Fragments of IOpen

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