Non-finite Axiomatizability of Generalized Medvedev Logics

We introduce a generalized form of Medvedev logics obtained by removing the greatest element from finite products of rooted Kripke frames with a top. We show that, before removing the top, the intermediate logic characterized by such finite products is exactly KC. Classical Medvedev logic is characterized by topless products of 2-chains, and a theorem of Maksimova, Skvortsov and Shehtman establishes that it is not finitely axiomatizable. Motivated by this result, Nick Bezhanishvili conjectured that non-finite axiomatizability extends to topless products of arbitrary finite chains and, more generally, to topless products of finite rooted frames with a top. We prove that every such generalized Medvedev logic is not finitely axiomatizable, thereby settling both conjectures in the affirmative. In 2003, van Benthem, Guram Bezhanishvili, and Gehrke introduced Cheq, the logic of chequered sets, and we show that whenever Cheq is a sublogic of a generalized Medvedev logic, the latter is not finitely axiomatizable over Cheq. Finally, we investigate the order structure of generalized Medvedev logics. We prove that there are at least countably many distinct generalized Medvedev logics and that no least such logic exists. These results extend the classical theory of Medvedev logic and clarify the behaviour of intermediate logics generated by topless product constructions.