You might wonder, given that there are so many different ways to match up infinitely many things, how we can know that there is no matching that catches everything\. I will now prove that, no matter how you try to match paths \(ways of walking\) and tiles, you will miss some tiles\. Since we have already seen that the number of tiles in a sidewalk two tiles wide is the same as the number of tiles in a sidewalk one tile wide, I will show that any matching between paths and tiles in a sidewalk one tile wide misses some paths\. I will do this by creating a path that does not match the path we have chosen for any tile\.

Should be "two tile wide", right?

## Comments

Eric Bruylant

Which instance of that are you referring to (there are five)?