Some weak fragments of HA and certain closure properties
Morteza Moniri, Mojtaba Moniri · Journal of Symbolic Logic · 2002
Abstract We show that Intuitionistic Open Induction iop is not closed under the rule DNS(Ǝ1). 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 VR and IP. We include some remarks on the classical worlds in Kripke models of iop.