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Journal of Logic and Computation 2006 16(2):199-204; doi:10.1093/logcom/exi085
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Vol. 16 No. 2, © The Author, 2006. Published by Oxford University Press. All rights reserved.

Original Articles

An Independence Result for Intuitionistic Bounded Arithmetic

Morteza Moniri

Department of Mathematics, Shahid Beheshti University, Evin, Tehran, Iran, and Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran. Email: ezmoniri{at}ipm.ir

It is shown that the intuitionistic theory of polynomial induction on positive {Pi}1b (coNP) formulas does not prove the sentence ¬¬{forall}x, yexistz ≤ y(x ≤ |y| -> x = |z|). This implies the unprovability of the scheme ¬¬PIND({sum}1b+) in the mentioned theory. However, this theory contains the sentence {forall}x, y¬¬existz ≤ y(x ≤ |y| -> x = |z|). The above independence result is proved by constructing an {omega}-chain of submodels of a countable model of S2 + {Omega}3 + ¬exp such that none of the worlds in the chain satisfies the sentence, and interpreting the chain as a Kripke model.

Keywords: Bounded arithmetic, intuitionistic logic, Kripke model, NP, polynomial hierarchy, polynomial induction

Received 2 December 2004.


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