# Short

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 $$\mathbb{N}$$.

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