To provide a little relaxation from the tension of the elections in the US this week.
A Man by the name George Cantor was playing with numbers and came up with what became to be known as the Cantor sets. The idea is in reference to the existence of real numbers.
The idea is that no matter how close to each other two numbers are, there is always a third number in between. This can also be accomplished visually with a slight variation by subdividing a line (which, by definition, has an infinite number of points). One can make 2 cuts in the line so that there are 3 sections of the same length. The middle section is discarded leaving out only the 2 side thirds. It is important to realize that this operation is not physical because there is no scissor small enough to cut a small section after a few iterations.
As this operation is repeated in each section, the number of segments multiplies rapidly while the length of the segments diminishes rapidly. The paradox is that regardless of how many times the cut-and-discard operation is repeated, it can be repeated again, and again. After a very large of repetitions (closing in to infinity) the number of segments is infinity.
Now, instead of 2 cuts, one can cut 3, 4 or more times and follow the same pattern. The end is the same, yet it is easy to see that in each case the segments will vary in length. We could, in fact, cut a very large number of times (closing in to infinity) and repeat the same type of operation, and still get at some point a very large number of segments.
The point is that there seems to be many sets with a different very large number of segments. In fact, it seems that the number of different infinities is also infinitive. Obviously, every segment will be very small, of an infinitesimal length.
Isn't it relaxing to think about these things? No wonder how some mathematician (the real ones, I am not even that interested in it) need to have a special brain to understand these things and why some are not considered normal.