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Paper IPM / M / 45  


Abstract:  
Buss asked in [1] whether every intuitionistic theory is, for some classical theory T, that of all Tnormal Kripke structures H (T) for which he gave an r.e. axiomatization. In the language of
arithmetic Iop and Lop denote PA^{−} plus Open Induction or Open LNP, iop and lop are their intuitionistic deductive closures. We show H (Iop)=lop is recursively axiomatizable and lop \vdash_{i c} \dashv iop, while i∀_{1} \not\vdash lop. If iT proves
PEM_{atomic} but not totality of a classically provably total Diophantine
function of T, then H(T) ⊄ eq i T and so iT ∉ range (H). A result due to Wehmeier then implies iΠ_{1} ∉ range (H). We prove Iop is not ∀_{2}conservative over i∀_{1}. If Iop ⊆ T ⊆ I∀_{1}, then iT is not closed under MR_{open} or Friedman's translation, so iT ∉ range (H). Both Iop and I∀_{1} are closed under the negative
translation.
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