Journal of Symbolic Logic

Degree spectra of prime models

Barbara F. Csima

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Abstract

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

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

Dates
First available in Project Euclid: 19 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1082418536

Digital Object Identifier
doi:10.2178/jsl/1082418536

Mathematical Reviews number (MathSciNet)
MR2058182

Zentralblatt MATH identifier
1069.03025

Citation

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


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References

  • 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.