Geodesic
Modal Translations of Heyting and Peano Arithmetic
Publications de l'Institut Mathématique, _N_S_47 (1990) no. 61, p. 13 .

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First-order Heyting arithmetic is embedded by various modal translations in modal extensions of first-order Peano arithmetic which are included in Peano S4. Peano arithmetic is embedded by analogous modal translations in an S5-like extension of Heyting arithmetic. This last system is included in the modal extension of Heyting arithmetic where the necessity operator is equivalent to double negation and where Peano arithmetic can be embedded by a modal translation which amounts to a usual double-negation translation.
Classification : 03B45 03F30 03F55 03F25
Keywords: Heyting arithmetic, Peano arithmetic, modal translations, double-negation translation 
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     author = {Kosta Do\v{s}en},
     title = {Modal {Translations} of {Heyting} and {Peano} {Arithmetic}},
     journal = {Publications de l'Institut Math\'ematique},
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     year = {1990},
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Kosta Došen. Modal Translations of Heyting and Peano Arithmetic. Publications de l'Institut Mathématique, _N_S_47 (1990) no. 61, p. 13 . http://geodesic.mathdoc.fr/item/PIM_1990_N_S_47_61_a2/