Limits of applicability of Gödel’s second incompleteness theorem
The celebrated Gödel’s second incompleteness theorem is the result that roughly speaking says that no strong enough consistent theory could prove its own consistency. In this talk I will first give an overview of the current state of research on the limits of applicability of the theorem. And second I will present two recent results: first is due to me [1] and the second is due to Albert Visser and me [2]. The first result is an example of a weak natural theory that proves the arithmetization of its own consistency. The second result is a general theorem with the flavor of Second Incompleteness Theorem that is applicable to arbitrary weak first-order theories rather than to extension of some base system. Namely the theorem states that no finitely axiomatizable first-order theory one-dimensionally interprets its own extension by predicative comprehension.
References
- [1]
- Fedor Pakhomov, A weak set theory that proves its own consistency, Preprint, arXiv:1907.00877, 2019, .
- [2]
- Fedor Pakhomov and Albert Visser, Finitely axiomatized theories lack self-comprehension, Preprint, arXiv:2109.02548, 2021, .
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