Journal of Mathematical Logic 19 (1):1950001 (2019)

Authors
Sam Sanders
Ruhr-Universität Bochum
Abstract
We study the logical and computational properties of basic theorems of uncountable mathematics, including the Cousin and Lindelöf lemma published in 1895 and 1903. Historically, these lemmas were among the first formulations of open-cover compactness and the Lindelöf property, respectively. These notions are of great conceptual importance: the former is commonly viewed as a way of treating uncountable sets like e.g. [Formula: see text] as “almost finite”, while the latter allows one to treat uncountable sets like e.g. [Formula: see text] as “almost countable”. This reduction of the uncountable to the finite/countable turns out to have a considerable logical and computational cost: we show that the aforementioned lemmas, and many related theorems, are extremely hard to prove, while the associated sub-covers are extremely hard to compute. Indeed, in terms of the standard scale, a proof of these lemmas requires at least the full extent of second-order arithmetic, a system originating from Hilbert–Bernays’ Grundlagen der Mathematik. This observation has far-reaching implications for the Grundlagen’s spiritual successor, the program of Reverse Mathematics, and the associated Gödel hierarchy. We also show that the Cousin lemma is essential for the development of the gauge integral, a generalization of the Lebesgue and improper Riemann integrals that also uniquely provides a direct formalization of Feynman’s path integral.
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DOI 10.1142/s0219061319500016
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References found in this work BETA

Systems of Predicative Analysis.Solomon Feferman - 1964 - Journal of Symbolic Logic 29 (1):1-30.
Open Questions in Reverse Mathematics.Antonio Montalbán - 2011 - Bulletin of Symbolic Logic 17 (3):431-454.
In the Light of Logic.G. Aldo Antonelli - 2001 - Bulletin of Symbolic Logic 7 (2):270-277.
Ordinal Notations Based on a Weakly Mahlo Cardinal.Michael Rathjen - 1990 - Archive for Mathematical Logic 29 (4):249-263.
Reverse Mathematics and Π21 Comprehension.Carl Mummert & Stephen G. Simpson - 2005 - Bulletin of Symbolic Logic 11 (4):526-533.

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Citations of this work BETA

Splittings and Disjunctions in Reverse Mathematics.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (1):51-74.
Pincherle's Theorem in Reverse Mathematics and Computability Theory.Dag Normann & Sam Sanders - 2020 - Annals of Pure and Applied Logic 171 (5):102788.
Reverse Mathematics and Parameter-Free Transfer.Benno van den Berg & Sam Sanders - 2019 - Annals of Pure and Applied Logic 170 (3):273-296.
Reverse Mathematics of Topology: Dimension, Paracompactness, and Splittings.Sam Sanders - 2020 - Notre Dame Journal of Formal Logic 61 (4):537-559.

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