Geodesic
On~modal logics axiomatizing provability
Izvestiya. Mathematics , Tome 27 (1986) no. 3, pp. 401-429.

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On the basis of the concept of the trace of a modal logic, introduced earlier by the author, a classification of arithmetically complete modal logics is given. It is proved that, between the least and the greatest arithmetically complete logics, there are continuum many logics which are not arithmetically complete. Bibliography: 17 titles. 
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S. N. Artemov. On~modal logics axiomatizing provability. Izvestiya. Mathematics , Tome 27 (1986) no. 3, pp. 401-429. http://geodesic.mathdoc.fr/item/IM2_1986_27_3_a0/

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