Relational Semantics for the Turing Schmerl Calculus

TL;DR

This paper introduces a universal relational model for the Turing Schmerl Calculus, enhancing understanding of provable interrelations in Turing progressions through a modification of Ignatiev's model.

Contribution

It defines a new universal model for TSC, based on a modification of Ignatiev's well-known model for GLP, bridging modal logic and Turing progressions.

Findings

The model $ extbf{J}$ is proven to be universal for $ extbf{TSC}$.

The model $ extbf{J}$ is a slight modification of Ignatiev's model $ extbf{I}$.

The model effectively captures provable interrelations in Turing progressions.

Abstract

In arXiv:1604.08705 the authors 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. The current paper defines a model $\mathcal{J}$ which is proven to be a universal model for $\textbf{TSC}$. The model $\mathcal{J}$ is a slight modification of the intensively studied $\mathcal{I}$ : Ignatiev's universal model for the closed fragment of G\"odel L\"ob's polymodal provability logic $\textbf{GLP}$.