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
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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 $\mathsf {RCA}_{0}$:
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Cousin’s lemma for continuous functions is equivalent to $\mathsf {WKL}_{0}$;
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Cousin’s lemma for Baire class 1 functions is equivalent to $\mathsf {ACA}_{0}$;
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Cousin’s lemma for Baire class 2 functions, or for Borel functions, is equivalent to $\mathsf {ATR}_{0}$ (modulo some induction).
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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