Hostname: page-component-848d4c4894-x24gv Total loading time: 0 Render date: 2024-05-10T18:51:30.448Z Has data issue: false hasContentIssue false
  • English
  • Français

Interpretability over peano arithmetic

Published online by Cambridge University Press:  12 March 2014

Claes Strannegård
Claes Strannegård*
Affiliation:
Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 Cs Utrecht, The Netherlands, E-mail: claes.strannegard@phil.uu.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 64 , Issue 4 , December 1999 , pp. 1407 - 1425
Copyright
Copyright © Association for Symbolic Logic 1999

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]Berarducci, A., The interpretability logic of Peano arithmetic, this Journal, vol. 55 (1990), pp. 1059–1089.Google Scholar
[2]de Jongh, D. and Veltman, F., Provability logics for relative interpretability, [8], 1990, pp. 31–42.CrossRefGoogle Scholar
[3]Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1991.Google Scholar
[4]Japartdze, G. and de Jongh, D., The logic of provability, Handbook of proof theory (Buss, S., editor), Elsevier, 1998.Google Scholar
[5]Kaye, R., Models of Peano arithmetic, Oxford Logic Guides 15, Clarendon Press, Oxford, 1991.CrossRefGoogle Scholar
[6]Magari, R., The diagonalizable algebras, Studio Logica, vol. 34 (1975), pp. 305–313.CrossRefGoogle Scholar
[7]Orey, S., Relative interpretations, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 7 (1961), pp. 146–153.CrossRefGoogle Scholar
[8]Petkov, P. (editor), Mathematical logic, Proceedings of the Heyting 1988 summer school in Varna, Bulgaria, Boston, Plenum Press, 1990.Google Scholar
[9]Shavrukov, V., The logic of relative interpretability over Peano arithmetic, Technical Report 5, Stekhlov Mathematical Institute, Moscow, 1988, in Russian.Google Scholar
[10]Shavrukov, V., A note on the diagonalizable algebras of P A and ZF, Annals of Pure and Applied Logic, vol. 61 (1993), pp. 161–173.CrossRefGoogle Scholar
[11]Shavrukov, V., Subalgebras of diagonalizable algebras of theories containing arithmetic, Dissertationes mathematicae (Rozprawy matematycne), vol. CCCXXIII (1993).Google Scholar
[12]Shavrukov, V., Undecidability of diagonalizable algebras, this Journal, vol. 62 (1997), pp. 79–116.Google Scholar
[13]Smoryński, C., Self-reference and modal logic, Universitext, Springer, New York, 1985.CrossRefGoogle Scholar
[14]Solovay, R.M., Provability interpretations of modal logic, Israel Journal of Mathematics, vol. 25 (1976), pp. 287–304.CrossRefGoogle Scholar
[15]Strannegård, C., Arithmetical realizations of modal formulas, Ph.D. thesis, University of Göteborg, 1997.Google Scholar
[16]Visser, A., Interpretability logic, [8], 1990, pp. 175–209.CrossRefGoogle Scholar
[17]Visser, A., An overview of interpretability logic, Technical Report LGPS 174, Department of Philosophy, Utrecht University, 1997.Google Scholar
[18]Zambella, D., On the proofs of arithmetical completeness of interpretability logic, The Notre Dame Journal of Formal Logic, vol. 32 (1992), pp. 542–551.Google Scholar
[19]Zambella, D., Shavrukov's theorem on the subalgebras of diagonalizable algebras for theories containing I Δ0 + Exp, The Notre Dame Journal of Formal Logic, vol. 35 (1994), pp. 147–157.CrossRefGoogle Scholar