Skip to main content
SpringerLink
Log in

The polytopologies of transfinite provability logic

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract

Provability logics are modal or polymodal systems designed for modeling the behavior of Gödel’s provability predicate and its natural extensions. If Λ is any ordinal, the Gödel-Löb calculus GLP Λ contains one modality [λ] for each λ < Λ, representing provability predicates of increasing strength. GLP ω has no non-trivial Kripke frames, but it is sound and complete for its topological semantics, as was shown by Icard for the variable-free fragment and more recently by Beklemishev and Gabelaia for the full logic. In this paper we generalize Beklemishev and Gabelaia’s result to GLP Λ for countable Λ. We also introduce provability ambiances, which are topological models where valuations of formulas are restricted. With this we show completeness of GLP Λ for the class of provability ambiances based on Icard polytopologies.

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. Abashidze, M.: Ordinal completeness of the Gödel-Löb modal system. In: Intensional logics and the logical structure of theories, pp. 49–73. Tbilisi (1985, in Russian)

  2. Bagaria, J., Magidor, M., Sakai, H.: Reflection and indescribability in the constructible universe. Israel J. Math. (2013). Accepted for publication

  3. Beklemishev L.D.: Provability algebras and proof-theoretic ordinals, I. Ann. Pure Appl. Log. 128, 103–124 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Beklemishev L.D.: Kripke semantics for provability logic GLP. Ann. Pure Appl. Log. 161, 756–774 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Beklemishev, L.D.: Ordinal completeness of bimodal provability logic GLP. In: Proceedings of the 8th International Tbilisi conference on Logic, Language, and Computation, TbiLLC’09, pp. 1–15. Springer, Berlin, Heidelberg (2011)

  6. Beklemishev, L.D., Bezhanishvili, G., Icard, T.F.: On topological models of GLP. In: Schindler, R. (ed.) Ways of Proof Theory, pp. 133–152. Ontos Verlag, Heusendamm bei Frankfurt, Germany (2010)

  7. Beklemishev, L.D., Fernández-Duque, D., Joosten, J.J.: On provability logics with linearly ordered modalities. Studia Logica (2013). doi:10.1007/s11225-013-9490-7

  8. Beklemishev, L.D., Gabelaia, D.: Topological interpretations of provability logic. ArXiv, 1210.7317 [math.LO] (2012)

  9. Beklemishev L.D., Gabelaia D.: Topological completeness of the provability logic GLP. Ann. Pure Appl. Log. 164(12), 1201–1223 (2013)

    Article  MathSciNet  Google Scholar 

  10. Bezhanishvili G., Esakia L., Gabelaia D.: Some results on modal axiomatization and definability for topological spaces. Stud. Log. 81(3), 325–355 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bezhanishvili G., Morandi P.J.: Scattered and hereditarily irresolvable spaces in modal logic. Arch. Math. Log. 49(3), 343–365 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  12. Blass A.: Infinitary combinatorics and modal logic. J. Symb. Log. 55(2), 761–778 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fernández-Duque D., Joosten J.J.: Models of transfinite provability logic. J. Symb. Log. 78(2), 543–561 (2011)

    Article  Google Scholar 

  14. Fernández-Duque, D., Joosten, J.J.: Well-orders in the transfinite Japaridze algebra. ArXiv, 1212.3468 [math.LO] (2011)

  15. Fernández-Duque D., Joosten J.J.: Hyperations, Veblen progressions, and transfinite iteration of ordinal functions. Ann. Pure Appl. Log. 164(7–8), 785–801 (2013)

    Article  MATH  Google Scholar 

  16. Icard T.F.: A topological study of the closed fragment of GLP. J. Log. Comput. 21(4), 683–696 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Icard T.F., Joosten J.J.: Provability and interpretability logics with restricted realizations. Notre Dame J. Formal Log. 53(2), 133–154 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ignatiev K.N.: On strong provability predicates and the associated modal logics. J. Symb. Log. 58(1), 249–290 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  19. Japaridze, G.: The polymodal provability logic. In: Intensional Logics and the Logical Structure of Theories: Material from the Fourth Soviet-Finnish Symposium on Logic, pp. 16–48. Telavi (1988, in Russian)

  20. Joosten, J.J.: Interpretabilty formalized. PhD Tesis (2004)

  21. Pohlers W.: Proof Theory, The First Step into Impredicativity. Springer, Berlin (2009)

    MATH  Google Scholar 

  22. Segerberg, K.: An essay in classical modal logic. Uppsala Philosophical Studies, pp. 1–3 (1971)

  23. Solovay R.M.: Provability interpretations of modal logic. Israel J. Math. 28(3–4), 33–71 (1976)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Instituto Tecnológico Autónomo de México, Mexico City, Mexico

    David Fernández-Duque

Authors
  1. David Fernández-Duque
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Fernández-Duque.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Fernández-Duque, D. The polytopologies of transfinite provability logic. Arch. Math. Logic 53, 385–431 (2014). https://doi.org/10.1007/s00153-014-0371-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-014-0371-1

Keywords

Mathematics Subject Classification

Search

Navigation