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On the proof theory of the modal logic for arithmetic provability

Published online by Cambridge University Press:  12 March 2014

Daniel Leivant
Daniel Leivant*
Affiliation:
Department of Computer Science, Cornell University, Ithaca, New York 14853
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Extract

The modal logic GL has been found by Solovay [13] to formalize the provable propositional properties of the provability-predicate for Peano's Arithmetic PA (cf. §1 below). We give several sequential calculi for GL, compare their merits, and use one calculus to syntactically derive several metamathematical results about GL.

Some of our results have been proved model theoretically, and similar proofs are fairly straightforward for several of the remaining ones (G. Boolos and the referee have provided such proofs for 4.1, 4.3 and 5.1 below). However, our syntactic techniques often yield more concise and obviously constructive proofs, they offer additional insight into the nature of the systems considered, and are easily adaptable to systems for which semantical analysis is problematic.

I am indebted to G. Boolos and to the referee for their valuable advice. The referee has suggested the rule GL of §3 below as an axiomatization of GL; the resulting sequential calculus has allowed a definite improvement of our original presentation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 46 , Issue 3 , September 1981 , pp. 531 - 538
Copyright
Copyright © Association for Symbolic Logic 1981

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References

REFERENCES

[1]Boolos, G., The unprovability of consistency, Cambridge University Press, Cambridge, 1979.Google Scholar
[2]Boolos, G., Reflection principles and iterated consistency assertions, this Journal, vol. 44 (1979), pp. 33–35.Google Scholar
[3]Czermak, J., Interpolation theorems for modal logics (abstract), this Journal, vol. 39 (1974), p. 416.Google Scholar
[4]Curry, H. B., Foundations of mathematical logic, McGraw-Hill, New York, 1963.Google Scholar
[5]Fine, K., Failures of the interpolation lemma in quantified modal logic, this Journal, vol. 44 (1979), pp. 201–206.Google Scholar
[6]Gabbay, D. M., Craig's, interpolation lemma for modal logics, Conference in Mathematical Logic, London, 1970, Lecture Notes in Mathematics, no. 255, Springer, Berlin, 1972, pp. 111–127.Google Scholar
[7]Kleene, S. C., Introduction to metamathematics, Noordhoff, Groningen, 1952.Google Scholar
[8]Löb, M. H., Solution of a problem of Leon Henkin, this Journal, vol. 20 (1955), pp. 115–118.Google Scholar
[9]Manaster, A. B., Completeness, compactness and undecidability, Prentice-Hall, Englewood Cliffs, N. J., 1975.Google Scholar
[10]Smoryński, C., Calculating self-referential statements. I: Explicit calculations, to appear in Studio Logica.Google Scholar
[11]Smoryński, C., Beth's theorem and self-referential sentences, Logic Colloquium '77 (Macintyre, et al., Editors), North-Holland, Amsterdam, 1978.Google Scholar
[12]Smullyan, R., First-order logic, Springer, Berlin, 1968.CrossRefGoogle Scholar
[13]Solovay, R., Provability interpretations of modal logics, Israel Journal of Mathematics, vol. 25 (1976), pp. 287–304.CrossRefGoogle Scholar
[14]Takeuti, G., Proof theory, North-Holland, Amsterdam, 1978.Google Scholar
[15]Zeman, J. J., Modal logic, Oxford University Press, Oxford, 1973.Google Scholar
[16]Sambin, G. and Valentini, S., A modal sequent calculus for a fragment of arithmetic, preprint, 1979.Google Scholar