Infinity and Cardinality

Infinity is a topic which is generally rather hard to grasp. Our world does not readily provide anything infinite for us to study in the Sciences. Indeed, if some structure was infinite, we would not be able to study it fully; indeed, we would not be able to find all its edges. It would have no shape, no measurable volume, no mass, energy or velocity which we would be in any way able to comprehend. And yet, in mathematics, the idea of infinity can actually have meaning, which shows that our imagination is not solely confined to the extent of our experiences. More than that, infinity can actually help us understand the finite physical, chemical, biological etc. structures with which scientists deal all the time. Infinity helps us understand sequences, which in turn  helps in the understanding of limits, giving rise to the ideas of calculus, on which most contemporary Science and Engineering methods rely.

So, what is infinity? In general, we often treat it as a number. We may put it in the limits of a sum, a product or an integral, we may even give it as the result of these things. However, infinity is not, in fact a number. (This assertion is true in the reals. There are formal systems, like the hyperreal numbers, where infinity is a number. These systems are not what I am talking about here.)

Probably the most important work on infinity was done by Georg Cantor, when he formalised the ideas of infinite sets. Indeed, the best way to think of infinity is as a measure of the elements of an infinite set. As for finding such an infinite set, this is simple. We can simply take the positive integers (also known as the natural numbers), 1, 2, 3... and so on. We call this set N and consider the number of elements it contains. Obviously, it cannot be empty, as it contains the number 1. It cannot have only 1 element, as it contains the numbers 1 and 2. Similarly, it cannot contain only 100 elements, as every number from 1 to 101 is in N. For any positive integer we think of, we can find that many elements in N, and more. This is the very idea of infinity: something that is not bounded, which is larger than any number.

As we can see, this set of the natural numbers is infinite, in that it has infinitely many elements. Naturally, this means that any set which contains the natural numbers and others as well (a "superset" of N, as it is called) would also be of infinite size. We can take, for example, the set of all integers, Z. This includes the natural numbers (since they are the positive integers), as well as the negative integers and zero. Intuitively we can see that this must be a bigger set than N. Unfortunately, our intuition is wrong. When it comes to infinite sets, the only way to decide if Z is larger than N is to examine whether a bijection (one-to-one and onto function) exists between the two sets. If such a function can be found, then automatically the sets must have the same "size". Note that the natural numbers are essentially the numbers we count with, 1, 2, 3 and so on. So, finding a bijection between Z and N is the same as finding some way of listing all the elements of Z, without missing any, in a certain order. If such a sequence exists for Z, then Z and N have the same "size". This sequence is extremely easy to come up with: 0, 1, -1, 2, -2... and so on. Obviously this ordering will go through every single integer, much in the same way that 1, 2, 3... goes through every natural number.

The above shows us that Z has, in a sense, the same number of elements as N. Except that, as they both have infinitely many elements, we are not really talking about a number of elements. Because of this, the idea of cardinality was formulated. Cardinality can be thought of as the size of infinite sets, like the naturals or the integers. so, our example above shows that in fact both N and Z have the same cardinality, even if one is a subset of the other. This is a very important concept; infinity does not behave in the same way that other numbers do. Cutting infinity in half or doubling it does not actually change its size at all.

At this point one might think that all infinite sets must have the same cardinality as the natural numbers. After all, it should be easy to find a sequence that lists all the elements in a set. This is often the case; sets like the rational numbers, the algebraic numbers, the set of polynomials with integer coefficients and so on are certainly all countable (i.e. have the same cardinality as N). But there are sets which are actually much larger. The set of real numbers is actually uncountable. Any attempt to make a comprehensive list of all the reals, will result in failure. Indeed any such list can be shown to be missing some real numbers (in fact, an infinite amount of them). The proof for this uses a rather elegant method devised by Cantor, known as Cantor's diagonal argument.. I do not propose to present it here, as it is readily available online in other places. However, it should be clear from all this that the subject of cardinalities and the nature of infinity is one with a lot of substance.