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

Simple! There are `ℵ₁` (aleph one) reals between `0` and `1`, and there are `2·ℵ₁` reals between `0` and `2`. The thing about `ℵ₁` or any `ℵₓ`, is that `2·ℵₓ = ℵₓ`. Reason being is that for every number between `0` and `2`, you could multiply it by `0.5` and get a number between `0` and `1`, meaning the sets are the same size. You can do the same for the number of reals between `0` and `4` (`4·ℵ₁`). If you take any real from this set and multiply it by `0.25`, you get a real from the set between `0` and `1`. `∴ 4·ℵ₁ = ℵ₁`.

Fun fact: even though there are `ℵ₁` reals between `0` and `1`, there are only `ℵ₀` (aleph null) integers overall. These sets are *not* of the same size.