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.)
This is tricky because the reals are uncountable.
If they were numbered in a countable (orderable) way you could just put a little 8 at each whole number on the x-axis. Another fun way would be to put each in one of the loops of the previous.
Some ways to cheat:
1) hide them at infinity: assign each positive real to a real from 0 to 360, then translate them each to an infinite distance from the origin at this angle.
2)Shrink each to a point (expansion of magnitude 0.) Then you can just line then up on the x-axis.
If there is a solution (which there may, I think, be) the assertion I have a problem with is that you 'drew' it.
Posted by Jer
on 2005-02-15 16:43:39