Year: 2017
Type: preprint
Abstract: In this paper we present a new proof of Solovay's theorem on arithmetical completeness of Gödel-Löb provability logic GL. Originally, completeness of GL with respect to interpretation of $\Box$ as provability in PA was proved by R. Solovay in 1976. The key part of Solovay's proof was his construction of an arithmetical evaluation for a given modal formula that made the formula unprovable PA if it were unprovable in GL. The arithmetical sentences for the evaluations were constructed using certain arithmetical fixed points. The method developed by Solovay have been used for establishing similar semantics for many other logics. In our proof we develop new more explicit construction of required evaluations that doesn't use any fixed points in their definitions. To our knowledge, it is the first alternative proof of the theorem that is essentially different from Solovay's proof in this key part.
Source: arXiv (Cornell University)
Author: Pakhomov, Fedor
Institution: Steklov Mathematical Institute
Language: English
Subfield: Artificial Intelligence
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
Funder: Russian Science Foundation
Awards: G4065927754, G7131804313
OpenAlex ID: W2953299604