Reflection ranks and ordinal analysis

Journal of Symbolic Logic 86 (4):1350-1384 (2021)
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Abstract

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderedness phenomenon by studying a coarsening of the consistency strength order, namely, the$\Pi ^1_1$reflection strength order. We prove that there are no descending sequences of$\Pi ^1_1$sound extensions of$\mathsf {ACA}_0$in this ordering. Accordingly, we can attach a rank in this order, which we call reflection rank, to any$\Pi ^1_1$sound extension of$\mathsf {ACA}_0$. We prove that for any$\Pi ^1_1$sound theoryTextending$\mathsf {ACA}_0^+$, the reflection rank ofTequals the$\Pi ^1_1$proof-theoretic ordinal ofT. We also prove that the$\Pi ^1_1$proof-theoretic ordinal of$\alpha $iterated$\Pi ^1_1$reflection is$\varepsilon _\alpha $. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles.

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James Walsh
New York University

Citations of this work

Truth, disjunction, and induction.Ali Enayat & Fedor Pakhomov - 2019 - Archive for Mathematical Logic 58 (5-6):753-766.
Reflection ranks and ordinal analysis.Fedor Pakhomov & James Walsh - 2021 - Journal of Symbolic Logic 86 (4):1350-1384.

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References found in this work

Reflection Principles and Their Use for Establishing the Complexity of Axiomatic Systems.Georg Kreisel & Azriel Lévy - 1968 - Zeitschrift für Mathematische Logic Und Grundlagen der Mathematik 14 (1):97--142.
Truth, disjunction, and induction.Ali Enayat & Fedor Pakhomov - 2019 - Archive for Mathematical Logic 58 (5-6):753-766.
Reflection ranks and ordinal analysis.Fedor Pakhomov & James Walsh - 2021 - Journal of Symbolic Logic 86 (4):1350-1384.
Aspects of Incompleteness.Per Lindström - 1999 - Studia Logica 63 (3):438-439.

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