ℋ-theories, fragments of HA and PA -normality

Archive for Mathematical Logic 41 (1):101-105 (2002)
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For a classical theory T, ℋ(T) denotes the intuitionistic theory of T-normal (i.e. locally T) Kripke structures. S. Buss has asked for a characterization of the theories in the range of ℋ and raised the particular question of whether HA is an ℋ-theory. We show that Ti∈ range(ℋ) iff Ti = ℋ(T). As a corollary, no fragment of HA extending iΠ1 belongs to the range of ℋ. A. Visser has already proved that HA is not in the range of H by different methods. We provide more examples of theories not in the range of ℋ. We show PA-normality of once-branching Kripke models of HA + MP, where it is not known whether the same holds if MP is dropped.

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