Although A Brief History of Infinity has been around a while, it is still one of my best selling titles and I probably get more letters and emails about it than anything else. I think it reflects the timeless fascination of infinity. Any road up, I thought I'd bring a little brightness to your Friday with a paradox of infinity that didn't make it into the book, though it does appear on the Popular Science website.
We start by thinking of the number line - let's say for simplicity, all the numbers from 0 upwards. So we've got a line, rather like the edge of a ruler, starting from zero and heading off to infinity, featuring all the numbers and fractions along its length.
Now we know the rational fractions (n/m where n and m are whole numbers) have the same cardinality as the counting numbers, thanks to a proof by Cantor (it's in the book). Simply put, this means you can match off each of the of the rational fractions with a positive integer. They are the same 'size' of infinite set. And we're going to use another set of fractions alongside them - the sequence 1/2, 1/4, 1/8, 1/16... It is simple enough to show that these also have the same cardinality: you can match one of these with each of the positive integers too. And we can prove that the sum of the whole series 1/2+1/4+1/8+1/16... is just 1. With these 'given's the fun begins.
Imagine we wanted to protect the whole number line from getting wet. What we are going to do is issue each rational fraction along the line an umbrella. The umbrella will be a simple T shape. The first umbrella we give out is 1/2 a unit of the number line across the T. The second umbrella is 1/4 of a unit of the number line across and so on. Once every rational fraction has an umbrella, then the whole number line is covered. The umbrella extends half its width in either direction - so, for instance, the first umbrella will cover all numbers for 1/4 of a unit to its left and 1/4 of a unit to its right. Note that this too is a rational fraction - and adding it to or subtracting it from the starting point (itself a rational fraction) will reach another rational fraction.
Okay so far? Each umbrella spans from its starting point to a rational fraction on either side of it. Now bearing in mind we've issued an umbrella to every rational fraction, the whole number line is covered, because there's at least a meeting of umbrellas and in most cases an overlap.
We've covered the whole line from 0 to infinity with our umbrellas. But, remember how wide the umbrellas were. Their widths form the infinite series 1/2+1/4+1/8... so with no overlaps, the maximum amount of the number line those umbrellas can cover is 1 unit - and with overlaps they will cover even less. A set of items with a width of just 1 covers a line that goes all the way to infinity.