Бесконечные множества несводимых модальностей в нормальных модальных логиках.

Abstract

The first mathematical results in modal logic was connected with the discovery of phenomenon of modality reduction: in many logics the infinite set of all iterated modalities is equivalent to finite sets (its own set for every concrete logic). The question about whether this reduction is always possible and how can we establish it (in algorithmic, criterion or another way) are not studied enough up to now. We will show that this problem has no simple solution: algorithm for its solution does not exist, a set of logics with infinite sets nonequivalent modalities has rather fuzzy bounds — set of maximal in inclusion normal modal logics each of which has an infinite set nonequivalent modalities is a continuum.
Some open problems are discussed.