# Provability Logics of Hierarchies

## TL;DR

This paper extends provability logic to hierarchies of theories using infinitely many modalities, defining hierarchical modal theories and establishing their soundness and completeness with provability interpretations.

## Contribution

It introduces hierarchical modal logics with infinitely many modalities and develops their provability interpretations, extending classical provability logic to theory hierarchies.

## Key findings

- Defined hierarchical modal theories such as K4, KD4, GL, S4 with infinitely many modalities.
- Established soundness and completeness theorems for these hierarchical provability logics.
- Provided canonical provability interpretations for the hierarchical modal theories.

## Abstract

The branch of provability logic investigates the provability-based behavior of the mathematical theories. In a more precise way, it studies the relation between a mathematical theory $T$ and a modal logic $L$ via the provability interpretation which interprets the modality as the provability predicate of $T$. In this paper we will extend this relation to investigate the provability-based behavior of a hierarchy of theories. More precisely, using the modal language with infinitely many modalities, $\{\Box_n\}_{n=0}^{\infty}$, we will define the hierarchical counterparts of some of the classical modal theories such as $\mathbf{K4}$, $\mathbf{KD4}$, $\mathbf{GL}$ and $\mathbf{S4}$. Then we will define their canonical provability interpretations and their corresponding soundness-completeness theorems.