Some Weak Fragments of HA and Certain Closure Properties

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+¬Ly(2y> x), where Ly(2y> x) is universally quantified on x. On the other hand, we prove that iop is equivalent with the intuitionistic theory axiomatized by P A − 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, (P A − ) i ⊢ Lyϕ(y). We observe that the theories iop, i∀1 and iΠ1 are closed under Friedman’s translation by negated formulas and so under V R and IP. We include some remarks on the classical worlds in Kripke models of iop