The Greeks were one of the first to discover infinity. They originally used the word apeiron which meant unbounded in their context. “For the Greeks, the original chaos out of which the world was formed was apeiron” (Allen). This seemed to be an uncomfortable topic for the Greeks to discuss because they believed that a number without limits is imperfect. The Greeks had three traditional observations that, “time seems without end, space and time can be unendingly subdivided, and space is without bound” (Allen). This leads to one of Zeno’s paradoxes of motion that an arrow in flight is really stationary. The hypothesis of this paradox is that time is composed of instantaneous moments. His reasoning behind this is that we can never say when ‘now’ really happens. When we say ‘now’, it automatically slips into the ‘not now’ the very moment it is being said. The arrow must be in place ‘A’ at time ‘A’. From our previous statement, since time ‘A’ is actually in time ‘B’, then the arrow has no true movement according to time. Time is also something that may only exist in our mind if we think about it as an objective unit of measurement. Therefore, the arrow appears to be moving in our minds, but it is never really moving. Relating this back to the concept of infinite, an infinite number of instances of time can be said to accompany each ‘moment’ of the arrows flight. Thus, the flying arrow is a stationary object. This may be a different theory to wrap our heads around, but if we truly think about the nature of time and space, then it may not be as simple as it appears. “For example, we know that time seems to flow at different rates for different people. Modern physics even describes phenomena like time dilation. Therefore, how can we be sure that there is an all-encompassing objective time, i.e. a single time that flows uniformly and is the same for everyone?” (Tassone). I can relate to this style of thinking in a sense that when talking with friends or classmates, they may feel that a fifty-minute lecture breezed by while I may feel it went on for a long time.
Galileo Galilei was involved with examining geometry and how circles related to polygons. The conclusion he made about a regular polygon was that if it were to have infinite sides, then it would be a circle (Aczel). Like many others that I will discuss, he was able to increase his knowledge about cardinality by applying the topic of infinity. In relation to his studies of geometry, he determined that the number of elements in natural numbers is the same as the number of elements in a set of its squares. This discovery was furthered by him realizing that two sets of different sizes can have the same number of elements.
Fast forward a few hundred years and we now learn about another person who had a significant contribution to the idea of infinity: Georg Ferdinand Ludwig Philipp Cantor. Cantor was born in St. Petersburg Russia and attended the University of Halle where at the early age of thirty-five, became a professor. His focus was on the subject of transfinite set theory. This eventually led to his work on the concept of infinity. He used set theory and transfinite arithmetic; Cantor believed that counting to infinity could eventually become uncountable. His famous continuum hypothesis was, “that 2ℵ0 = ℵ1 where ℵ0 represents the smallest infinity, countable infinity, and ℵ1 represents the infinity just larger than that, an uncountable infinity, which Cantor referred to as the absolute” (Aczel). After many years, he was unsuccessful in either proving or disproving his hypothesis. Cantor suffered from depression and bipolar disorders and many believed that his downfall was from this hypothesis that he was unable to solve.
