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 (based on comprehension axioms), 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.

中文翻译：

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论不可数的数学和基础意义

我们研究了不可数数学基本定理的逻辑和计算性质，包括 1895 年和 1903 年发表的 Cousin 和 Lindelöf 引理。从历史上看，这些引理分别是开盖紧致性和 Lindelöf 性质的第一个公式。这些概念在概念上非常重要：前者通常被视为一种将不可数集合（例如 [公式：参见文本]）视为“几乎有限”的方式，而后者允许人们将不可数集合（例如 [公式：参见文本）处理] 为“几乎可数”。这种将不可数减少到有限/可数的做法证明具有相当大的逻辑和计算成本：我们表明，上述引理和许多相关定理极难证明，而相关的子覆盖极难计算. 确实，就标准尺度（基于理解公理）而言，这些引理的证明至少需要完全范围的二阶算术，该系统源自 Hilbert-Bernays 的 Grundlagen der Mathematik。这一观察结果对 Grundlagen 的精神继承者、逆向数学计划以及相关的哥德尔层次结构具有深远的影响。我们还表明，Cousin 引理对于规范积分的发展是必不可少的，它是 Lebesgue 和不正确 Riemann 积分的推广，它也唯一地提供了 Feynman 路径积分的直接形式化。这一观察结果对 Grundlagen 的精神继承者、逆向数学计划以及相关的哥德尔层次结构具有深远的影响。我们还表明，Cousin 引理对于规范积分的发展是必不可少的，它是 Lebesgue 和不正确 Riemann 积分的推广，它也唯一地提供了 Feynman 路径积分的直接形式化。这一观察结果对 Grundlagen 的精神继承者、逆向数学计划以及相关的哥德尔层次结构具有深远的影响。我们还表明，Cousin 引理对于规范积分的发展是必不可少的，它是 Lebesgue 和不正确 Riemann 积分的推广，它也唯一地提供了 Feynman 路径积分的直接形式化。