Covering The Reals

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Here's an interesting paradox.

We know that the rational numbers are countably infinite. That means we can list them in order: $r_1,{\quad}r_2,{\quad}r_3,$ etc.

So cover the first with an umbrella of size 1/2, the second with an umbrella of size 1/4, the third with an umbrella of size 1/8, the fourth with an umbrella of size 1/16, and so on ...

The umbrellas are open.

Clearly all the rationals are covered.

Even more, consider some rational. Its umbrella is of rational size, so its endpoints are also rationals, and hence also covered. Thus the umbrellas all overlap, and so all the real numbers must be covered and kept dry.

Or not.

You see, the umbrellas are, in total, of length 1. They overlap, so the total amount of number line covered is strictly less than 1.

So the number line is, in fact, entirely wet.

How does that work ??!!

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