Advertisement

Archive for Mathematical Logic

, Volume 56, Issue 5–6, pp 555–584 | Cite as

Interpretability suprema in Peano Arithmetic

  • Paula HenkEmail author
  • Albert Visser
Open Access
Article
  • 161 Downloads

Abstract

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic \(\mathsf {ILM}\) of Peano Arithmetic (\(\mathsf {PA}\)). It is well-known that any theories extending \(\mathsf {PA}\) have a supremum in the interpretability ordering. While provable in \(\mathsf {PA}\), this fact is not reflected in the theorems of the modal system \(\mathsf {ILM}\), due to limited expressive power. Our goal is to enrich the language of \(\mathsf {ILM}\) by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a (nonstandard) provability predicate satisfying the axioms of the provability logic \(\mathsf {GL}\).

Keywords

Provability logic Interpretability Peano Arithmetic 

Mathematics Subject Classification

03F45 03F25 03F30 

References

  1. 1.
    Berarducci, A.: The interpretability logic of Peano arithmetic. J. Symb. Logic 55, 1059–1089 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Boolos, G.: The Logic of Provability. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  3. 3.
    Ehrenfeucht, A., Feferman, S.: Representability op recursively enumerable sets in formal theories. Archiv für mathematische Logik und Grundlagenforschung 5(1), 37–41 (1961). doi: 10.1007/BF01977641 zbMATHGoogle Scholar
  4. 4.
    Feferman, S.: Arithmetization of metamathematics in a general setting. Fundam. Math. 49, 35–92 (1960)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Guaspari, D.: Partially conservative extensions of arithmetic. Trans. Am. Math. Soc. 254, 47–68 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hájek, P.: On interpretability in set theories I. Commun. Math. Univ. Carolinae 12, 73–79 (1971)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hájek, P., Pudlák, P.: Metamathematics of First-Order Arithmetic. Springer, Berlin (1998)zbMATHGoogle Scholar
  8. 8.
    Hájková, M., Hájek, P.: On interpretability in theories containing arithmetic. Fundam. Math. 76(2), 131–137 (1972). http://eudml.org/doc/214455
  9. 9.
    Henk, P., Pakhomov, F.: Slow and Ordinary Provability for Peano Arithmetic. ArXiv e-prints (2016)Google Scholar
  10. 10.
    Henk, P., Shavrukov, V. Yu.: A Solovay function for the least 1-inconsistent subtheory of PA. Prepublication Series. Report No. PP-2016-31. ILLC, University of Amsterdam (2016)Google Scholar
  11. 11.
    Hilbert, D., Bernays, P.: Grundlagen der Mathematik II. Springer, Berlin (1939). Second edition: 1970Google Scholar
  12. 12.
    Jeroslow, R.G.: Consistency statements in formal theories. Fundam. Math. 72, 17–40 (1971)MathSciNetzbMATHGoogle Scholar
  13. 13.
    de Jongh, D., Jumelet, M., Montagna, F.: On the proof of Solovay’s theorem. Stud. Log. 50, 51–70 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    de Jongh, D., Veltman, F.: Provability logics for relative interpretability. In: Mathematical logic, Proceedings of the Heyting 1988 summer school in Varna, Bulgaria, pp. 31–42. Plenum Press, Boston (1990)Google Scholar
  15. 15.
    Lindström, P.: On certain lattices of degrees of interpretability. Notre Dame J. Formal Logic 25(2), 127–140 (1984). doi: 10.1305/ndjfl/1093870573 MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lindström, P.: Some results on interpretability. In: F. Jensen, M. B.H., M. K.K. (eds.) Proceedings of the 5th Scandinavian Logic Symposium 1979, pp. 329–361. Aalborg University Press, Aalborg (1979)Google Scholar
  17. 17.
    Lindström, P.: On Parikh provability—an exercise in modal logic. In: H. Lagerlund, S. Lindström, R. Sliwinski (eds.) Modality Matters: Twenty-Five Essays in Honour of Krister Segerberg, Uppsala Philosophical Studies 53, pp. 279–288 (2006)Google Scholar
  18. 18.
    Löb, M.: Solution of a problem of Leon Henkin. J. Symb. Logic 20, 115–118 (1955)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Montagna, F.: On the algebraization of a Feferman’s predicate. Stud. Log. 37, 221–236 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Montague, R.: Theories incomparable with respect to relative interpretability. J. Symb. Logic 27(2), 195–211 (1962). http://projecteuclid.org/euclid.jsl/1183734432
  21. 21.
    Ono, H.: Reflection principles in fragments of Peano arithmetic. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 33, 317–333 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Orey, S.: Relative interpretations. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 7, 146–153 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Parikh, R.: Existence and feasibility in arithmetic. J. Symb. Logic 36(3), 494–508 (1971). http://www.jstor.org/stable/2269958
  24. 24.
    Shavrukov, V.: The logic of relative interpretability over Peano arithmetic (in Russian). Tech. Rep. No.5, Steklov Mathematical Institute, Moscow (1988)Google Scholar
  25. 25.
    Shavrukov, V.: A smart child of Peano’s. Notre Dame J. Formal Logic 35, 161–185 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Smoryński, C.: Arithmetic analogues of McAloon’s unique Rosser sentences. Arch. Math. Logic 28, 1–21 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Solovay, R.: Provability interpretations of modal logic. Israel J. Math. 25, 287–304 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Švejdar, V.: Degrees of interpretability. Comment. Math. Univ. Carolinae 19, 789–813 (1978)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Tarski, A., Mostowski, A., Robinson, R.: Undecidable Theories. North-Holland, Amsterdam (1953)zbMATHGoogle Scholar
  30. 30.
    Visser, A.: Peano’s smart children: A provability logical study of systems with built-in consistency. Notre Dame J. Formal Logic 30, 161–196 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Visser, A.: An inside view of EXP. J. Symb. Logic 57, 131–165 (1992)CrossRefzbMATHGoogle Scholar
  32. 32.
    Visser, A.: An Overview of Interpretability Logic. In: M. Kracht, M. de Rijke, H. Wansing, M. Zakharyaschev (eds.) Advances in Modal Logic, vol 1, CSLI Lecture Notes, no. 87, pp. 307–359. Center for the Study of Language and Information, Stanford (1998)Google Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Institute for Logic, Language, and InformationUniversity of AmsterdamAmsterdamThe Netherlands
  2. 2.Philosophy, Faculty of HumanitiesUtrecht UniversityUtrechtThe Netherlands

Personalised recommendations