Now, let be equal to with its middle third removed. Mathematically, . Let the following be an illustration of :
If we continue in this manner, always removing the middle thirds of the remaining intervals, then we will obtain the following sequence:
Let be any number in . Define the Cantor set as the limit of as approaches infinity. Mathematically, .
Clearly, comprises intervals of length each (convince yourself of this fact). It therefore follows that the Cantor set has a length of
Intuitively, if an interval has a length of zero, then said interval must contain nothing at all; it must be the empty set, right?
Nevertheless, observe that the endpoints of the intervals are preserved through the sequence. In other words, observe that the points zero and one are preserved; all contain zero and one. It therefore follows that preserves two points, preserves four points, preserves eight points, and so on. Generally, preserves points. Therefore, the Cantor set preserves
points. In other words, a set with a length of zero can have infinitely many points. How can this be?
This exercise clearly demonstrates that our intuition of the mathematical infinity is insufficient to understand it, and that the need for rigorous tools and careful approaches to dissect it are of the utmost essence.