Topological solution to the continuum hypothesis

DATE: 12 November 2019

MSC classes:

03E50, 26A03, 97E60, 11Z05


For over a century, the study of transfinite ordinal and cardinal values has grown into a vast body of compelling research into the logic of infinite quantities. However, one of the most central notions of the field has remained an open problem since the beginning. The continuum hypothesis, which attempts to define specific bounds on countable and uncountable cardinalities, has become nearly an intractable problem, with the combined works of Cohen and Gödel showing that a proof for or against the hypothesis is independent of standard Zermelo-Fraenkel set theory, with or without the axiom of choice. This paper presents an approach to the problem through a straightforward geometric argument, resulting in theorems that reframe the continuum hypothesis in a topological context. These results are consistent with classical set theory and present both a resolution to the long-standing conundrum and the means for explicitly evaluating topological elements via cardinalities.

DOI: 10.13140/RG.2.2.18091.80162/1

ARTICLE: Fundamental representation of spinors via an algebraic calculus

DATE: 24 July 2019

MSC classes: 15A66 (Primary) 03E50, 20G45, 13NXX, 11Z05 (Secondary)

ABSTRACT: Here it is shown that by equipping nonordered sets of various dimensionality with a novel algebra, one can conveniently describe relationships that have been especially difficult to study—in particular, the mathematics of spinors, which underpin the physics of fermionic particles. In addition to a Clifford algebra, the described formalism produces a linear-algebraic version of differential calculus, resulting in a field generalization of the dual numbers—a quotient ring known for its property of automatic differentiation. Consequently, several new relations of spinors emerge and therefore inspire deep questions regarding the most fundamental particles of nature, while also providing an intuition for their underlying mechanics. Moreover, these results are derived in relation to notions of transfinite cardinality, motivating new theorems on the topology of the infinitesimal and countably infinite with which these particles are associated, and culminating in a resolution of the long-standing continuum hypothesis.

DOI: 10.13140/RG.2.2.19321.01126

ARTICLE: On ordinal dynamics and the multiplicity of transfinite cardinality

DATE: 1 June 2018

MSC classes: 11Z05 (Primary) 97E60, 54A25, 28A80 (Secondary)

ABSTRACT: This paper explores properties and applications of an ordered subset of the quadratic integer ring Z[(1+sqrt{5})/2]. The numbers are shown to exhibit a parity triplet, as opposed to the familiar even/odd doublet of the regular integers. Operations on these numbers are defined and used to generate a succinct recurrence relation for the well-studied Fibonacci diatomic sequence, providing the means for generating analogues to the famed Calkin-Wilf and Stern-Brocot trees. Two related fractal geometries are presented and explored, one of which exhibits several identities between the Fibonacci numbers and golden ratio, providing a unique geometric expression of the Fibonacci words and serving as a powerful tool for quantifying the cardinality of ordinal sets. The properties of the presented set of numbers illuminate the symmetries behind ordinals in general, as well as provide perspective on the natural numbers and raise questions about the dynamics of transfinite values. In particular, the first transfinite ordinal ω is shown to be logically consistent with a value whose cardinality is dual: both zero and one. Considerations of these points and opportunities for further study are discussed.

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