You can write the letter O a lot of times. In fact, you could write it a (countably) infinite number of times. In fact, you could write it an uncountably infinite number of times. To do so, draw every circle with its center at the center of the paper, and whose radius is less than the radius of the paper. To keep the circles from touching, don't draw any circle whose radius is a rational value. Why does that work? If I say there are two circles that touch, you point out that between any two irrational numbers there is a rational number, and you didn't draw that circle, so between the two circles you did draw there is a "blank" circle!
You can draw the letter X a lot of times. In fact, you can draw the letter X a countably infinite number of times. But you cannot draw it an uncountably infinite number of times.
To see that you can draw the X (countably) infinitely many times, we'll pick a sequence of points at which the X's will be centered. The first point is at the center of the paper, the second is halfway to the lower right edge, the third is one quarter away from the lower right edge and so on. Now it is possible to draw a sequence of X's, each centered on one of the points, and half as big as the previous one, so that no X touches another. And we've hardly used most of the paper at all!
Is there room to go further, that is, to draw an uncountable number of X's on the paper? Sadly, there is not. One way to see this is as follows. Note that if two X's are at least one inch in size (measuring the distance from the center of an X to the end of an arm) then it is impossible for them to be placed on a piece of paper so that they do not touch, but so that their centers are closer than one inch. In this way, an X essentially draws a circle around itself such that no X of the same or greater size can be centered there.
But this is very telling. Because if we draw a bunch of X's on the paper, we have now shown that, for any finite size, there can be no more than finitely many X's that are that size or greater. The set of X's can be described as the union of the sets of X's of size greater than 1, greater than 1/2, greater than 1/4, and so on. If every one of these sets is finite, and we are unioning a countable number of such sets, then the resulting set can have cardinality that is at most countably infinite, and can never be uncountable. So at most countably many X's can fit on the paper!
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