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 solution - the number of curves must be countable(?) by ronen)

I disagree with you here, ronen.  I agree that it is impossible to fill (i.e., cover all rational points of) an entire component of an eight with finitely many eights, but one can certainly specify an arrangement of infinitely many eights to fill an entire component. 

owl gives the correct proof here.

Posted by David Shin on 2005-02-23 07:19:29