Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T13:57:41.326Z Has data issue: false hasContentIssue false
  • English
  • Français

Closed Fragments of Provability Logics of Constructive Theories

Published online by Cambridge University Press:  12 March 2014

Albert Visser
Albert Visser*
Affiliation:
Department of Philosophy, Faculty of Humanities, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands, E-mail: Albert.Visser@phil.uu.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we give a new proof of the characterization of the closed fragment of the provability logic of Heyting's Arithmetic. We also provide a characterization of the closed fragment of the provability logic of Heyting's Arithmetic plus Markov's Principle and Heyting's Arithmetic plus Primitive Recursive Markov's Principle.

Keywords

provability logicconstructive arithmetic
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 1081 - 1096
Copyright
Copyright © Association for Symbolic Logic 2008

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]Boolos, G., On deciding the truth of certain statements involving the notion of consistency, this Journal, vol. 41 (1976), pp. 779–781.Google Scholar
[2]Boolos, G., The logic of provability, Cambridge University Press, Cambridge, 1993.Google Scholar
[3]Boolos, G. and Sambin, G., Provability: the emergence of a mathematical modality, Studia Logica, vol. 50 (1991), pp. 1–23.CrossRefGoogle Scholar
[4]de Jongh, D. H. J. and Visser, A., Embeddings of Heyting algebras, In Hodges, et al. [6], pp. 187–213.Google Scholar
[5]Friedman, H., One hundred and two problems in mathematical logic, this Journal, vol. 40 (1975), pp. 113–129.Google Scholar
[6]Hodges, W., Hyland, M., Steinhorn, C., and Truss, J. (editors), Logic: from foundations to applications, Clarendon Press, Oxford, 1996.CrossRefGoogle Scholar
[7]MacLane, S., Categories for the working mathematician, Graduate Texts in Mathematics, no. 5, Springer, New York, 1971.Google Scholar
[8]Smoryński, C., Applications of Kripke models, Metamathematical investigations of intuitionistic arithmetic and analysis (Troelstra, A. S., editor), Springer Lecture Notes 344, Springer, Berlin, 1973, pp. 324–391.CrossRefGoogle Scholar
[9]Smoryński, C., Self-reference and modal logic, Universitext, Springer, New York, 1985.CrossRefGoogle Scholar
[10]Troelstra, A. S., Metamathematical investigations of intuitionistic arithmetic and analysis, Springer Lecture Notes 344, Springer Verlag, Berlin, 1973.CrossRefGoogle Scholar
[11]Troelstra, A. S. and van Dalen, D., Constructivism in mathematics, vol 1, Studies in Logic and the Foundations of Mathematics, vol. 121, North Holland, Amsterdam, 1988.Google Scholar
[12]van Benthem, J. F. A. K., Doctoral Thesis, 1974, Manuscript, Mathematical Institute, University of Amsterdam.Google Scholar
[13]Visser, A., On the completeness principle, Annals of Mathematical Logic, vol. 22 (1982), pp. 263–295.CrossRefGoogle Scholar
[14]Visser, A., Evaluation, provably deductive equivalence in Heyting's Arithmetic of substitution instances of propositional formulas, Logic Group Preprint Series 4, Department of Philosophy, Utrecht University, 1985.Google Scholar
[15]Visser, A., Propositional combinations of Σ-sentences in Heyting's Arithmetic, Logic Group Preprint Series 117, Department of Philosophy, Utrecht University, 1994.Google Scholar
[16]Visser, A., Substitutions of -sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 227–271.CrossRefGoogle Scholar
[17]Visser, A., Predicate logics of constructive arithmetical theories, this Journal, vol. 71 (2006), no. 4, pp. 1311–1326.Google Scholar