Proceedings of the American Mathematical Society Series B

Published by the American Mathematical Society since 2014, this gold open access, electronic-only journal is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 2330-1511

The 2024 MCQ for Proceedings of the American Mathematical Society Series B is 0.95.

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Cousin’s lemma in second-order arithmetic
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by Jordan Mitchell Barrett, Rodney G. Downey and Noam Greenberg;
Proc. Amer. Math. Soc. Ser. B 9 (2022), 111-124
DOI: https://doi.org/10.1090/bproc/111
Published electronically: April 8, 2022

Abstract:

Cousin’s lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin’s lemma for various classes of functions, using Friedman and Simpson’s reverse mathematics in second-order arithmetic. We prove that, over $\mathsf {RCA}_{0}$:

  1. Cousin’s lemma for continuous functions is equivalent to $\mathsf {WKL}_{0}$;

  2. Cousin’s lemma for Baire class 1 functions is equivalent to $\mathsf {ACA}_{0}$;

  3. Cousin’s lemma for Baire class 2 functions, or for Borel functions, is equivalent to $\mathsf {ATR}_{0}$ (modulo some induction).

References
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Bibliographic Information
  • Jordan Mitchell Barrett
  • Affiliation: Victoria University of Wellington, P.O. Box 600, Wellington, 6140, New Zealand
  • MR Author ID: 1414645
  • Email: math@jmbarrett.nz
  • Rodney G. Downey
  • Affiliation: Victoria University of Wellington, P.O. Box 600, Wellington, 6140, New Zealand
  • MR Author ID: 59535
  • Email: rod.downey@vuw.ac.nz
  • Noam Greenberg
  • Affiliation: Victoria University of Wellington, P.O. Box 600, Wellington, 6140, New Zealand
  • MR Author ID: 757288
  • ORCID: 0000-0003-2917-3848
  • Email: greenberg@msor.vuw.ac.nz
  • Received by editor(s): May 5, 2021
  • Received by editor(s) in revised form: October 18, 2021
  • Published electronically: April 8, 2022
  • Additional Notes: The second and third authors were partially supported by the Marsden Fund of New Zealand. Many of the results in this paper are also contained in Barrett’s honours thesis.
  • Communicated by: Vera Fischer
  • © Copyright 2022 by the authors under Creative Commons Attribution-NonCommercial 3.0 License (CC BY NC 3.0)
  • Journal: Proc. Amer. Math. Soc. Ser. B 9 (2022), 111-124
  • MSC (2020): Primary 03B30, 03F35, 03D78, 26A39
  • DOI: https://doi.org/10.1090/bproc/111
  • MathSciNet review: 4405506