Mathematics
The Cantor Set and Uncountability
Quick fact
The Cantor set contains uncountably many points, yet its total length is exactly zero—a striking demonstration that uncountability is not about filling space but about the number of points.
Why this is interesting
You know that the real numbers are uncountable, but can an uncountable set still be so small that it has no length at all? Meet the Cantor set, a set of infinitely many points that takes up zero space on the number line.
Read the full explanation
Understanding The Cantor Set and Uncountability
Imagine you have a line segment from 0 to 1, the unit interval. Begin by removing the middle third, leaving two smaller segments. Now remove the middle third of each of those two segments, leaving four segments. Continue this process forever: at each step, remove the middle third of every remaining segment. What remains at the end is the Cantor set.\n\nAt first glance, it seems like after infinitely many removals almost everything is gone. But the Cantor set is not empty—it contains infinitely many points. In fact, it contains as many points as the entire real number line! Yet the total length of the remaining pieces shrinks to zero: after the first removal, total length is 2/3; after the second, 4/9; and so on. The total length tends to 0. So we have a huge (uncountable) collection of points that somehow occupies zero length.
A deeper explanation
Why is the Cantor set uncountable? The key is to represent each remaining point in its ternary (base-3) expansion. Initially, every real number in [0,1] has a ternary expansion like 0.12021... When we remove the middle third, we eliminate all numbers whose first ternary digit is 1. In the next step, we eliminate numbers whose second digit is 1, and so on. Therefore, the Cantor set consists exactly of those numbers that can be written using only the digits 0 and 2 in their ternary expansion.\n\nNow, map each such ternary number to a binary number by changing every 2 to a 1. This gives a one-to-one correspondence between the Cantor set and the set of all binary sequences (sequences of 0s and 1s). By Cantor's diagonal argument, the set of all binary sequences is uncountable, hence the Cantor set is uncountable.\n\nSimultaneously, the measure (length) of the Cantor set is zero because the total length of the removed intervals sums to 1, the original length. Thus, the Cantor set is a perfect set (closed and with no isolated points) that is nowhere dense and has measure zero—yet it is uncountable. This serves as a powerful counterexample to the intuitive assumption that uncountable sets must contain an interval or have positive length.