序动力学与超限基数的重数
On ordinal dynamics and the multiplicity of transfinite cardinality
摘要：
本文研究了二次整数环$\mathbb{Z}\left[\frac{1+\sqrt{5}}{2}\right]$的一个有序子集的性质及其应用.这些数字显示出奇偶三重态，而不是常见的偶数/奇数二重态。这些数字上的操作被定义和用于生成一个简洁的递归关系的研究斐波那契双原子序列，提供了生成类似物的著名的卡尔金-威尔夫和斯特恩-布罗科特树的手段。两个相关的分形几何提出和探讨，其中一个展示了几个之间的斐波那契数和黄金比例的身份，提供了一个独特的几何表达的斐波那契字，并作为一个强大的工具，量化的基数序数集。所提出的一组数字的属性照亮了一般序数背后的对称性，以及提供对自然数的看法，并提出了有关超限值的动态问题。特别是，第一超限序数$ω$被证明是逻辑上一致的值，其基数是双重的：零和一。考虑这些点和进一步研究的机会进行了讨论。
英文摘要：
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 $ω$ 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
