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Some abstract versions of Gödel's second incompleteness theorem based on non-classical logics

We study abstract versions of Gödel's second incompleteness theorem and formulate generalizations of Löb's derivability conditions that work for logics weaker than the classical one. We isolate the role of contraction rule in Gödel's theorem and give a (toy) example of a system based on modal logic without contraction invalidating Gödel's argument.