Infinity has been a concept that has fascinated me since the days I graduated college. The most amazing realization for me was that infinity was a relative concept. One normally thinks of infinity as absolute, that infinity is the ultimate in terms of size, extension, and grasp. Well, the truth is that there is an infinite number of infinities!
Infinity is a Real Mind-Blower
When I came to the above realization in my upper senior year in college, I left set theory class in a state of stupefaction. I remember strolling out of the classroom and walking in utter amazement over the import of what I had just learned. To say this experience was mind-blowing would be a serious understatement. The day this happened, we were working on the proof that the Natural Numbers, that is the set {1, 2, 3,...}, was not as infinite as the set of real numbers, which includes as subset the set of Natural Numbers, the set of all fractions, and the set of all non-terminating, non-repeating decimals, the so called irrationals.
This fact seemed totally counter-intuitive as infinity is well...infinite. If a set was clearly infinite, then intuitively it would seem, such set should be able to muster muscle with anything---including another infinite set. Yet this is not true and here, by a simple argument, we shall show why. Before we get to this argument, we start with a discussion of finite sets. It is easy to show that two sets are of the same size simply by being able to pair off elements of each representative set. For example, the set {1, 2, 3} has the same size as the set {Alice, George, Sandy}. Obviously, both sets have three elements, albeit distinct.
One-to-One Correspondence
To make this size comparison more precise, we form a pairing, let us say, by assigning one element from one set to another element of another set. Thus we could pair 1 with Sandy, 2 with George, and 3 with Alice. The way we pair does not matter; what matters is that every element of the first set has a partner in the second set. In mathematics, such a pairing is called a one-to-one correspondence. For sets that have the same number of elements, we should be able to make such a pairing; yet for sets that do not have the same number of elements, we should not. That is, if one set has more elements than another, then no matter how we assay to form a one-to-one pairing or correspondence, then there will always be elements of one set left over, or that do not have a corresponding partner in the other set.
The above argument is easy with finite sets, yet becomes a bit trickier with infinite sets. This notwithstanding, the principle remains the same: if we can form a pairing so that all elements are "tied up," so to speak, then the two infinite sets display the same type of infinity. When this occurs, we say the sets are infinitely co-numerous. Another term used for this equality of infinities, so to speak, is that the sets have the same cardinality, or same cardinal number.
The Natural Numbers are infinitely co-numerous with the integers and the fractions, yet not with the set of real numbers. Although these proofs quite easy, we will not show them here. Rather we will show that the Natural Numbers do not display the same kind of infinity as the set of real numbers; essentially proving that there is at least two kinds of distinct infinities. What is then extrapolated from this result---although not shown here (I guess you will have to take this on faith)---is that there is an infinite number of distinct infinities!
The Superiority of the Real Numbers
To show that the real numbers display an infinity greater than that of the natural numbers is quite easy. Although easy, the argument is subtle, and I will do my best to make this truth self-evident. Since the Natural Numbers can serve as a listing, that is, we can use these numbers to list items 1, 2, 3, 4,...etc., then the irrationals should be resistant to such a listing. By showing that we cannot list all such real numbers, then we will have shown that you cannot place them in one-to-one correspondence with the naturals. In this specific argument we will use the irrationals (those non-terminating non-repeating decimals) between 0 and 1, for if we can show that a subset of the reals cannot be placed in one-to-one correspondence with the naturals, then certainly the entire set of reals cannot be.
We create a partial list of these infinite decimal irrationals to show what this set looks like:
0.1383848484440138...
0.744839810483740...
0.3330284874740320983...
This listing goes on forever in both directions; that is, the decimal expansion and the number of items in the listing continue forever. Remember that these are irrational numbers and by definition, their decimals do not end and do not repeat. For if either of those cases occurred, they would not be irrational numbers but rational ones.
Cantor's Diagonal Argument
Now to show that we could never list all these irrationals between 0 and 1, we need to generate at least one irrational that is not on the list. Here is where a very clever device comes in, made famous by the mathematician Georg Cantor, who employed it. We go along the diagonal of our list, taking the digit in that position. Thus for our list above, we would form the irrational number 0.143...We now alter this number by adding 1 to each digit in its expansion, obtaining 0.254...
At this point, this is where the subtlety of the argument comes into play. We claim that this number is no where on our list. Remember that our listing supposedly enumerated all the irrationals between 0 and 1. Yet now we have produced an irrational that is nowhere on our list. The reason is simple: if this number were on our list, then it would have to match one of the numbers digit by digit. Yet this is impossible based on its construction, for we have changed this irrational by adding 1 to each of its digits.
Consequently, this new irrational cannot match let us say the 3rd number on our list because we have added 1 to the 3rd position. Thus our new irrational 0.254... does not match 0.3330284874740320983... in the 3rd position because we have a 4 instead of a 3. This is the case if we try to match this new irrational with any on the list: the numbers will always differ in at least the nth position (that is to say along the diagonal) because of the way we created this number. Thus our supposed listing of the irrationals between 0 and 1 is incomplete and no matter how we try, we can never accomplish a complete listing because there will always be irrationals left over.
Infinity of Infinities
Consequently the irrationals between 0 and 1, as a subset of the reals, can never be put into one-to-one correspondence with the natural numbers. This is a euphemistic way of saying that the naturals are mere pussy cats in terms of the type of infinity that they display. The reals are the real machos, having an infinity greater than the naturals. Indeed the proof just completed shows that a small interval of the reals, that between 0 and 1, has a bigger infinity than that of all the naturals. Indeed very strange things happen when we enter the realm of infinite sets.
Even more amazing than this result is that by extension, we can prove that an infinite number of infinities can be generated mathematically, although envisioning such sets, or even finding sets of mathematical objects that display such infinities are unknown. Wow, if that is not a mouthful!
As you can clearly understand now, my stupefaction upon leaving set theory class that day was quite justified given the import of the results we have just explored. Such is the amazing world of mathematics, and the reason why we should never cease to explore this fascinating subject, as even the layperson can find enlightening tidbits that help explain the world around him.
Joe Pagano - Joe is a prolific writer, author, poet, and linguist, who has published over 200 articles and several books on a broad array of ...
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