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On the Herbrand notion of consistency for finitely axiomatizable fragments of bounded arithmetic theories

Published online by Cambridge University Press:  12 March 2014

Leszek Aleksander Kołodziejczyk
Leszek Aleksander Kołodziejczyk*
Affiliation:
Warsaw University, Institute of Mathematics, Banacha 2, 02-097 Warszawa, Poland. E-mail: [email protected]
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Abstract

Modifying the methods of Z. Adamowicz's paper Herbrand consistency and bounded arithmetic [3] we show that there exists a number n such that ⋃mSm (the union of the bounded arithmetic theories Sm) does not prove the Herbrand consistency of the finitely axiomatizable theory S3n

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 2 , June 2006 , pp. 624 - 638
Copyright
Copyright © Association for Symbolic Logic 2006

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References

REFERENCES

[1]Adamowicz, Z., private communication.Google Scholar
[2]Adamowicz, Z., On tableaux consistency in weak theories, Preprint 618, Institute of Mathematics of the Polish Academy of Sciences, 2001.Google Scholar
[3]Adamowicz, Z., Herbrand consistency and bounded arithmetic, Fundamenta Mathematicae, vol. 171 (2002), pp. 279–292.CrossRefGoogle Scholar
[4]Adamowicz, Z. and Zbierski, P., On Herbrand consistency in weak arithmetic, Archive for Mathematical Logic, vol. 40 (2001). pp. 399–413.CrossRefGoogle Scholar
[5]Buss, S. R. and Ignjatović, A., Unprovability of consistency statements in fragments of bounded arithmetic, Annals of Pure and Applied Logic, vol. 74 (1995), pp. 221–244.CrossRefGoogle Scholar
[6]Hájek, P. and Pudlák, P., Metamathematics of First-Order Arithmetic. Springer-Verlag, 1993.CrossRefGoogle Scholar
[7]Kraíček, J., Bounded Arithmetic, Propositional Logic, and Complexity Theory, Cambridge University Press, 1995.CrossRefGoogle Scholar
[8]Pudlák, P., Cuts, consistency statements, and interpretations, this Journal, vol. 50 (1985), pp. 423–441.Google Scholar
[9]Pudlák, P., A note on bounded arithmetic, Fundamenta Mathematicae, vol. 136 (1990), pp. 85–89.CrossRefGoogle Scholar
[10]Pudlák, P., Consistency and games, Logic colloquium '03 (Stoltenberg-Hansen, V. and Väänänen, J., editors), Lecture Notes in Logic, vol. 24, ASL and AK Peters, 2006, pp. 244–281.CrossRefGoogle Scholar
[11]Wilkie, A. J. and Paris, J. B., On the scheme of induction for bounded arithmetic formulas, Annals of Pure and Applied Logic, vol. 35 (1987), pp. 261–302.CrossRefGoogle Scholar
[12]Wilkie, A. J. and Paris, J. B., On the existence of end-extensions of models of bounded induction, Logic, Methodology, and Philosophy of Science VIII (Moscow 1987) (Fenstad, J. E., Frolov, I. T., and Hilpinen, R.. editors), North-Holland, 1989, pp. 143–161.Google Scholar
[13]Willard, D. E., How to extend the semantic tableaux and cut-free versions of the second incompleteness theorem almost to Robinson's arithmetic Q. this Journal, vol. 67 (2002). pp. 465–496.Google Scholar