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 re: additional thoughts by Steve Herman)

Well, on second initial thought, I think that Larry's scheme doesn't work, because every figure eight drawn has a finite decimal expansion.  We haven't drawn any figure eights corresponding to transcendentals, or irrational numbers, or even to repeating decimals like 1/3. 

This is analgous to marking a line at every tenth of an inch, and then every hundredth, and then every 1000th, etc. 
1/3 never gets marked!

I need to think on this a little bit more, but I don't have a scheme yet that would allow David Shin to accomplish this.  I wonder if I can prove that he has suceeded in making an impossible assertion.

p.s. -- I was thinking about this while out walking, and came in prepared to post this message.  I was chagrined to see that Charlie has is a step ahead, again today. 

Posted by Steve Herman on 2005-02-15 20:08:30