"Infinity is a floorless room with no walls or ceiling." "Anonymous For those who have followed my series on infinity, this article gets to the crux of the matter concerning the existence of different types of infinity. Having read the two previous articles, you get the sense that the topic of infinity is a strange one indeed. Throughout the ages both philosophers and mathematicians have been debating this topic and puzzling over its implications. From the fifth century Eleatics (the Greek philosophers from Elea: Zeno, Parmenides, and Melissus) all the way through to the famous German mathematician Georg Cantor, who is known as the father of modern set theory, great thinkers as these have pondered and labored feverishly in trying to nail down a precise formulation for this seemingly surreal idea. As the quote above illustrates, infinity defies our concept of dimension in that we cannot put a bound around it and yet, based on the work of Cantor, the idea of only one type of infinity can be shown to be untrue. This concept is so mystifying and at the same time enervating to contemplate that Cantor saw his health fail in the face of the constant denunciation that he, in espousing these ideas, received from his contemporaries. What a price to pay for such forward progress in mathematics, as Cantor's work led to critical foundations in both functional analysis and topology, two higher branches of this discipline. At any rate, the proof that the real numbers are more numerous"that they display a "bigger" infinity"than the counting numbers is quite simple. The implications of this proof are mind-quickening and the extensions of such proof yield a whole hierarchy of transfinite numbers. Before we get to the proof (which is quite simple and far unlike the proof I studied in college, which was quite elaborate and required lots of coffee and a minimum of three aspirins to understand), I want to make some preliminary comments and edify you on a couple of points regarding the real numbers and what we mean by them. The field of real numbers consists of all the counting numbers {1, 2, 3,...}, the negatives of the counting numbers {...-3, -2, -1}, all the fractions (what we mathematicians call rational numbers"because they are sane), and the numbers like square root of 2, square root of 3, the number pi, and 0. The claim we are making here is that there are more decimal numbers between the interval 0 and 1, that is numbers like 0.12, 0.0498, etc. than there are all the counting numbers {1, 2, 3,...}. At first blush, it would appear that since the set of counting numbers is infinite, and infinity means that there are no limits, that there is no end, that there are no bounds"you get the picture"then we should have the same number of elements between 0 and 1 as there are counting numbers. Ah, but there's the rub, as this is not true; and for those of you who have been thinking ahead, the reason may have already dawned upon you. Georg Cantor finally proved this fact using his famous diagonal proof, but we will use an approach that is even simpler. The method is also based on Cantor's idea of "pairing elements," which is known as a "one-to-one correspondence."(Both these concepts were discussed in Part II of this article.) Basically, we show how we can pair each element of the counting numbers with an element in the interval 0 to 1. For example, we could pair 1 with 0.25 and 2 with 0.354. If we do this in such a way as to show that every counting number is "paired" or tied to a different number between 0 and 1, then we will have shown that all the counting numbers have been matched with a distinct group of numbers from this interval. Once we have done that, we then show that there are still many numbers between 0 and 1 that have no "dates," so to speak; that is, that there are unmatched numbers from the interval in question. This would mean that there are still more numbers in this set and therefore prove our argument. Isn't mathematics grand! So how do we do this? Very simply. Now watch carefully as the simplicity of this will astound you. We set up the following one-to-one correspondence between the set {1, 2, 3,...} and the interval 0 to 1, as follows: we pair 1 with 0.1; 2 with 0.11; 3 with 0.111; and we do this forever. Now every counting number is tied to a unique number in the interval 0 to 1. Clearly, any number 0.1, 0.11, etc., is in the interval in question and each one differs from the next by the next place over. For example, 0.1 and 0.11 differ by one one-hundredth; 0.11 and 0.111 by one one-thousandth, and so on. Since this pattern goes on forever in a manner that keeps every number in the form 0.1111111...... within the interval, we have exhausted every possible counting number. Ah, but what about a number like 0.2 or 0.046? The possibilities are endless. Since every counting number is already paired with a number in the interval 0 to 1, these two new numbers have no representation in this one-to-one pairing. Consequently, there must be many more numbers in the interval 0 to 1 then all the counting numbers, and hence we have established indubitably that there exists more than one type of infinity. Wow! When I first learned this fact and its necessary extension, which leads to the existence of infinitely many infinities, my mind expanded so much and I blew so many circuits that I had a headache for three days! Chew on this tidbit for a bit and see whether you now think that the existence of God is so hard to fathom. |
