Archive for Mathematical Logic 42 (8):791-796 (2003)

We construct ω-framed Kripke models of i∀1 and iΠ1 non of whose worlds satisfies ∀x∃y(x=2y∨x=2y+1) and ∀x,y∃zExp(x, y, z) respectively. This will enable us to show that i∀1 does not prove ¬¬∀x∃y(x=2y∨x=2y+1) and iΠ1 does not prove ¬¬∀x, y∃zExp(x, y, z). Therefore, i∀1⊬¬¬lop and iΠ1⊬¬¬iΣ1. We also prove that HA⊬lΣ1 and present some remarks about iΠ2
Keywords Fragments of Heyting Arithmetic  Kripke Models  exp
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DOI 10.1007/s00153-003-0189-8
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