 17.

    A. I. Stukachev, “Degrees of presentability of structures. I,” Algebra i logika 46, 763 (2007) [Algebra and Logic 46, 419 (2007)].

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to V. G. Puzarenko.

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Russian Text © The Author(s), 2018, published in Matematicheskie Trudy, 2018, Vol. 21, No. 2, pp. 3–60.

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Avdeev, R.R., Puzarenko, V.G. A Computable Structure with Non-Standard Computability. Sib. Adv. Math. 29, 77–115 (2019). https://doi.org/10.3103/S1055134419020019

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Keywords

  • admissible set
  • hyperadmissible set
  • hereditarily finite superstructure
  • recursively saturated model
  • computable model
  • decidable model
  • Σ-reducibility
  • Σ-definability