Notre Dame Journal of Formal Logic

An Effective Analysis of the Denjoy Rank

Linda Westrick

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Notre Dame J. Formal Logic, Volume 61, Number 2 (2020), 245-263.

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

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1585965655

Digital Object Identifier
doi:10.1215/00294527-2020-0006

Mathematical Reviews number (MathSciNet)
MR4092534

Zentralblatt MATH identifier
07222690

Subjects
Primary: 03E15: Descriptive set theory [See also 28A05, 54H05]
Secondary: 26A39: Denjoy and Perron integrals, other special integrals

Keywords
Denjoy totalization coanalytic ranks

Citation

Westrick, Linda. An Effective Analysis of the Denjoy Rank. Notre Dame J. Formal Logic 61 (2020), no. 2, 245--263. doi:10.1215/00294527-2020-0006. https://projecteuclid.org/euclid.ndjfl/1585965655


Export citation

References

  • [1] Ash, C. J., and J. Knight, Computable Structures and the Hyperarithmetical Hierarchy, vol. 144 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 2000.
  • [2] Cenzer, D., and R. D. Mauldin, “On the Borel class of the derived set operator,” Bulletin de la Société Mathématique de France, vol. 110 (1982), pp. 357–80.
  • [3] Cenzer, D., and R. D. Mauldin, “On the Borel class of the derived set operator, II,” Bulletin de la Société Mathématique de France, vol. 111 (1983), pp. 367–72.
  • [4] Dougherty, R., and A. S. Kechris, “The complexity of antidifferentiation,” Advances in Mathematics, vol. 88 (1991), pp. 145–69.
  • [5] Greenberg, N., A. Montalbán, and T. A. Slaman, “Relative to any non-hyperarithmetic set,” Journal of Mathematical Logic, vol. 13 (2013), art. ID 1250007.
  • [6] Holický, P., S. P. Ponomarev, L. Zajíček, and M. Zelený, “Structure of the set of continuous functions with Luzin’s property (N),” Real Analysis Exchange, vol. 24 (1998/99), pp. 635–56.
  • [7] Kechris, A. S., and W. H. Woodin, “Ranks of differentiable functions,” Mathematika, vol. 33 (1986), pp. 252–78.
  • [8] Lempp, S., “Hyperarithmetical index sets in recursion theory,” Transactions of the American Mathematical Society, vol. 303 (1987), pp. 559–83.
  • [9] Saks, S., Theory of the Integral, 2nd revised edition, Dover, New York, 1964.
  • [10] Walsh, S., “Definability aspects of the Denjoy integral,” Fundamenta Mathematicae, vol. 237 (2017), pp. 1–29.
  • [11] Westrick, L. B., “A lightface analysis of the differentiability rank,” Journal of Symbolic Logic, vol. 79 (2014), pp. 240–65.