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

L. D. Beklemishev; D. S. Shamkanov

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

P. 15–29.

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.

Language: English

Keywords: Goedel theorem fixed points substructural logics

In book

Liber Amicorum Alberti. A Tribute to Albert Visser

L.: College Publications, 2016.

Explorations in subexponential non-associative non-commutative linear logic

Blaisdell E., Kanovich M., Stepan L. Kuznetsov et al., , in: EPTCS 381: Proceedings Modalities in substructural logics: Applications at the interfaces of logic, language and computation (AMSLO 2023), Ljubljana, Slovenia, August 7-8, 2023, Electronic Proceedings in Theoretical Computer Science.: Open Publishing Association, 2023. P. 4–19.

In a previous work we introduced a non-associative non-commutative logic extended by multimodalities, called subexponentials, licensing local application of structural rules. Here, we further explore this system, exhibiting a classical one-sided multi-succedent classical analogue of our intuitionistic system, following the exponential-free calculi of Buszkowski, and de Groote, Lamarche. A large fragment of the intuitionistic calculus ...

Added: November 23, 2023

EPTCS 381: Proceedings Modalities in substructural logics: Applications at the interfaces of logic, language and computation (AMSLO 2023), Ljubljana, Slovenia, August 7-8, 2023, Electronic Proceedings in Theoretical Computer Science

Open Publishing Association, 2023.

Added: November 23, 2023

Commutative action logic

Stepan L. Kuznetsov, Journal of Logic and Computation 2023 Vol. 33 No. 6 P. 1437–1462

We prove undecidability and pinpoint the place in the arithmetical hierarchy for commutative action logic, i.e. the equational theory of commutative residuated Kleene lattices (action lattices), and infinitary commutative action logic, the equational theory of *-continuous commutative action lattices. Namely, we prove that the former is Σ01�10-complete and the latter is Π01�10-complete. Thus, the situation is the ...

Added: March 7, 2023

Об эквивалентности некоторых утверждений о неподвижных точках сжатий

Semenov P., Функциональный анализ и его приложения 2017 Т. 51 № 4 С. 92–95

It is shown that a series of recent (2012–2016) generalizations of the notion of contraction (F-contraction, weak F-contraction, etc.) in fact reduce to known notions of contraction (due to Browder, Boyd and Wong, Meir and Keeler, etc.). ...

Added: October 2, 2018

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

Lev Beklemishev, Shamkanov D., / Series math "arxiv.org". 2016. No. 1602.05728.

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. ...

Added: March 13, 2016