Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T06:05:33.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Prime models of computably enumerable degree

Published online by Cambridge University Press:  12 March 2014

Rachel Epstein
Rachel Epstein*
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA, E-mail: rachel@math.uchicago.edu, URL: http://www.math.uchicago.edu/~rachel/
Get access Rights & Permissions [Opens in a new window]

Abstract

We examine the computably enumerable (c.e.) degrees of prime models of complete atomic decidable (CAD) theories. A structure has degree d if d is the degree of its elementary diagram. We show that if a CAD theory T has a prime model of c.e. degree c, then T has a prime model of strictly lower c.e. degree b, where, in addition, b is low (b′ = 0′), This extends Csima's result that every CAD theory has a low prime model. We also prove a density result for c.e. degrees of prime models. In particular, if c and d are c.e. degrees with d < c and c not low2 (c″ > 0″), then for any CAD theory T, there exists a c.e. degree b with d < b < c such that T has a prime model of degree b, where b can be chosen so that b′ is any degree c.e. in c with d′ ≤ b′. As a corollary, we show that for any degree c with 0 < c < 0′, every CAD theory has a prime model of low c.e. degree incomparable with c. We show also that every CAD theory has prime models of low c.e. degree that form a minimal pair, extending another result of Csima. We then discuss how these results apply to homogeneous models.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 4 , December 2008 , pp. 1373 - 1388
Copyright
Copyright © Association for Symbolic Logic 2008

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

Csima, B. F. [2004], Degree spectra of prime models, this Journal, vol. 69, pp. 430–442.Google Scholar
Csima, B. F., Hirschfeldt, D. R., Knight, J. F., and Soare, R. I. [2004], Bounding prime models, this Journal, vol. 69, pp. 1117–1142.Google Scholar
Goncharov, S. S. [1978], Strong constructivizability of homogeneous models (Russian), Algebra i Logika, vol. 17, pp. 363–388; translated in Algebra and Logic, vol. 17 (1978), pp. 247–263.Google Scholar
Goncharov, S. S. and Nurtazin, A. T. [1973], Constructive models of complete decidable theories, Algebra and Logic, vol. 12, pp. 67–77, (English translation).CrossRefGoogle Scholar
Harrington, L. [1974], Recursively presentable prime models, this Journal, vol. 39, pp. 305–309.Google Scholar
Harrington, L. and Soare, R. I. [1992], Games in recursion theory and continuity properties of capping degrees, Set theory of the continuum (Judah, H., Just, W., and Woodin, H., editors), Springer-Verlag, New York, pp. 39–62.CrossRefGoogle Scholar
Hirschfeldt, D. R. [2006], Computable trees, prime models, and relative decidability, Proceedings of the American Mathematical Society, vol. 134, pp. 1495–1498.CrossRefGoogle Scholar
Knight, J. F. [1986], Degrees coded in jumps of orderings, this Journal, vol. 51, pp. 1034–1042.Google Scholar
Lachlan, A. H. [1966], Lower bounds for pairs of recursively enumerable degrees, Proceedings of the London Mathematical Society, vol. 16, pp. 537–769.CrossRefGoogle Scholar
Lange, K. [2008], The degree spectra of homogeneous models, this Journal, vol. 73, pp. 1009–1028.Google Scholar
Lange, K. [ip]. The reverse mathematics of homogeneous models, in preparation.Google Scholar
Martin, D. A. [1966], Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12, pp. 295–310.CrossRefGoogle Scholar