You have an ink stamp that is so amazingly precise that, when inked and pressed down on the plane, it makes every circle whose radius is an irrational number (centered at the center of the stamp) black.
Is it possible to use the stamp three times and make every point in the plane black?
If it is possible, where would you center the three stamps?
(In reply to re:
It is one thing to say "You can prove that there exists an irrational number between any two unequal rational numbers, and that there exists a rational number between any two unequal irrational numbers" (which you've demonstrated is true), but quite another (and misleading) to say "Actually, looking at a single radius from a single point, the distances must actually alternate rational and irrational points. "
The idea of alternation would imply equal numbers in the sense of 1-to-1 correspondence. But there are many more irrational numbers on the radius than rational. There are Aleph-1 of the former, but only Aleph-null of the latter. The numbers are intimately mixed, but not alternating.
Edited on December 2, 2003, 8:58 am
Posted by Charlie
on 2003-12-02 08:57:43