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 be arithmetically sound and complete for a natural interpretation, named the formalized Turing progressions (FTP) interpretation.
"The Logic of Turing Progressions." Notre Dame J. Formal Logic 61 (1) 155 - 180, January 2020. https://doi.org/10.1215/00294527-2019-0037