Let X be a compact metric space. A closed set K $\subseteq$ X is located if the distance function d(x, K) exists as a continuous real-valued function on X; weakly located if the predicate d(x, K) $>$ r is Σ 0 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL (...) 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets. (shrink)

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. (shrink)

We develop fundamental aspects of the theory of metric, Hilbert, and Banach spaces in the context of subsystems of second-order arithmetic. In particular, we explore issues having to do with distances, closed subsets and subspaces, closures, bases, norms, and projections. We pay close attention to variations that arise when formalizing definitions and theorems, and study the relationships between them.