Turing-Taylor expansions for arithmetic theories

Abstract

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories. Turing progressions based on $n$-provability give rise to a $\Pi_{n+1}$ proof-theoretic ordinal. As such, to each theory $U$ we can assign the sequence of corresponding $\Pi_{n+1}$ ordinals $\langle |U|_n\rangle_{n>0}$. We call this sequence a \emph{Turing-Taylor expansion} of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories of Peano Arithmetic to Ignatiev's universal model for the closed fragment of the polymodal provability logic ${\mathbf{GLP}}_\omega$. In particular, in this first draft we observe that each point in the Ignatiev model can be seen as Turing-Taylor expansions of formal mathematical theories. Moreover, each sub-theory of Peano Arithmetic that allows for a Turing-Taylor expression will define a unique point in Ignatiev's model.