In Mathematics, Infinity is not a thing, it is many things. Transfinite arithmetic produces an infinity of infinities! In fact, Infinity opens up so many degrees of freedom that paradoxical behavior seemingly becomes the norm.
This monograph introduces the reader to this fascinating topic and
submerges him/her into its mysteries and the paradoxes that result from a careful analysis of the Infinite. The monograph culminates with the introduction, and heuristic proof, of the famous Banach-Tarski Paradox.
A background in High School Algebra and Analytic Geometry will be helpful, as such courses get one used to thinking in such terms. But, really, the only mathematics required is some fancy arithmetic and an ability to perhaps visualize 3-D rotations in space. Mostly, thinking logically and rigorously should suffice.
It is fitting to proclaim here that everything of interest in this monograph ultimately traces back to the intellectual efforts of one man: Georg Cantor [1845-1918], a famous 19th century German mathematician, is the creator of transfinite set theory and the theory of transfinite numbers.
Cantor remains one of the most imaginative and controversial figures in the long history of mathematics. He was controversial mostly because his work created a major upheaval in mathematics during its day. In fact, some of his peers attempted to prohibit him from publishing his work on Transfinite Set Theory! Such was their fear of its disruption to Mathematics as a discipline. Cantor culminated his life’s work in a two volume set [1895 and 1897] referred to as his Beitrage.
Luckily, Cantor’s detractors lost their philosophical assault on what they viewed as heretical mathematics. Cantor’s influence survives to this day as his research not only established an entirely new field of mathematics, but it also has deep importance in the fields of topology, number theory, analysis, the theory of functions, as well as the entire field of modern logic.
This brief exposition and condensation of some of the fascinating and fun results that flow from a small glimpse into Cantor’s world is in his honor.
