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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:

* EQN:r_1,\;r_2,\;r_3, /etc./

So cover

* the first rational with an umbrella of size 1/2,

* the second rational with an umbrella of size 1/4,

* the third rational with an umbrella of size 1/8,

* the fourth rational 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 we can look at the rationals under

its edges. Clearly they are rational, so they're covered with

umbrellas, and these umbrellas overlap.

This shows that all the real numbers must be covered and kept dry.

Or not.

The umbrellas are, in total, of length 1. They overlap,

so the amount of

than 1.

So the

How does that