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May 2020 An Effective Analysis of the Denjoy Rank
Linda Westrick
Notre Dame J. Formal Logic 61(2): 245-263 (May 2020). DOI: 10.1215/00294527-2020-0006

Abstract

We analyze the descriptive complexity of several Π11-ranks from classical analysis which are associated to Denjoy integration. We show that VBG, VBG*, ACG, and ACG* are Π11-complete, answering a question of Walsh in case of ACG*. Furthermore, we identify the precise descriptive complexity of the set of functions obtainable with at most α steps of the transfinite process of Denjoy totalization: if || is the Π11-rank naturally associated to VBG, VBG*, or ACG*, and if α<ω1ck, then {FC(I):|F|α} is Σ2α0-complete. These finer results are an application of the author’s previous work on the limsup rank on well-founded trees. Finally, {(f,F)M(I)×C(I):FACG*andF'=fa.e.} and {fM(I):fis Denjoy integrable} are Π11-complete, answering more questions of Walsh.

Citation

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Linda Westrick. "An Effective Analysis of the Denjoy Rank." Notre Dame J. Formal Logic 61 (2) 245 - 263, May 2020. https://doi.org/10.1215/00294527-2020-0006

Information

Received: 31 October 2017; Accepted: 4 November 2019; Published: May 2020
First available in Project Euclid: 4 April 2020

zbMATH: 07222690
MathSciNet: MR4092534
Digital Object Identifier: 10.1215/00294527-2020-0006

Subjects:
Primary: 03E15
Secondary: 26A39

Rights: Copyright © 2020 University of Notre Dame

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Vol.61 • No. 2 • May 2020
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