Abstract

We show that Intuitionistic Open Induction iop is not closed under the rule DNS(∃ - 1 ). This is established by constructing a Kripke model of iop + $\neg L_y(2y > x)$ , where $L_y(2y > x)$ is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by PA - plus the scheme of weak ¬¬LNP for open formulas, where universal quantification on the parameters precedes double negation. We also show that for any open formula φ(y) having only y free, $(PA^-)^i \vdash L_y\varphi(y)$ . We observe that the theories iop, i∀ 1 and iΠ 1 are closed under Friedman's translation by negated formulas and so under VR and IP. We include some remarks on the classical worlds in Kripke models of iop