Georg Cantor was a German mathematician who is partially credited with creating the foundation of modern general topology. Topology is a branch of mathematics that starts with logic and set theory, and applies measurement and classification such as ordering, shape, size, dimension. It is considered to be a foundational branch of mathematics in that many of the principles and theories have applications elsewhere. It is also the subject that made me want to know more, to keep studying mathematics. Georg Cantor and other mathematicians of his time had discovered theories that confounded logic as we know it, but were principally true nonetheless. One such theory deals with the question of “how many” and the set of all real numbers greater than 0 but less than 1.

Cantor Set - First Three Iterations

To begin with, start listing fractions beginning with 1/2 and continue to increase the denominator by 1 each fraction. The list goes like this: 1/2, 1/3, 1/4, 1/5, … and so on. Since there is no “largest denominator”, these fractions go on forever. There is an infinite number of them, and they are all contained between 0 and 1. Now the concept of infinity was not new in 1883. What indeed was new, was that not all infinities are equal. The above example is what mathematicians call a “countable” infinity. The definition of countable simply means that it has the same “quantity” as the integers themselves.

Now throw in irrational numbers as well. Take all of the numbers between 0 and 1. How many of these are there? Well, it was discovered that to even try to list them is impossible. In fact, take any two numbers in that interval. There will always be an infinite number between them… no matter how close together they are. Once a set of objects is uncountable, it begins to defy “common sense” logic. For example, it can be shown that there are just as many numbers between 0 and 1 as there are on the entire real number line. While this seems contradictory, it is the nature of microscopic infinity.

To put it in perspective, consider flying in an airplane high above the landscape. Start descending to the point where you can just make out a thin thread that is a highway stretching below. No individual cars can be seen; it appears to be just a thin line of light. As you descend, you begin to make out tiny shapes moving on the highway. The closer you get, the more possible it is to make out individual cars and discern spaces between them. As you approach, gaps between them increase and fewer are present in your field of vision. That is what the integers look like. If the cars represented all rational numbers, you would be curious to find that no matter how close you get, the row of cars appears unchanged. With each zoom, what before was invisible becomes visible; cars are in single file but just as dense and packed in. Still, you are seeing a countably infinite set. Even these are a mere drop compared to the ocean of the “reals”. The real number line has layers upon layers; they will not be placed in a line, for there are far too many of them. Endless microscopic rows appear, though moving along a thread.

Some had presumed that to be uncountably large, a set logically must have enough density so that its length (one-dimensional) or volume (multi-dimensional) would be greater than 0. But in 1883, Cantor showed this to be false. He presented the mathematical world with a set that had infinitely infinite number of members but whose length is 0. It was called the Cantor set. His set involved the number line, removing middle third of each remaining segment (see picture above) until all that remained was still uncountable but had total length 0. Here is a demonstration of a 2-dimensional example of a Cantor-type set:

Sierpinski Triangle

If this pattern were to be repeated an infinite number of times, it can be proven that the “black” portion that remains is an uncountable number of “dots” with area = 0. It is called the Sierpinski Triangle and is one of the most basic examples of what is called a fractal. Fractal Explained Hence the Cantor Set, infinitely complex but measure of 0, was born.

Come to think of it, House Majority Leader Eric Cantor and Georg Cantor have at least two things in common: both defy common sense and both arrive at seemingly contradictory conclusions. Perhaps there is a family connection.