On the mathematical and foundational significance of the uncountable

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 (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.

Splittings and Disjunctions in Reverse Mathematics

“…By the results in [50,51], Z Ω 2 proves HBU, but Z ω 2 ≡ ∪ k Π 1 k -CA ω 0 cannot. The importance and naturalness of HBU is discussed in Section 4.…”

“…The Lindelöf lemma. We study splittings and disjunctions for the Lindelöf lemma LIN from [50]. We stress that our formulation of HBU and LIN is faithful to the original theorems from 1895 and 1903 by Cousin ([15]) and Lindelöf ([36]).…”

“…, f n such that the set of ∪ i≤n [f i F (f i )] still covers 2 N . By [50,Theorem 3.3], HBU is equivalent to the same compactness property for C, as follows:…”

“…To be or not to be continuous. The results in Section 3 and [48][49][50][51] identify huge differences between second-and higher-order RM. As discussed in the previous section, these results trace back to the fact that higher-order (resp.…”

“…In particular, we obtain splittings and disjunctions involving (higher-order) WWKL 0 , the Big Five, and Z 2 , among others. We similarly treat the covering theorems Cousin's lemma and Lindelöf 's lemma studied in [50]. Our main results are in Section 3, while a summary may be found in Section 5; our base theories are generally conservative over WKL 0 (or are strictly weaker).…”