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(3):my proof by SilverKnight)
You wrote:
Your assertion (that "Any point D chosen from the complement of S(A,B) will give you three points (A,B,D) where you can stamp and cover the plane.") is not an immediate result of the claim.
But it is! The definition of S(A,B) includes all points C such that stamping at C will yield at least one white point. So if D is in the complement, it yields no white points.
You wrote:
You must demonstrate that, in the non-empty complement, there exists a point that, if stamped, will ensure that the set of non-black points (are not only countable, but) equals the NULL SET
That is what S(A,B) means: S(A,B) is the set of points where you get ANY white points at all. If D is in the complement of S(A,B) then stamping at A, B, and D will yield NO white points.
The question of measures came in at the size of S. I showed that nearly any choice of D will give you NO white points remaining.
re(4):my proof
