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(2): by Charlie)

You are correct. Alteration was a poor word choice; I meant only to address Benjamin's claim that there can exist an unbroken continuity of either rational or irrational numbers. SK said it better; between any two given unequal irrational points, there are an infinite number of both rational and irrational points; likewise between any two unequal rational points.

Of course, there is not a one-to-one correspondence between rational and irrational numbers, which would imply the same cardinality. The limit of the ratio of irrational to rational numbers in an interval as the infinitesimal size of that interval approaches zero, which would intuitively be 1, is not (I think that ratio will be Ν₁/Ν₀, the ratio of the cardinalities of the sets, but I am not sure). 'Alteration' in this case does not imply a one-to-one ratio because we cannot delimit every possible point on an infinite segment or within a finite interval.

Also, I did type the word 'not' in the last sentence of my previous post, but for some reason the <i> </i> tags made the word disappear altogether (I will not bother editing the post, but the curious can look at the source of that page to confirm). But yes, that should read, "the answer is not that a single stamp will cover an entire plane" (I will not risk the use of italics this time).

Posted by DJ on 2003-12-02 22:36:09