Journal of Symbolic Logic 67 (1):91-103 (2002)

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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
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DOI http://projecteuclid.org/euclid.jsl/1190150031
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References found in this work BETA

Basic Proof Theory.Roy Dyckhoff - 2001 - Bulletin of Symbolic Logic 7 (2):280-280.
Logic and Structure.Melvin Fitting - 1986 - Journal of Symbolic Logic 51 (3):826-827.
Fragments of $HA$ Based on $\Sigma_1$ -Induction.Kai F. Wehmeier - 1997 - Archive for Mathematical Logic 37 (1):37-49.
Classical and Intuitionistic Models of Arithmetic.Kai F. Wehmeier - 1996 - Notre Dame Journal of Formal Logic 37 (3):452-461.
Primes and Their Residue Rings in Models of Open Induction.Angus Macintyre & David Marker - 1989 - Annals of Pure and Applied Logic 43 (1):57-77.

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