Hostname: page-component-76fb5796d-5g6vh Total loading time: 0 Render date: 2024-04-30T07:02:55.069Z Has data issue: false hasContentIssue false
  • English
  • Français

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: andrews@math.wisc.edu
Julia F. Knight
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, USA, E-mail: knight.1@nd.edu
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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