We continued our study of the Cantor set by spending some time thinking about its properties, in particular how it has a length of zero yet still has an (uncountable) infinity of points contained inside.

That the length is zero is fairly easy to see from the fact that we remove 1/3 of the set in the first step, 2/9 in the second, 4/27 in the third, 8/81 in the fourth, and so on. That sum 1/3 + 2/9 + 4/27 + 8/81 + … forms an infinite geometric series, the sum of which is 1. And since the length of the original segment is also 1, the length of the “final” version of the Cantor Set is 0.

Yet it clearly contains an infinity of points! With each stage, we create endpoints of segments that never get removed, and an infinite number of stages produces an infinite number of endpoints. But not only that, I claimed the Cantor set is uncountably infinite, which required some explanation of the realization that some infinities are bigger than other infinities.

Additional viewing:

- Hilbert’s Hotel (TED Ed)
- Cantor’s Diagonal Argument (Numberphile)