The Cantor Set

This is the first of two problems; the second problem is How Long is the Cantor Set? . We have published these problems as they provide a good opportunity for pupils to practise using fractions, and also to meet the idea of infinity.

Mathematicians are interested in the Cantor set because it is a good example of an object whose existence is not at first clear.

If you aren't familiar with the idea of countability, then you may be interested in the article All about Infinity. For now, we shall be rather vague and only try to give an indication of the ideas involved, as making them rigorous would take rather more time. Roughly speaking, an infinite set is uncountable if its size is a large infinity.

It is not too hard to see that the Cantor set consists of precisely those numbers between $0$ and $1$ that contain only $0$s and $2$s (no $1$s) in their ternary expansion (like a decimal expansion, but in base $3$). There is a slight technical detail here, because some numbers may be written in two ways: a number that only has$0$s and $2$s apart from a single $1$ at the end may be written using only $0$s and $2$s. For example, $0.21$ (remember that this is in base $3$!) may also be written as $0.20\dot{2}$. (This is precisely analogous to the fact that, in base $10$, $1$ is the same as $0.\dot{9}$.)

From this, it is possible to use the same diagonal argument as is used for real numbers to show that the Cantor set is uncountable. That is, it contains infinitely many points, and the infinity is really large.

Mathematicians have developed a more formal and rigorous version of the idea of length, called Lebesgue measure. The Lebesgue measure of a line is the same as its usual length, but it is also possible to find the Lebesgue measure of other sets. For example, it is possible to find the Lebesgue measure of the set of rational numbers: amazingly, it has Lebesgue measure $0$! However, there are only countably many rational numbers, so in some sense there aren't very many of them, and we perhaps shouldn't be surprised that they don't take up any space.

Having introduced this notion of Lebesgue measure, a natural question is ``Is there an uncountable set of Lebesgue measure $0$?''. That is, is there an enormously large set that still somehow contrives to take up no space? The answer? Yes, there is: the Cantor set is such a set. (This is the object mentioned earlier whose existence is of interest to mathematicians.)

See How Many Elements Are There in the Cantor Set? for a proof of the uncountability of the Cantor set.