Fragments of IOpen

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