Infinite Sets

As the name would suggest, an infinite set is a set that is not finite. Infinite sets turn out to play an important role in the advanced study of calculus -- and you can probably imagine why: R, the set of all real numbers, is an infinite set. [Is this obvious? Sure! You know a non-finite subset of R, namely: 1, 2, 3, 4, ... .]

What you may not know is that infinite sets come in different "sizes". Much of advanced mathematical analysis is taken up with the study of various notions of the "size" of sets.

Let's look at a few examples.

Example 1

Consider the collection of all finite length "strings" of 0's and 1's; for example,

10, 01011, 1011001110, 001, 1101, ... , and so on,

are all elements of this collection. This collection is an infinite set. Why? Because given any finite list of such strings, we can always name another one that is not on the list. For instance, given

we could just concatenate the strings (in other words, stick them all together) to define a new string

We know that is a new string, not on our original list, because it's longer than any of the previous strings. This observation means that the set of all strings of 0's and 1's cannot be finite.

On the other hand, in spite of the fact that this set is infinite, it is still possible to enumerate or list all the elements of the set -- to literally count its elements.

Neal Carothers -