Boy this math stuff can be hard to understand. I mean concepts like infinity, differential calculus, set theory. Good grief. John von Neumann, the famous mathematician, said "Young man, in mathematics, you don't understand things, you just get used to them." That's exactly the way I feel sometimes when I myself read and study about this most curious subject. I guess the reason why I excel at explaining concepts in mathematics is that I have experienced so much frustration in trying to understand this subject from another's perspective that I go out of my way to break things down so that even a baby could understand. Well, not a baby, but you know what I mean. Often I go and do research on the internet and read articles there. It is sad that well written, intelligible articles are hard to find. And this is coming from someone who has studied mathematics at the graduate level!

I remember back in graduate school, how hard it was to read the textbooks and make any sense out of what was being taught. The stuff literally was layered with abstraction after abstraction. The theoretical material was explained in a way that would daunt even the most scholarly. No wonder people never really get past basic mathematics, and no wonder this subject is easily despised by so many. Goodness gracious, even mathematicians find the material mind boggling. However, one can come to understand the material if"and only if"it is broken down in a way that humans can comprehend. Granted, mathematics is hard. Especially when one starts to delve into concepts like infinity and the like. Given this, I still say that if you have the right teacher to explain something, progress can be made in this subject. It is indeed true that in mathematics particularly, the teacher can make you or break you. Thus you should not feel too bad and think too little of yourself if you had trouble with mathematics in school.

Now let's get back to this concept of infinity and the case made that this is not a universal concept. What is meant by this is that there is not a unique infinity. There are actually"and this can be shown mathematically"infinitely many kinds of infinity. (Wow, if this is not an amazing fact then I don't know what is.) To get you to understand this, ultimately I am going to show that the set of numbers {1, 2, 3, ...}, which go on forever, cannot be paired with the decimal numbers between the interval 0 and 1.

Let's examine this a little more carefully. If I take a set of objects, let us say, the set of odd numbers between 1 and 9, then I have the set {3, 5, 7}; a set consisting of three members. If I now take the set of three boys named {John, Jules, and Steven}, I have a set of three boys. How can we determine which set has more elements? Well, it is quite intuitive that we can pair off the elements from each set and we see that no element from the set {3, 5, 7} would be unpaired with an element from the set of three boys. Of course, there are a couple of ways we could do the pairing but the main thing is that each set has the same number of elements, although these elements are not necessarily the same thing. What we are interested in is that the number of elements of each set is the same and is equal to three. We call three the cardinality of each set. Since the number three is finite, the cardinality of these sets is finite.

Clearly for the set of counting numbers {1, 2, 3, ...}, we can not associate a number which represents its cardinality. If we chose a billion to represent the cardinality, this would be wrong because the set has elements which surpass one billion. Similarly if we chose any predetermined number, regardless of its size, we can always go out further in this set and find a bigger number. Mathematicians call the cardinality of the set of counting numbers "aleph nought," which is the Hebrew letter aleph with the subscript of 0. This turns out to be the first "transfinite" number, or number which is not finite. Some very strange and bizarre things happen when we enter the realm of transfinite numbers. For one, it seems as though certain infinite sets have more elements than another infinite set, when in actuality both sets have the same number of elements! What? How can this be? Follow this example carefully. Take the counting numbers {1, 2 ,3...} and the set of odd counting numbers {1, 3, 5...}. It would seem that there should be more counting numbers than odd counting numbers, but in fact, this is not the case. How can we show this?

The technique of pairing elements is the way we show that two different infinite sets have the same cardinality. This pairing is called a "one-to-one" correspondence. For the two finite sets mentioned above it works like this: we pair 1 with John, 2 with Jules, and 3 with Steven. Therefore each element of the first set is paired with exactly one element from the second set. There is no spillage, or leftover elements that are not paired. That is all there is to the notion of one-to-one correspondence.

It makes sense then if we can pair each element of one infinite set with each element of another infinite set so that there is no spillage, then the two sets, though infinite, must have the same number of elements, or cardinality. To show how this is done with the counting numbers and odd counting numbers is very simple: pair 1 with 1, 2 with 3, 3 with 5, 4 with 7, 5 with 9, and so on, ad infinitum. Since the odd numbers go on forever, we can go further out in this set to pair the elements from the counting numbers. Thus even though at first it seems that the set {1, 2, 3,...} is more numerous than the set {1, 3, 5, ...}, this we now see is entirely untrue. Strange realities indeed exist when it comes to infinite sets!

In Part III, I will show you how and why the decimal numbers between the interval 0 and 1 are more numerous"that is display an infinity which is greater"than the set of counting numbers, though the latter go on forever. Very strange indeed! What bizarre things lurk in this world of transfinite numbers.