Skip to main content
SpringerLink
Log in

Generic generalized Rosser fixed points

  • Published:
Studia Logica Aims and scope Submit manuscript

Abstract

To the standard propositional modal system of provability logic constants are added to account for the arithmetical fixed points introduced by Bernardi-Montagna in [5]. With that interpretation in mind, a system LR of modal propositional logic is axiomatized, a modal completeness theorem is established for LR and, after that, a uniform arithmetical (Solovay-type) completeness theorem with respect to PA is obtained for LR.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. N. Artyomov, Arithmetically complete modal theories, (Russian), Semiotics and Information Science no. 14 (Russian) 1980, pp. 115–133. Akad. Nauk SSSE, Vsesojuz, Inst. Naucn. i Tehn. Informatcii, Moscow.

    Google Scholar 

  2. C. Bernardi, The fixed point theorem for diagonizable algebras, Studia Logica 34 (1975), pp. 239–251.

    Google Scholar 

  3. C. Bernardi, The uniqueness of the fixed point in every diagonizable algebra, Studia Logica 35 (1976), pp. 335–343.

    Google Scholar 

  4. C. Bernardi and M. Mirolli, The hyperdiagonizable algebras, to appear in Algebra Universalis.

  5. C. Bernardi, and F. Montagna, Equivalence relations induced by extensional formulae: Classification by means of a new fixed point property, Fundamenta Mathematicae 124 (1984), pp. 221–233.

    Google Scholar 

  6. G. Boolos, The Unprovability of Consistency, Cambridge University Press, Cambridge 1979.

    Google Scholar 

  7. G. Boolos, Extremely undecidable sentences, Journal of Symbolic Logic 47 (1982), pp. 191–196.

    Google Scholar 

  8. D. H. J. de Jongh, A simplification of a completeness proof of Guaspari and Solovay, this volume pp. 187–192.

  9. D. Guaspari and R. M. Solovay, Rosser Sentences, Annals of Mathematical Logic 16 (1979), pp. 81–99.

    Google Scholar 

  10. F. Montagna, On the diagonizable algebra of Peano arithmetic, Bolletino Unione Matematica Italiana (5) 16-B (1979), pp. 795–812.

    Google Scholar 

  11. F. Montagna, Relatively precomplete numerations and arithmetic, Journal of Philosophical Logic 11 (1982), pp. 419–430.

    Google Scholar 

  12. G. Sambin, An effective fixed point theorem in intuitionistic diagonizable algebra, Studia Logica 35 (1976), pp. 345–361.

    Google Scholar 

  13. C. Smorynski, The Incompleteness Theorems, Handbook of Mathematical Logic (J. Barwise, ed.), North-Holland, Amsterdam (1977), pp. 827–865.

    Google Scholar 

  14. C. Smorynski, Beth's Theorem and self-referential sentences, Logic Colloquium 1977, (A. Macintyre, L. Pacholski, J. Paris, eds.), North-Holland, Amsterdam (1978).

    Google Scholar 

  15. C. Smorynski, Calculating self-referential statements, Fundamenta Mathematicae 109 (1980), pp. 189–210.

    Google Scholar 

  16. C. Smorynski, Self-Reference and Modal Logic, Springer-Verlag, New York 1985.

    Google Scholar 

  17. R. M. Solovay, Provability interpretations of modal logics, Israel Journal of Mathematics 25 (1976), pp. 287–304.

    Google Scholar 

  18. A. Visser, Aspects of Diagonalization and Provability, Dissertation, Utrecht (1981).

  19. A. Visser, The provability logics of recursively enumerable theories extending Peano arithmetic at arbitrary theories extending Peano arithmetic, Journal of Philosophical Logic 13 (1984).

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Universiteit van Amsterdam, Roetersstraat 15, 1018, WB Amsterdam

    Dick H. J. de Jongh & Franco Montagna

  2. Dipartimento di Matematico, Universita di Siena, Via Del Capitano 15, 53100, Siena

    Dick H. J. de Jongh & Franco Montagna

Authors
  1. Dick H. J. de Jongh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Franco Montagna
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

This paper supersedes: Franco Montagna, Extremely undecidable sentences and generic generalized Rosser's fixed points, Rapporto Matematico, No. 95, Siena, 1983.

Rights and permissions

Reprints and permissions

About this article

Cite this article

de Jongh, D.H.J., Montagna, F. Generic generalized Rosser fixed points. Stud Logica 46, 193–203 (1987). https://doi.org/10.1007/BF00370381

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00370381

Keywords

Search

Navigation