# Solovay's completeness without fixed points

## TL;DR

This paper offers a new, fixed-point-free proof of Solovay's arithmetical completeness theorem for G"odel-L"ob provability logic, simplifying the original construction and broadening understanding of the logic's semantics.

## Contribution

It introduces an alternative proof of Solovay's theorem that avoids using fixed points, providing a more explicit and potentially more accessible construction.

## Key findings

- New proof of Solovay's theorem without fixed points
- Explicit construction of arithmetical evaluations
- Broader applicability to similar modal logics

## Abstract

In this paper we present a new proof of Solovay's theorem on arithmetical completeness of G\"odel-L\"ob 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.