Some Results in Calculus and Set Theory through the Directed Set Method

Abstract

So far, real numbers have not dealt with sums of arbitrary series or given answers to every infinite operation application like . They surrender to the paradigm that not every continuous function is differentiable, and no one is Riemann integrable. Likewise, set theory does not hint at existing number-like entities that are as many as ordinals but retain field axioms or a set universe expressed as functions on natural numbers with finite sets as its values, which is very unfortunate as there is no analysis for such entities. This paper aims to unravel the blockades that hinder the solutions to such problems through something described as follows. Suppose in a world of sequences of balls, every single color from red, green, and blue appears separately. They are evaluated term by term by how they end up, and such a sequence is “green” if, in the end, all of them are green balls. Such a sequence is darker than another if the former is eventually always darker than the latter. However, if red and green boxes appear one after another forever, then it is more appropriate to describe them as red–green, namely “yellow,” a color previously unknown. The above is how new colors occur in the presence of endless sequences of colored balls. Such new objects would serve as solutions to the above problems. This research aims to supply calculus with more than just real numbers, called constructed numbers, so that undefined notions such as undefined derivatives or integrals, divergent sequences, can be given values in the hope that through these values, they can be further studied in more detail. Another goal is to explore the Zemelo–Fraenkel set theory to view it one step closer to the authors’ conjecture that everything in its universe and mathematics are expressible as sequences of finite structures. Sets defined there are likewise called constructed sets. Objects such as constructed numbers or sets they called directed set objects or constructed objects. These objects may not be completely tangible, but like a telescope that can zoom the moon’s surface as sharply as anybody wishes but will never touch it. The directed set method that produces constructed objects is the novelty of this study.

Keywords: directed set objects, constructed derivatives, constructed Riemann integrals, sums of arbitrary series.

https://doi.org/10.55463/issn.1674-2974.50.10.19

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