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Working paper

Lev Beklemishev, Daniyar Shamkanov. Some abstract versions of Gödel's second incompleteness theorem based on non-classical logics, ArXiv: 1602.05728, Feb. 2016.

arxiv.org. math. Cornell University, 2016. No. 1602.05728.
Lev Beklemishev, Shamkanov D.
We study abstract versions of G\"odel's second incompleteness theorem and formulate generalizations of L\"ob's derivability conditions that work for logics weaker than the classical one. We isolate the role of contraction rule in G\"odel's theorem and give a (toy) example of a system based on modal logic without contraction invalidating G\"odel's argument.