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| Paper IPM / M / 770 |
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| Abstract: | |
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This paper proves some independence results for weak fragments of
Heyting arithmetic by using Kripke models. We present a necessary
condition for linear Kripke models of arithmetical theories which
are closed under the negative translation and use it to show that
the union of the worlds in any linear Kripke model of HA
satisfies PA. We construct a two-node PA-normal Kripke
structure which does not force iΣ2. We prove
i∀1\nvdash i∃1, i∃1\nvdashi∀1, iΠ2\nvdash iΣ2 and iΣ2\nvdashiΠ2. We use Smorynski's operation Σ′ to show
HA\nvdash lΠ1.
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