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Some weak fragments of HA and certain closure properties

Published online by Cambridge University Press:  12 March 2014

Morteza Moniri  and
Mojtaba Moniri
Morteza Moniri
Affiliation:
School of Mathematics, IPM, P.O.Box 19395-5746, Tehran, Iran, E-mail: [email protected]
Mojtaba Moniri
Affiliation:
Mathematics Department, Tarbiat Modarres University, P.O. BOX 14115-175, Tehran, Iran, E-mail: [email protected], [email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 91 - 103
Copyright
Copyright © Association for Symbolic Logic 2002

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