Cantor’s ternary set, or more briefly, the Cantor set, is a well-known construction that illustrates some interesting topological properties of the real numbers. To describe it briefly, we define an iterative process as follows: Let \( C_0 \) be the interval from \( 0 \) to \( 1\). Then \( C_1 \) is obtained by removing the middle third from \( C_0 \), leaving the two intervals from \( 0 \) to \( \frac{1}{3} \) and \( \frac{2}{3} \) to \( 1 \). The set \( C_2 \) is obtained by removing the two middle thirds from \( C_1 \). So \( C_2 \) will then contain four intervals, each of length \( \frac{1}{9}. \) In general, the set \( C_n \) will contain \( 2^{n} \) intervals, each of length \( \left(\frac{1}{3}\right)^{n} \). The Cantor set, denoted \( C \), is then defined as

\(C=\displaystyle\bigcap_{n=0}^{\infty} C_n. \)

The Cantor set satisfies many curious properties. For instance, in a certain sense, it has length zero. This is more or less because at stage \( n \), it has a total length of \( \left(\frac{2}{3}\right)^{n} \), which approaches \( 0 \) as \( n \) approaches infinity. However, from the perspective of cardinality, it can be shown that the Cantor set is uncountable. This is actually fairly simple to see, using the following argument. In order to be an element of the Cantor set, you must be a member of every set \( C_n \). Going from \( C_0 \) to \( C_1 \), you have to be either ‘left’ or ‘right’ of the middle third. Let’s say hypothetically that you are ‘left.’ Then going from \( C_1 \) to \( C_2 \), again, you must be either ‘left’ or ‘right’ of the middle third. It’s clear that this process repeats indefinitely. If we represent ‘left’ and ‘right’ by \( 0 \) and \( 1 \), then we see that elements of the Cantor set can be put into one-to-one correspondence with infinite binary sequences. Thus, by Cantor’s diagonal argument, the Cantor set must be uncountable. So although it has measure zero, it is an uncountable set!

One can also try to “hear” the Cantor set in the following way. Start with a particular frequency, let’s say, middle C. Define \( C_0 \) as sustaining that frequency for some duration, let’s say, one second. Then choose an interval of modulation, for instance, a tone. Then \( C_1 \) would be the triplet D-C-D, since we have modulated the outer thirds by a tone. We can proceed similarly to the general form of \( C_n \), and in principle, we could even define the full Cantor set \( C \), although it would be difficult to program. The following audio tracks illustrate a few of these ideas.

Here’s Cantor for \( n = 0, 1, 2, 3, 4: \)

Cantor with \( n = 6: \)

Inverted Cantor:

Cantor with the modulation factor as a perfect fourth:

A mash-up of many of these ideas:

I hope this post allows you to hear the Cantor set better!

(Note: Once again, I’ve coded all the musical examples in Gamma.