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.)

Am I missing something here?

The ‘official’ solution refers to a set, C, of disjoint figure ‘8’s drawn in the plane. It then uses the fact that they are disjoint to prove that C is a countable set.

However, the original problem was to do this for a set, C, which contained a figure ‘8’ for every positive real number. The ‘proof’ seems to take as a starting point the assumption that it is possible for this uncountable number of ‘8’s to exist on the plane in a disjoint arrangement. Unlike the concentric circles, there seems to be no evidence for this assumption, and without it the given ‘proof’ cannot begin.

Posted by Harry on 2010-07-24 20:52:05