Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-31T19:33:10.928Z Has data issue: false hasContentIssue false
  • English
  • Français

Models of arithmetic and upper bounds for arithmetic sets

Published online by Cambridge University Press:  12 March 2014

Alistair H. Lachlan  and
Robert I. Soare
Alistair H. Lachlan
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, British Columbia, CanadaV5A 1S6, E-mail: [email protected]
Robert I. Soare
Affiliation:
Department of Mathematics, University of Chicago, Chicago, Illinois 60637, E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We settle a question in the literature about degrees of models of true arithmetic and upper bounds for the arithmetic sets. We prove that there is a model of true arithmetic whose degree is not a uniform upper bound for the arithmetic sets. The proof involves two forcing constructions.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 977 - 983
Copyright
Copyright © Association for Symbolic Logic 1994

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]Knight, J. F., Degrees of models with prescribed Scott set, Proceedings of the U.S.-Israel Workshop on Model Theory in Mathematical Logic: Classification Theory, Chicago, December 15–19, 1985 (John Baldwin, editor), Lecture Notes in Mathematics, vol. 1292. Springer-Verlag, Berlin. Heidelberg, and New York, 1987, pp. 182–191.Google Scholar
[2]Knight, J. F., A metatheorem for constructions by finitely many workers, this Journal, vol. 55 (1990), pp. 787–804.Google Scholar
[3]Knight, J., Lachlan, A. H., and Soare, R. I., Two Theorems On Degrees Of Models Of Arithmetic, this Journal, vol. 49 (1984), pp. 425–436.Google Scholar
[4]Lerman, M., Upper bounds for the arithmetical degrees, Annals of Pure and Applied Logic, vol. 29 (1985), pp. 225–253.CrossRefGoogle Scholar
[5]Macintyre, A. and Marker, D., Degrees of recursively saturated models, Transactions of the American Mathematical Society, vol. 282 (1984), pp. 539–554.CrossRefGoogle Scholar
[6]Marker, D., Degrees of models of true arithmetic, Proceedings of the Herbrand Symposium: Logic Colloquium, 1981 (J. Stern, editor), North-Holland, Amsterdam, 1982, pp. 233–242.Google Scholar
[7]Scott, D., Algebras of sets binumerable in complete extensions of arithmetic, Recursive Function Theory, Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, RI, 1961, pp. 117–121.Google Scholar
[8]Solovay, R. M., Degrees of models of true arithmetic, preliminary version, unpublished manuscript (1983).Google Scholar