Covering The RealsYou are currentlybrowsing as guest. Click here to log in 

Here's an interesting paradox.
We know that the rational numbers are countably infinite. That means we can list them in order:
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 numberline covered is strictly less than 1.
So the numberline is, in fact, entirely wet.
How does that work ??!!
Last change to this page Full Page history Links to this page 
Edit this page (with sufficient authority) Change password 
Recent changes All pages Search 