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arXiv · 1704.07678

Provability Logics of Hierarchies

Amirhossein Akbar Tabatabai
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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.

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Amirhossein Akbar Tabatabai. 2017-04-20. Provability Logics of Hierarchies. https://arxiv.org/abs/1704.07678

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