Logic Journal of IGPL Advance Access originally published online on July 10, 2007
Logic Journal of IGPL 2008 16(1):1-13; doi:10.1093/jigpal/jzm013
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An Introduction to Basic Arithmetic
Department of Mathematics, Sharif University of Technology, P.O. Box 11365-9415, Tehran, Iran. E-mail: mardeshir{at}sharif.edu
E-mail: bardyaa{at}gmail.com
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Abstract |
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We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA over intuitionistic logic IQC, and of Peano Arithmetic PA over classical logic CQC. It turns out that The Friedman translation is applicable to BA. Using this translation, we prove that BA is closed under a restricted form of the Markov rule. Moreover, it is proved that all nodes of a finite Kripke model of BA are classical models of 
, a fragment of PA with Induction restricted to the formulas made up of 
, 
and/or 
. We also study an interesting extension of BQC, called EBQC, which is the extension by the axiom schema 
. We show that this extension behaves very like to IQC, and the corresponding arithmetic, EBA shows similarities to HA.
Mathematics Subject Classification: Primary 03F30; secondary 03F50.
Key Words: Basic Logic • Basic Arithmetic • Completeness • Heyting Arithmetic • Kripke models
Received for publication 15 June 2006. Revision received 13 March 2007. Accepted for publication 5 May 2007.