An Introduction to Basic Arithmetic

We study Basic Arithmetic BA, which is the basic logic BQC equivalent of Heyting Arithmetic HA overintuitionistic logic IQC, and of Peano Arithmetic PAover classical logic CQC. It turns out that The Friedmantranslation is applicable to BA. Using this translation, we provethat 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 $\mathrm{I}\exists_{1}^{+}$,a fragment of PA with Induction restricted to the formulas made upof ∃, ∧ 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 thecorresponding arithmetic, EBA shows similarities to HA. Mathematics Subject Classification: Primary 03F30; secondary03F50.