Recall that the set \(\mathbb{N}\) is \(\{0, 1, 2, 3, …\}\). The set \((0,1)\) is all the real numbers between, but not including, zero and one. We can think of real numbers as decimals, and so we can think of the set \((0,1)\) as all the decimals between zero and one.

In Basics of Set Theory, we saw that making a one-to-one correspondence of a set with \(\mathbb{N}\) is the same as making a list of the elements in that set, so that everything appears exactly once.

We won't give all the details here (for that, check out the In-Depth explanation), but will explain the line of reasoning. The classic, and best, way to prove this result is the following: we show that every time you try to list all the elements of \((0,1)\), you miss one.