There are infinitely many real numbers between 0 and 1. Are there twice as many between 0 and 2, or are the two amounts equal?

Question

I know the actual technical answer. I’m looking for a witty parallel that has a low chance of triggering an infinite “why?” procedure in a child.

In: 1335

Answer 1

The reason this is counterintuitive is because it brings into contrast two measurements of mathematical size: cardinality and volume. The interval between 0 and 2 has twice the volume, but the same cardinality.

The first thing to understand is that a single number takes up no space. The reason this is true is because we can contain it in an arbitrarily small ball. Think about 0, for instance. The interval (-0.1, 0.1) contains 0, and the volume of this interval is 0.2. The interval (-0.01, 0.01) also contains 0 and has size 0.02. We can continue this process, and squeeze 0 into a smaller and smaller ball. Now the mathematical concept of a limit comes into it. Because we can fit 0 into a ball of arbitrarily small volume, 0 itself must have 0 volume.

The thing that is hard to understand is that even though an individual number has 0 volume, if we look at all the numbers between 0 and 1, that set has volume 1. This phenomenon is one example of how our intuition between “count” and “volume” breaks down when dealing with infinite sets.