Yet More on the Cantor Set

Recall our construction

and so on...

We start with the interval [0,1] and, in the first step, we remove 1 interval of length 1/3; in the second step, we remove 2 more intervals of length 1/9; in the third step we remove 4 more intervals of length 1/27; and so on. In general, at the n-th stage, we remove another intervals, each of length .

Adding the lengths of all the intervals removed in the construction of the Cantor set, we get

This is a geometric series, and it's easy to find the sum:

Thus, the total length of all the intervals we removed from [0,1] in the construction of the Cantor set equals 1 -- the same as the length of the interval we started with! In other words, we removed everything!? Is this weird, or what?

We've shown that the set [0,1] \ , the complement of the Cantor set, has "total length", or measure, equal to 1. Thus, the Cantor set must have measure zero. This is yet another way of saying that the Cantor set is a very small set: It has "total length" equal to 0.

A slightly more direct (but perhaps less convincing) approach here is to note that is contained in , which consists of intervals, each of length . Thus, for each n, the Cantor set can be "covered" by intervals of total length , a number which tends to 0 as n increases.

We know that the Cantor set is a "big" set -- after all, it has the same "size" as [0,1] itself, at least in one sense -- and yet it must also be a "small" set, since it's the result of removing a set of "length 1" from [0,1]. Incroyable!

The dilemma, if you will, centers around the fact that we've employed two different notions of "size". Evidently, "total length" and "cardinality" are not equivalent notions of size -- a set can be very small in the first sense while being very large in the second. But this sort of "confusion" is precisely what makes the Cantor set so interesting! Don't you agree?


Neal Carothers -