Fragments of IOpen.

In this paper we consider some fragments of IOpen (Robinson arithmetic Q with induction for quantifier-free formulas) proposed by Harvey Friedman and answer some questions he asked about these theories. We prove that I (lit) is equivalent to IOpen and is not finitely axiomatizable over Q , establish some inclusion relations between I (=) , I (≠) , I (⩽) and I (⩽̸) . We also prove that the set of diophantine equations solvable in models of I (=) is (algorithmically) decidable. [ABSTRACT FROM AUTHOR]