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 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 ` 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. —————————————————————————2000 Mathematics Subject Classification: 03F30, 03F55, 03H15.