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Spectra of Atomic Theories

Published online by Cambridge University Press:  12 March 2014

Uri Andrews  and
Julia F. Knight
Uri Andrews
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706-1388, USA, E-mail: [email protected]
Julia F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA, E-mail: [email protected]
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Abstract

For a countable structure , the spectrum is the set of Turing degrees of isomorphic copies of . For a complete elementary first order theory T, the spectrum is the set of Turing degrees of models of T. We answer a question from [1] by showing that there is an atomic theory T whose spectrum does not match the spectrum of any structure.

Keywords

Primary 03C57
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 4 , December 2013 , pp. 1189 - 1198
Copyright
Copyright © Association for Symbolic Logic 2009

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References

REFERENCES

[1] Andrews, U. and Miller, J. S., Spectra of theories and structures, Proceedings of the American Mathematical Society , to appear.Google Scholar
[2] Ash, C. J. and Knight, J. F., Computable structures and the hyperarithmetical hierarchy, Elsevier Science, 2000.Google Scholar
[3] Csima, B. F. and Kalimullin, I. Sh., Degree spectra and immunity properties, Mathematical Logic Quarterly, vol. 56 (2010), pp. 67–77.CrossRefGoogle Scholar
[4] Diamondstone, D., Greenberg, N., and Turetsky, D., Inherent enumerability of strong jump-traceability, pre-print.Google Scholar
[5] Goncharov, S. S., Harizanov, V. S., Laskowski, M. C., Lempp, S., and McCoy, C. F. D., Trivial strongly minimal theories are model complete after naming constants, Proceedings ofthe American Mathematical Society, vol. 131 (2003), pp. 3901–3912.CrossRefGoogle Scholar
[6] Greenberg, N., Montalbán, A., and Slaman, T. A., Relative to any non-hyperarithmetic set, pre-print.Google Scholar
[7] Khisamiev, N. G., Constructive Abelian p-groups, Siberian Advances in Mathematics, vol. 2 (1992), pp. 68–113, (English translation).Google Scholar
[8] Khisamiev, N. G., Constructive abelian groups, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., Remmel, J. B., and Marek, V. W., editors), vol. 2, Elsevier, 1998, pp. 1177–1231.Google Scholar
[9] Knight, J. F., Nonarithmetical ℵ0-categorical theories with recursive models, this Journal, vol. 59 (1994), pp. 106–112.Google Scholar
[10] Knight, J. F., Degrees of models, Handbook of recursive mathematics (Ershov, Yu. L., Goncharov, S. S., Nerode, A., Remmel, J. B., and Marek, V. W., editors), vol. I, Elsevier, 1998, pp. 289–309.Google Scholar
[11] Knight, J. F., True approximations and models of arithmetic, Models and computability (Cooper, B. and Truss, J., editors), Cambridge University Press, 1999, pp. 255–278.CrossRefGoogle Scholar
[12] Lachlan, A. H. and Soare, R. I., Models of arithmetic and upper bounds for arithmetic sets, this Journal, vol. 59 (1994), pp. 977–983.Google Scholar
[13] Lachlan, A. H. and Soare, R. I., Models of arithmetic and subuniform bounds for the arithmetic sets, this Journal, vol. 63 (1998), pp. 59–72.Google Scholar
[14] Lerman, M. and Schmerl, J., Theories with recursive models, this Journal, vol. 44 (1979), pp. 59–76.Google Scholar
[15] Macintyre, A. and Marker, D., Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), pp. 539–554.CrossRefGoogle Scholar
[16] Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Proceedings of Symposia in Pure Mathematics, vol. 5 (1962), pp. 117–121.CrossRefGoogle Scholar
[17] Slaman, T. A., Relative to any nonrecursive set, Proceedings ofthe American Mathematical Society, vol. 126 (1998), pp. 2117–2122.CrossRefGoogle Scholar
[18] Solovay, R. M., Degrees of models of true arithmetic, unpublished manuscript, 1983.Google Scholar
[19] Solovay, R. M., personal correspondence, 1991.Google Scholar
[20] Wehner, S., Enumerations, countable structures, and Turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131–2139.CrossRefGoogle Scholar