Abstract
We build a decidable structure \(\mathcal {M}\) such that \(\mathcal {M}\) is a prime model of the theory \(Th(\mathcal {M})\) and \(\mathcal {M}\) has no degree of autostability relative to strong constructivizations.
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Fröhlich, A., Shepherdson, J.C.: Effective procedures in field theory. Philos. Trans. Roy. Soc. London. Ser. A. 248, 407–432 (1956)
Mal’tsev, A.I.: Constructive algebras. I. Russ. Math. Surv. 16, 77–129 (1961)
Mal’tsev, A.I.: On recursive abelian groups. Sov. Math. Dokl. 32, 1431–1434 (1962)
Fokina, E.B., Kalimullin, I., Miller, R.: Degrees of categoricity of computable structures. Arch. Math. Logic. 49, 51–67 (2010)
Csima, B.F., Franklin, J.N.Y., Shore, R.A.: Degrees of categoricity and the hyperarithmetic hierarchy. Notre Dame J. Formal Logic. 54, 215–231 (2013)
Bazhenov, N.A.: Degrees of categoricity for superatomic Boolean algebras. Algebra Logic. 52, 179–187 (2013)
Anderson, B.A., Csima, B.F.: Degrees that are not degrees of categoricity. Notre Dame J. Formal Logic. (to appear)
Fokina, E., Frolov, A., Kalimullin, I.: Categoricity spectra for rigid structures. Notre Dame J. Formal Logic. (to appear)
Goncharov, S.S.: Degrees of autostability relative to strong constructivizations. Proc. Steklov Inst. Math. 274, 105–115 (2011)
Miller, R.: \(\mathbf{d}\)-computable categoricity for algebraic fields. J. Symb. Log. 74, 1325–1351 (2009)
Fokina, E.B., Harizanov, V., Melnikov, A.: Computable model theory. In: Downey, R. (ed.) Turing’s Legacy: Developments from Turing Ideas in Logic. Lecture Notes Logic, vol. 42, pp. 124–194. Cambridge University Press, Cambridge (2014)
Bazhenov, N.A.: Autostability spectra for Boolean algebras. Algebra Logic. 53, 502–505 (2014)
Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973)
Jockusch, C.G., Soare, R.I.: \(\Pi ^0_1\) classes and degrees of theories. Trans. Amer. Math. Soc. 173, 33–56 (1972)
Cenzer, D.: \(\Pi ^0_1\) classes in computability theory. In: Griffor, E.R. (ed.) Handbook of Computability Theory. Studies Logic Foundations Mathematics, vol. 140, pp. 37–85. Elsevier Science B.V., Amsterdam (1999)
Ash, C.J., Knight, J.F.: Computable Structures and the Hyperarithmetical Hierarchy. Elsevier Science B.V, Amsterdam (2000)
Ershov, Y.L., Goncharov, S.S.: Constructive Models. Kluwer Academic/Plenum Publishers, New York (2000)
Goncharov, S.S.: Countable Boolean Algebras and Decidability. Consultants Bureau, New York (1997)
Ershov, Y.L.: Decidability of the elementary theory of distributive lattices with relative complements and the theory of filters. Algebra Logic. 3, 17–38 (1964)
Steiner, R.M.: Effective algebraicity. Arch. Math. Logic. 52, 91–112 (2013)
Acknowledgements
The author is grateful to Sergey Goncharov and Svetlana Aleksandrova for fruitful discussions on the subject. This work was supported by RFBR (grant 14-01-00376), and by the Grants Council (under RF President) for State Aid of Leading Scientific Schools (grant NSh-860.2014.1).
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Bazhenov, N. (2015). Prime Model with No Degree of Autostability Relative to Strong Constructivizations. In: Beckmann, A., Mitrana, V., Soskova, M. (eds) Evolving Computability. CiE 2015. Lecture Notes in Computer Science(), vol 9136. Springer, Cham. https://doi.org/10.1007/978-3-319-20028-6_12
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DOI: https://doi.org/10.1007/978-3-319-20028-6_12
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