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.)

(In reply to additional thoughts by Larry)

This is pretty clever Larry, and works well, right up until the last sentence, which confuses me.  If you draw one for each integer, and then stuff in every fraction between it and the next integer, then you are done.  There is no need to 'have larger and larger '8's for the digits of the 1's column, the 10's column etc."

Posted by Steve Herman on 2005-02-15 17:41:01