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BIG IN REVERSE MATHEMATICS: MEASURE AND CATEGORY

Part of: Proof theory and constructive mathematics General logic

Published online by Cambridge University Press:  17 October 2023

SAM SANDERS
SAM SANDERS*
Affiliation:
DEPARTMENT OF PHILOSOPHY II RUHR-UNIVERSITÄT BOCHUM BOCHUM, GERMANY
*
E-mail: sasander@me.com
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Abstract

The smooth development of large parts of mathematics hinges on the idea that some sets are ‘small’ or ‘negligible’ and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely ‘measure zero’, originated with Lebesgue, while a second smallness notion, namely ‘meagre’ or ‘first category’, originated with Baire around the same time. The associated Baire category theorem is a central result governing the properties of meagre (and related) sets, while the same holds for Tao’s pigeonhole principle for measure spaces and measure zero sets. In this paper, we study these theorems in Kohlenbach’s higher-order Reverse Mathematics, identifying a considerable number of equivalent and robust theorems. The latter involve most basic properties of semi-continuous and pointwise discontinuous functions, Blumberg’s theorem, Riemann integration, and Volterra’s early work circa 1881. All the aforementioned theorems fall (far) outside of the Big Five of Reverse Mathematics, and we investigate natural restrictions like Baire 1 and quasi-continuity that make these theorems provable again in the Big Five (or similar). Finally, despite the fundamental differences between measure and category, the proofs of our equivalences turn out to be similar.

Keywords

Baire category theorempigeonhole principle for measure spacesReverse Mathematicshigher-order arithmeticweak continuity

MSC classification

Primary: 03B30: Foundations of classical theories (including reverse mathematics) 03F35: Second- and higher-order arithmetic and fragments
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 46
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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