Cousin’s lemma in second-order arithmetic

Barrett, Jordan; Downey, Rodney; Greenberg, Noam

doi:10.1090/bproc/111

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 $\mathsf {RCA}_{0}$:

Cousin’s lemma for continuous functions is equivalent to $\mathsf {WKL}_{0}$;
Cousin’s lemma for Baire class 1 functions is equivalent to $\mathsf {ACA}_{0}$;
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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Proceedings of the American Mathematical Society Series B

Cousin’s lemma in second-order arithmetic
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Abstract: