(sorry I don't know how to add pictures)
Two friends argue if larger circles have more points than smaller circles
Friend number 1 (a well known argument)
Say the circles are concentric. you cannot draw a line from the centre that cuts the bigger circle while that doesn't cut the inner circle so they have the same amounts of points.
Friend number 2 ( a dissenting voice)
Ok lets take concentric circles, he draws a line to the centre ( say along the x axis)
adds another line parallel to this line it cuts both circles 2 times)
adds another (cuts both circles 2 times)
and so on
two parallel lines are only tangent to the smaller circle while still cuting the bigger at two points.
and some paralel lines only cut the larger cirle.
Therefore all points on the smaller circle are related to some point on the bigger one, but some points on the bigger one are not related to point on the smaller one.
So the larger circle has more points.
Which friend is right or how do you convince them that they are both right?
Best Answer
Friend 1's argument does indeed show that the circles have the same number of points, as each ray from the center uniquely determines a point on the circle, and each point on the circle uniquely determines a ray from the center.
Friend 2's argument (if made carefully) shows that the bigger circle has at least as many points as the smaller circle, but does not show that the smaller circle has strictly fewer points.
It all comes down to the notion of cardinality of infinite sets, where things can get pretty counterintuitive. If you want to give friend 2 an example to show why that argument does not show strict size difference, consider the set of integers $\Bbb Z.$ Obviously the same size as itself, yes? Ah, but now consider the function $f:\Bbb Z\to\Bbb Z$ given by $f(x)=2x$. This function maps $\Bbb Z$ into $\Bbb Z$, but misses infinitely many integers! That does not mean, however, that $\Bbb Z$ is strictly larger than itself. It is simply a peculiar quirk of (many) infinite sets that they can be put into one-to-one correspondence with subsets of themselves. Thus, showing that the smaller circle is in one-to-one correspondence with only a part of the larger circle isn't enough to show that the larger circle has more points--though such an argument would work for finite sets.