The goal here is to sketch the argument for why \(|(0,1)| > |\mathbb{N}|\).

We want to show that \(|(0,1)| > |\mathbb{N}|\). In other words, we want to show that the sets are not the same size. This means we have to show that there is no one-to-one correspondence between \((0,1)\) and \(\mathbb{N}\).

We must show that no matter how we try, or how anyone else tries -- today or tomorrow or a thousand years from now -- there is no way of making a one-to-one correspondence.

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.

That is, given any potential candidate for a one-to-one correspondence -- assigning every element of \(\mathbb{N}\) to one element of \((0,1)\) and hoping to not missing any -- it is always possible to find an element of \((0,1)\) not on that list! Every list of the elements in \((0, 1)\) must miss something, and we prove this by, given such an list, constructing an element in \((0, 1)\) that we know cannot be on that particular list.

The way this is done is very clever, and if you're curious you can read about it in the In-Depth explanation. But the gist is that there can be no one-to-one correspondence, and in fact the set \((0, 1)\) must be bigger than \(\mathbb{N}\).