Dabbling in Infinity

Philosophers through the ages have pondered on and dabbled in the concept of infinity. In fact, the Calculus hinges its very essence on this most noble concept. Yet if sets of numbers can shatter our preconceived notions of this idea, which is more or less universally accepted as something real, what more can we uncover by plunging into the mysteries of numbers and mathematics in general?

In continuation of my discussion on infinity and its implications with the divine, I should mention that the concept of there existing infinities beyond imagination is quite difficult to comprehend. If you read my poem, "How Can this Be? (See the link below to this collection of poems) you read in verse the proof that shows clearly that there is no such thing as one kind of infinity. The extension of this most curious fact is that there are actually an infinite number of infinities!

Occasionally I wax metaphysical in conversations with my uncle and the other evening we were discussing some points regarding the spiritual realms. In passing, I brought up the topic of infinity and I asked him his impression of it. His response, which is typical of most people, is that infinity is just that—infinity something that never ends. But how do we make this vague notion somewhat more concrete? I pointed my uncle's attention to the set of natural, or counting numbers. This set comprises the familiar numbers {1, 2, 3, ...}. The numbers go on and on, falling like dominoes, and never reaching a "biggest one." This process is easy to grasp and presents no ordinary difficulty for the average person. What does become difficult to understand is why the infinity typified by this set of numbers is not unique.

Now let's delve a little more deeply into this curious set of numbers and the topic of infinity in general. This set of counting numbers obviously never ends. If you have ever seen a chronometer counting hundredths of a second, then you have seen how fast the digits representing the hundredths of a second whiz by, not appearing for any length of time sufficient to allow recognition of the appropriate digit. And this is for hundredths of a second. Imagine the same chronometer counting off thousandths of a second. Now imagine this going on from, let us say, ten thousand years ago and continuing for another ten thousand years, starting with 1 and such that each thousandth of a second would represent the next sequential counting number. Think of how far along in the set of counting numbers you would be. We could actually compute the number but we are only interested in trying to conceptualize how large potential infinity could be.

Now that we have this huge number in hand, we could do whatever we wanted with it to project ourselves much further out in the set of counting numbers. We could multiply it by itself ten times (the mathematical way of saying we can raise the number to the tenth power); we could multiply it by itself a hundred times, a thousand times, and so on. We could then take the largest product and do the same process all over again. How big is this set?
This set is so large—never ending in fact—that we should be able to use it to compare to anything else that is infinite, right? Wrong. And in a continuing article on this most fascinating subject, I will discuss how this notion of one universal infinity is completely wrong. Thus, if sets of numbers can shatter our preconceived notions of a concept like infinity, which is more or less universally accepted as something that is real, what more can we uncover by plunging into the mysteries of numbers and mathematics in general? Stay tuned......

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Joe is a prolific writer of self-help and educational material and an award-winning former teacher of both college and high school mathematics. Under the penname, JC Page, Joe authored Arithmetic Magic, the little classic on the ABC's of arithmetic. Joe is also the author of the original collection of poetry, Poems for the Mathematically Insecure, and the short but highly effective fraction troubleshooter Fractions for the Faint of Heart.