On the computational properties of basic mathematical notions

Abstract

We investigate the computational properties of basic mathematical notions pertaining to ${\mathbb R}\rightarrow {\mathbb R}$-functions and subsets of ${\mathbb R}$, like finiteness, countability, (absolute) continuity, bounded variation, suprema and regularity. We work in higher-order computability theory based on Kleene’s S1–S9 schemes. We show that the aforementioned italicised properties give rise to two huge and robust classes of computationally equivalent operations, the latter based on well-known theorems from the mainstream mathematics literature. As part of this endeavour, we develop an equivalent $\lambda $-calculus formulation of S1–S9 that accommodates partial objects. We show that the latter are essential to our enterprise via the study of countably based and partial functionals of type $3$.

The Biggest Five of Reverse Mathematics

“…This sequential notion goes back to Baire's equivalent definition of usco (see [5]) and the associated restriction of Jordan decomposition theorem is readily 8 seen to be equivalent to ACA 0 . A similar result can be obtained for Baire 1 * formulated using RM-codes for closed sets, in light of [78,Lemma 4.11].…”

“…Now, arithmetical comprehension suffices to derive Theorem 1.4 for various kinds of second-order representations of BV -functions in [57,69]. By contrast, the results in [75][76][77][78] show that the Jordan decomposition theorem is even 'explosive': combining with the Suslin functional from Π 1 1 -CA ω 0 (see Section A.3), one derives Π 1 2 -CA 0 . Fourth, we shall make use of the following notion of (open and closed) set, which was studied in e.g.…”

“…Fourth, we shall make use of the following notion of (open and closed) set, which was studied in e.g. [75][76][77][78]89].…”

Advantages of a Free Software Culture for Qualitative Researchers in the Social Sciences

“…Three great events that took place in the Nineteenth Century bore a direct impact on the development of science: 1) the onset of Thermodynamics in the year 1811 and the ensuing Entropy principle; 2) the flourishing of Biology as influenced by Darwin's works in year 1859, with the evolution and temporality principle; and lastly 3) the emergence of Chemistry in year 1871, and the resulting development of Set Theory in Mathematics between years 1874 and 1884 (Court, 2020;Normann & Sanders, 2022;Scerri, 2020;Starikov, 2021). This process evolved piecemeal and side by side with the divorce between Philosophy and Science in the pursuit of the truth.…”