Suppose that I drew an infinite number of disjoint closed curves in the plane (such as circles, squares, etc.). Suppose that I then tell you that there is one curve for each positive real number.

You would not have too much trouble believing my assertions at this point. For example, I could have drawn all circles with center at the origin. They are all disjoint, and for each positive real number x, there is a corresponding circle - namely, the circle of radius x.

But suppose that I also tell you that all the curves I drew were figure eights. Can you believe my assertions now?

(A figure eight is a curve in the plane obtained from the basic "8" shape by any combination of translation, rotation, expansion, or shrinking.)

Thinking about infinity can get very confusing.  Let me offer the following criterion to analyze a particular infinite set of figure eights:

"If I asked you to point to the figure eight corresponding to the number x, could you do it?"

It is especially important to consider irrational x. 

Posted by David Shin on 2005-02-15 19:59:16