Journal of Symbolic Logic

Degree spectra of prime models

Barbara F. Csima

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider the Turing degrees of prime models of complete decidable theories. In particular we show that every complete decidable atomic theory has a prime model whose elementary diagram is low. We combine the construction used in the proof with other constructions to show that complete decidable atomic theories have low prime models with added properties.

If we have a complete decidable atomic theory with all types of the theory computable, we show that for every degree d with 0 < d0’, there is a prime model with elementary diagram of degree d. Indeed, this is a corollary of the fact that if T is a complete decidable theory and L is a computable set of c.e. partial types of T, then for any Δ02 degree d > 0, T has a d-decidable model omitting the nonprincipal types listed by L.

Article information

J. Symbolic Logic, Volume 69, Issue 2 (2004), 430-442.

First available in Project Euclid: 19 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Csima, Barbara F. Degree spectra of prime models. J. Symbolic Logic 69 (2004), no. 2, 430--442. doi:10.2178/jsl/1082418536.

Export citation


  • C. C. Chang and H. J. Keisler Model theory, 3rd ed., Studies in Logic and the Foundations of Mathematics, vol. 73, North-Holland, Amsterdam,1990.
  • P. Clote A note on decidable model theory, Model theory and arithmetic, Lecture Notes in Mathematics, vol. 890, Springer, Berlin,1981, pp. 134--142.
  • B. F. Csima, D. R. Hirschfeldt, J. F. Knight, and R. I. Soare Bounding prime models, to appear.
  • A. S. Denisov Homogeneous $\bf 0'$-elements in structural pre-orders, Algebra and Logic, vol. 28 (1989), pp. 405--418.
  • B. N. Drobotun Numerations of simple models, Siberian Mathematical Journal, vol. 18 (1977), no. 5, pp. 707--716 (1978).
  • V. S. Harizanov Pure computable model theory, Handbook of recursive mathematics (Yu. L. Ershov, S. S. Goncharov, A. Nerode, and J. B. Remmel, editors), Studies in Logic and the Foundations of Mathematics, vol. 138--139, Elsevier Science, Amsterdam,1998, pp. 3--114.
  • C. G. Jockusch, Jr. and R. I. Soare $\Pi \sp0\sb1$ classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33--56.
  • J. F. Knight Degrees coded in jumps of orderings, Journal of Symbolic Logic, vol. 51 (1986), pp. 1034--1042.
  • D. Marker Model theory, Springer-Verlag, New York,2002.
  • T. S. Millar Foundations of recursive model theory, Annals of Mathematical Logic, vol. 13 (1978), pp. 45--72.
  • R. G. Miller The $\Delta\sp 0\sb 2$-spectrum of a linear order, Journal of Symbolic Logic, vol. 66 (2001), pp. 470--486.
  • T. Slaman Relative to any nonrecursive set, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2117--2122.
  • R. I. Soare Recursively enumerable sets and degrees: A Study of Computable Functions and Computably Generated Sets, Springer-Verlag, Heidelberg,1987.
  • S. Wehner Enumerations, countable structures and Turing degrees, Proceedings of the American Mathematical Society, vol. 126 (1998), pp. 2131--2139.