Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T19:19:25.126Z Has data issue: false hasContentIssue false
  • English
  • Français

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: lak@mimuw.edu.pl
Get access Rights & Permissions [Opens in a new window]

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

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]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