Today (March 3) is the birthday of Georg Cantor (1845-1918). Cantor was a German mathematician, best known as the inventor of set theory. Set theory is the branch of mathematics that studies collections of objects, known as sets. Cantor established the importance of one-to-one correspondence in sets and is responsible for the idea that “some infinities are bigger than others”. For example, the counting numbers (1, 2, 3, 4,…), known as natural numbers, is an infinite set. The real numbers is also an infinite set, but this set includes the natural numbers, as well as, negative integers, fractions, and irrational numbers, such as pi. Each number in the set of natural numbers corresponds to the natural numbers within the set of real numbers, but the set of real numbers includes many other numbers. Therefore, even though both sets are infinite, the set of real numbers is larger.
If you watched the documentary about fractals that I shared a while ago, you may recognize that Cantor Set, illustrated below. It was one of the earliest fractals, known at the time as “monsters”. The Cantor Set is formed by taking a line, removing the middle third, then removing the middle third of the remaining lines, and so on. It would seem that eventually there would be nothing left, but there is always a set of remaining points, just ever smaller and smaller.