Turing–Taylor Expansions for Arithmetic Theories

Studia Logica 104 (6):1225-1243 (2016)
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Abstract

Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \ proof-theoretic ordinal \ also denoted \. As such, to each theory U we can assign the sequence of corresponding \ ordinals \. We call this sequence a Turing-Taylor expansion or spectrum 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 \. In particular, 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 expansion will define a unique point in Ignatiev’s model.

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Joost Joosten
Universitat de Barcelona

Citations of this work

Münchhausen provability.Joost J. Joosten - 2021 - Journal of Symbolic Logic 86 (3):1006-1034.
On the inevitability of the consistency operator.Antonio Montalbán & James Walsh - 2019 - Journal of Symbolic Logic 84 (1):205-225.
The Logic of Turing Progressions.Eduardo Hermo Reyes & Joost J. Joosten - 2020 - Notre Dame Journal of Formal Logic 61 (1):155-180.

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