Some abstract versions of Gödel’s second incompleteness theorem based on non-classical logics. (English) Zbl 1418.03175

Beklemishev, Lev; Shamkanov, Daniyar

van Eijck, Jan (ed.) et al., Liber amicorum Alberti. A tribute to Albert Visser. London: College Publications. Tributes 30, 15-29 (2016).

Summary: 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 the contraction rule in Gödel’s theorem and give toy examples of systems based on modal logic without contraction invalidating Gödel’s argument.
For the entire collection see [Zbl 1367.03011].

MSC:

03F40	Gödel numberings and issues of incompleteness
03B45	Modal logic (including the logic of norms)
03F05	Cut-elimination and normal-form theorems