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)
I'm confused how you could say that between any two unequal irrational numbers there is a rational number and vice versa without also showing that the numbers alternate between rational and irrational. If the numbers didn't alternate then either;
1) there is a third type of number which is neither rational nor irrational (which I don't agree with), or,
2) there are/is (as least one case of) cases/a case where two irrational numbers are adjacent, i.e. with no rational value between them. I believe that this is what you're saying.
Poorly worded-ness aside - My question is how you can have a rational value between ~any~ two irrational values, while having consecutive irrational values? Now I understand that this can be associated with the idea that "there is no smallest positive number" (i.e. you can infinitely subdivide any region of the number line), but I don't see how this would imply either a majority of irrationals or conversely a majority of rationals - it just means they are arbitrarily close to one another in value.
Now let me say that I didn't verify either DJ's equations to show that this is the case nor Charlies assertion that there are Aleph-1 irrationals and Aleph-null rationals, so I don't really have a position on this. If I had to, I'd agree that there are more irrational, not that there are equal numbers, but my lack of understanding of infinity is unfortunately well documented...
alternation implies 1-1
