On ordinal dynamics and the multiplicity of transfinite cardinality

TL;DR

This paper investigates an ordered subset of quadratic integers revealing new parity properties, recurrence relations, and fractal geometries that deepen understanding of ordinal dynamics and transfinite cardinality.

Contribution

It introduces a novel set of quadratic integers with unique parity and geometric properties, linking them to Fibonacci sequences and transfinite ordinal concepts.

Findings

Defines operations generating Fibonacci analogues

Explores fractal geometries related to Fibonacci and golden ratio

Proposes a dual cardinality for the first transfinite ordinal 

Abstract

This paper explores properties and applications of an ordered subset of the quadratic integer ring $\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$. 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 $\omega$ 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.