Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-16T17:01:36.719Z Has data issue: false hasContentIssue false
  • English
  • Français

Modal analysis of generalized rosser sentences

Published online by Cambridge University Press:  12 March 2014

Vítězslav Švejdar
Vítězslav Švejdar*
Affiliation:
Mathematical Institute, Čsav, Zitná 25, 115 67 Praha, Czechoslovakia
Get access Rights & Permissions [Opens in a new window]

Abstract

A modal theory Z using the Guaspari witness comparison signs ≤, < is developed. The theory Z is similar to, but weaker than, the theory R of Guaspari and Solovay. Nevertheless, Z proves the independence of the Rosser fixed-point. A Kripke semantics for Z is presented and some arithmetical interpretations of Z are investigated. Then Z is enriched to ZI by adding a new modality sign for interpretability and by axioms expressing some facts about interpretability of theories. Two arithmetical interpretations of ZI are presented. The proofs of the validity of the axioms of ZI in arithmetical interpretations use some strengthening of Solovay's result about interpretability in Gödel-Bernays set theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 48 , Issue 4 , December 1983 , pp. 986 - 999
Copyright
Copyright © Association for Symbolic Logic 1983

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]Feferman, S., Arithmetization of metamathematics in a general setting, Fundamenta Mathematicae, vol. 49 (1960), pp. 35–92.CrossRefGoogle Scholar
[2]Guaspari, D., Partially conservative extensions of arithmetic, Transactions of the American Mathematical Society, vol. 254 (1979), pp. 41–68.CrossRefGoogle Scholar
[3]Guaspari, D. and Solovay, R.N., Rosser sentences, Annals of Mathematical Logic, vol. 16 (1979), pp. 81–99.CrossRefGoogle Scholar
[4]Hájek, P., Interpretability in theories containing arithmetic. II, Commentationes Mathematicae Universitatis Carolinae, vol. 22 (1981), pp. 667–688.Google Scholar
[5]Lindström, P., Some results on interpretability, Proceedings of the 5th Scandinavian Logic Symposium (Jensen, F.V., Mayoh, B.H., Moller, K.K., editors), Aalborg University Press, Aalborg, 1979.Google Scholar
[6]Shoenfield, J.R., Mathematical Logic, Addison-Wesley, Reading, Mass., 1967.Google Scholar
[7]Smoryński, C., Self-reference and modal logic, mimeographed lecture notes, Warsaw, 1980.Google Scholar
[8]Smoryński, C., A ubiquitous fixed-point calculationInterpretability (Szczerba, L. and Prazmowski, K., editors), Bialystok (to appear).Google Scholar
[9]Solovay, R., On interpretability in set theories (to appear).Google Scholar
[10]Švejdar, V., Degrees of interpretability, Commentationes Mathematicae Universitatis Carolinae, vol. 19 (1978), pp. 789–813.Google Scholar
[11]Tarski, A., Mostowski, A. and Robinson, R.M., Undecldable theories, North-Holland, Amsterdam, 1953.Google Scholar