Today is the 165th birthday of Georg Ferdinand Ludwig Philipp Cantor, the mathematician who indirectly inspired me to major in math. In my first few semesters of college, I was at best an indifferent student, finding little inspiration in the humanities majors I was bouncing around among, playing a prodigious amount of pinball, and attaining (according to rumor) history’s first-ever grade of C in Peter Regenstrief‘s Poltical Science 101. Then one day, my friend Bob Hyman happened to mention that some infinities are larger than others, and set my life on track. This—the vision of Georg Cantor—was something I had to know more about. Before long I was immersed in math.
What does it mean for some infinities to be larger than others? Well, for starters, some infinite sets can be listed, while others are too big to list. The natural numbers, for example, are already packaged as a list:
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The integers, by contrast (that is, the natural numbers plus their negatives) aren’t automatically listed because a list, by definition, has a starting point, whereas the integers stretch infinitely far in both directions. But we can fix that by rearranging them:
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So the integers can also be listed.