Who’s Got More Points?

August 1, 2011

So, the question is, which of the following has more points: a line segment or an infinite line? You might think the obvious answer is the infinite line. But it is not as simple as you think it is. A simple proof done by the mathematician Georg Cantor will show you the answer:

Imagine two semicircles of the same size and shape. A semicircle is pretty much half a circle. You agree, since both semicircles are the same, they both have the same number of points, right? Beneath the first semicircle, place a line segment that have the same size as the diameter of the semicircle. Beneath the other one, place a line that goes to infinity. Now, your j0b will be to match every single point on the border of the semicircle to every single point on the line segment and every single point in the second semicircle to every single point in the infinite line. The setup is done as shown in the figure below:

In the line segment, I match up every point of it to the semicircle by a straight line connection. By going from left to right (or any direction you want), you realize that you can match every single point on that semicircle to the line segment. For that reason, the line segment has as many points as the semicircle above it.

In the case of the infinite line, I have to do a cleverer maneuvering. From the center of the semicircle, I extend a line straight to the left (or right), and sweep around the semicircle counterclockwise. Doing so will match every point on the semicircle. Yes, even the one point in the infinite line that seems unimaginably distant. After all, all I would have to do is extend a line from the semicircle to that point. As you will see, no matter how far away the point is from the semicircle, it will touch a point on the semicircle. Interesting, it seems like an infinite line has the same amount of points as the semicircle. What is the implication of all of these things I have written?

Well, since the line segment has the same number of points as its corresponding semicircle, and so does the infinite line, and both semicircles have the same number of points, there is only one conclusion I can take. The infinite line has the same number of points as a line segment!

Mind Blown.

-This post has been inspired by the book To Infinite and Beyond: A Cultural History of the Infinite, so I have the book to thank-


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