The logic of Turing progressions

TL;DR

This paper introduces a logic that precisely characterizes the relationships between various Turing progressions based on different consistency notions, establishing soundness and completeness within an arithmetical framework.

Contribution

It presents a new logic that captures all relations between Turing progressions under natural consistency notions, with proven arithmetical soundness and completeness.

Findings

Logic exactly characterizes relations between Turing progressions

Proven arithmetical soundness and completeness of the logic

Defines the Formalized Turing progressions (FTP) interpretation

Abstract

Turing progressions arise by iteratedly adding consistency statements to a base theory. Different notions of consistency give rise to different Turing progressions. In this paper we present a logic that generates exactly all relations that hold between these different Turing progressions given a particular set of natural consistency notions. Thus, the presented logic is proven to arithmetically sound and complete for a natural interpretation, named the \emph{Formalized Turing progressions} (FTP) interpretation.