# Some abstract versions of Goedel’s Second Incompleteness Theorem based on non-classical logics

## Lev Beklemishev

### Steklov Mathematical Institute of Russian Academy of Sciences in Moscow

We study abstract versions of Goedel’s second incompleteness theorem and formulate generalizations of Loeb’s derivability conditions that work for logics weaker than the classical one. We isolate the role of the contraction and weakening rules in Goedel’s theorem and give a (toy) example of a system based on modal logic without contraction invalidating Goedel’s argument. On the other hand, Goedel’s theorem is established for a version of Peano arithmetic without contraction. (Joint work with Daniyar Shamkanov)

Lev Dmitrievich Beklemishev graduated from Moscow State University in 1989, got Ph.D. from Steklov Mathematical Institute of Russian Academy of Sciences in 1992, a D.Sci. degree in 1998. He is the winner of the Moscow Mathematical Society prize in 1994, A. von Humboldt Fellowship (Germany) in 1998. In 2000–2005 he was a faculty member of Utrecht University (the Netherlands). In 2006 L. Beklemishev was elected a corresponding member of Russian Academy of Sciences. Currently, he is a Chief Scientific Researcher at Steklov Institute Moscow, a Professor of Moscow University, a Yandex Professor of the Higher School of Economics, Moscow. His research interests include mathematical logic, proof theory, provability logic, modal logic, fragments of arithmetic.