2014/04/17 by Joost J. Joosten, Joosten, Joost J.
Mathematics · #FOS: Mathematics #Logic (math.LO) #math.LO
paper · pdf · doi:10.48550/arxiv.1404.4483
First draft
arxiv created 2015/08/03 · arxiv updated 2015/08/04
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 Πn+1 proof-theoretic ordinal. As such, to each theory U we can assign the sequence of corresponding Πn+1 ordinals ⟨ |U|n⟩n>0. We call this sequence a 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 GLPω. 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.