Hopefully we all remember Cantor as the guy to blame for ruining lives by delving too deeply into infinity. In addition to showing that some infinite sets are “larger” than others, we have a graphical representation of something called the Cantor set:
Not very impressive looking, is it? But it’s more interesting than it seems. For those who find it easier to work with numbers, imagine starting with the interval [0,1], and for others just imagine a perfectly straight wooden “ruler” 1 unit length long (inch, centimeter, whatever).
The Cantor Set is an iterated function system (IFS) which is a fancy way of saying that we take things one step at a time. We start with our interval [0,1] or 1 unit length, and we remove the middle third (as in the picture). Next, we remove the middle from the remaining two intervals/pieces of ruler (the intervals would be [0,1/3] & [2/3,1], in case you were wondering). We again take out the middle third, but now twice: we remove the middle third of each interval/piece of ruler. By the time we get to the fourth step, we have to remove 1/3 from intervals. We continue taking out the middle third of each remaining “level” of intervals/pieces of ruler, and the number of intervals at any level n from which we remove 1/3 is :
…and onward to infinity. Here’s the interesting part. Imagining that we really did this infinitely many times, how much have we removed? Well, at step n we remove 1/3 from remaining intervals/pieces of ruler, so we just let n go to infinity. We get this:
(the weird looking s-like thing is the uppercase Greek letter sigma and simply means “sum the following expression starting at n=1 and continue to infinity”).
I can almost here the groans and complains. I’ve forced people to confront algebra nightmares from high school for this? No. The interesting thing is how much is left. Think about it: every time we removed the middle of some number of remaining intervals/pieces of ruler. We remove the middle of intervals/pieces of ruler infinitely many times, each time leaving TWICE as much as we took away. So we started out with (a unit) 1, we removed a total amount/length 1, and we wind up with twice as much as we began with.
Why doesn’t my bank account work that way?