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A small reflection principle for bounded arithmetic

Published online by Cambridge University Press:  12 March 2014

Rineke Verbrugge  and
Albert Visser
Rineke Verbrugge*
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, 1018 Tv Amsterdam, The Netherlands
Albert Visser
Affiliation:
Department of Philosophy, University of Gothenburg, S-412 G8 Gothenburg, Sweden, E-mail: Albert.Visser@phil.RUU.nl
*
Department of Philosophy, University of Gothenburg, S-412 G8 Gothenburg, Sweden, E-mail: Rineke.Verbrugge@phil.gu.se
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Abstract

We investigate the theory IΔ0+Ω1 and strengthen [Bu86, Theorem 8.6] to the following: if NP ≠ co-NP, then Σ-completeness for witness comparison foumulas is not provable in bounded arithmetic. i.e.,

Next we study a “small reflection principle” in bounded arithmetic. We prove that for all sentences φ

The proof hinges on the use of definable cuts and partial satisfaction predicates akin to those introduced by Pudlák in [Pu86].

Finally, we give some applications of the small reflection principle, showing that the principle can sometimes be invoked in order to circumvent the use of provable Σ-completeness for witness comparison formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 3 , September 1994 , pp. 785 - 812
Copyright
Copyright © Association for Symbolic Logic 1994

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