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- Title
Cousin's lemma in second-order arithmetic.
- Authors
Barrett, Jordan Mitchell; Downey, Rodney G.; Greenberg, Noam
- 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 RCA0: Cousin's lemma for continuous functions is equivalent to WKL0; Cousin's lemma for Baire class 1 functions is equivalent to ACA0; Cousin's lemma for Baire class 2 functions, or for Borel functions, is equivalent to ATR0 (modulo some induction).
- Subjects
LEBESGUE integral; COUSINS; REVERSE mathematics; CONTINUOUS functions; GENERALIZATION
- Publication
Proceedings of the American Mathematical Society, Series B, 2022, Vol 9, p111
- ISSN
2330-1511
- Publication type
Article
- DOI
10.1090/bproc/111