The History of Infinity.

Infinity is today so well integrated into today's language that we can scarcely imagine many thoughts and expressions without it. However, despite its widespread use, infinity is one of those objects we scarcely understand. Most of us view time as infinite and space as infinitesimally decomposable and possibly infinite in breadth, even though both involve unmeasurable, unfathomable dimensions that defy comprehension. Yet, infinities (yes, there are several) are very, very useful to "tie" things together, to provide comprehensible models, and for the mathematician to provide a completion of mathematical theories that actually simplifies statements, proofs, and applications.

As there is no record of earlier civilizations regarding, conceptualizing, or discussing infinity, we will begin the story of infinity with the ancient Greeks. However, the ancient Greeks had immense difficulties with infinity. They never could quite accept it as we do today. But they did imagine potential infinity. So, for example, Aristotle (384 - 322 BCE) would allow the integers (1,2,3,...) to be potentially infinite but not actually infinite in quantity. It was Zeno of Elea (c. 490 - c. 430 BCE), who brought to the fore several contradictions between the discrete and the infinite. Applying finite reasoning to essentially infinite processes, Zeno provided four paradoxes that challenge students to this day. If one wishes to list failures of Greek mathematical thought and has the courage to do so, one must mention the Greek inability to deal with infinity and infinite process as among their greatest.

Even after infinity was defined by Cantor (1845 - 1918) it did not mean the the end of the paradox. More and more subtle paradoxes were constructed. For example, one of the simplest is Russell's paradox which was first used to establish the paradox of the greatest cardinal. It was later "purified" to reveal semantic difficulties with language itself. On the other hand, one of the most complex and mystifying paradoxes is the Banach-Tarski Theorem, an actual theorem, which allows the decomposition of a bowling ball and reassembly into a giant planet earth-sized bowling ball - and with just a finite number of pieces!

Infinity is fraught with issues that require the most careful study. This chapter is about them; it is about primitive concepts and the evolution to modern ideas. There are many excellent references, some of which are at the end of the first reading.

The History of Infinity