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| Paper IPM / M / 634 |
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| Abstract: | |
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In the first section of this paper we show that iΠ1 ≡ W¬¬lΠ1 and that a Kripke model which decides bounded formulas forces iΠ1 if and only if the union of the worlds in any path in it satisfies IΠ1. In particular, the union of the worlds in any path of a Kripke model
of HA models IΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of
Πm-formulas in a linear Kripke model deciding ∆0-formulas, it is necessary and sufficient that the model be Σm-elementary. This implies that if a linear Kripke model forces PEMprenex, then it forces PEM. We also show that, for each n\geqslant 1, iΦn does not prove H(IΠn). Here, Φn's are Burr's fragments
of HA.
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