A Finitely Supported Frame for the Turing Schmerl Calculus

TL;DR

This paper introduces a new universal frame for the Turing Schmerl Calculus, enhancing the modal definability of worlds and building on previous models to better understand provability interrelations.

Contribution

It presents a modified universal frame for TSC that improves modal definability and extends prior models based on Ignatiev's universal model.

Findings

The new frame $ ext{"H"}$ is a slight modification of $ ext{"J"}.

Each world in $ ext{"H"}$ is modal definable.

The model better captures provability interrelations.

Abstract

In arXiv:1604.08705 we introduced the propositional modal logic $\textbf{TSC}$ (which stands for Turing Schmerl Calculus) which adequately describes the provable interrelations between different kinds of Turing progressions. In arXiv:1709.04715 we defined a model $\mathcal{J}$ which is proven to be a universal model for $\textbf{TSC}$ based on the intensively studied Ignatiev's universal model for the closed fragment of $\textbf{GLP}$ (G\"odel L\"ob's polymodal provability logic). In the current paper we present a new universal frame $\mathcal{H}$, which is a slight modification of $\mathcal{J}$, and whose domain allows for a modal definability of each of its worlds.