I didn't write anything about paradoxes for a long time, so here is a little something. Lets look on the volume we get by rotating the graph of the function y=1/x, for x>1. The object we get is called Gabriel's Horn. It is easy enough to show that its volume is finite and in is equal to pi. If we cut the horn in any finite point a we will get that the volume is exactly:If we now take the limit when a goes to infinity we will get that the volume is indeed pi. However the surface area is infinite. For any finite a we will get that it is exactly:

But the limit of this expression is infinity. Now that we know this , we can go on to the description of the paradox. Suppose that you want to paint the Horn with finite amount of paint. Obviously, it is not possible because the surface area is infinite. But you can fill it with a finite amount of paint. Lets now suppose that Horn is made from a transparent plastic. In this case, filling it with paint is the same thing as painting it.

As a result we get that it is both impossible and possible to paint the Horn with finite amount of paint. So which one is true? The solution is in fact rather simple. Firstly lets look on the graph of y=1/2x. Obviously this is again Gabriel's horn, but in a scaled down version. Lets put it inside the original while it is still filled with paint. In this way we painted it from the outside. How did we do it? The answer to this is in the distribution of paint. The thickness of paint is given by g=1/x-1/2x=1/2x. And this is the whole trick. We can paint even an infinite surface, the only thing we need to worry about is allowing the thickness of paint to approach zero in a way similar to this example (we need the integral of the paint distribution to be finite).

Grothendieck seminar at the ENS

21 hours ago

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