Motivation behind Dedekind's cut set

I want to know the motivation behind Dedekind's real number construction. The motivation of such properties of the cut sets is not clear to me. BTW, I am new to real analysis and just have started reading the first chapter of Rudin.

what made Dedekind to thought of sets with some nice properties (what are the motivations behind such properties ? For example, why cut set should not have any greatest element ?). I read somewhere that initially Dedekind thought that any real number can be uniquely identified by rational numbers less than that. So, this was his starting point. From that the first and second property of cut set is understandable, but the third property (no greatest element) is not clear to me.

Solution 1:

Here's my best guess. Dedekind:

Hmmmmmm.

The rational number line is full of holes. At $\sqrt{2}$, there should be a number, but there isn't.

What does that even mean?

Hm.

Okay, there is no hole at $2$. We can make this precise by observing that $(2-\epsilon,2)_\mathbb{Q}$ is distinct from $(2-\epsilon,2]_\mathbb{Q},$ for all strictly positive $\epsilon$.

But there is a hole at $\sqrt{2}.$ We can make this more precise by observing that $(\sqrt{2}-\epsilon,\sqrt{2})_\mathbb{Q}$ has exactly the same elements as $(\sqrt{2}-\epsilon,\sqrt{2}]_\mathbb{Q}$ for all strictly positive $\epsilon$.

But wait. If we're trying to build the real numbers, then the notation $\sqrt{2}$ isn't 'allowed' yet. Okay, so lets make it precise like this.

There is a hole at $\sqrt{2},$ which means that $\{x \in \mathbb{Q} \mid x^2 < 2\}$ has exactly the same elements as $\{x \in \mathbb{Q} \mid x^2 \leq 2\}.$

But no, this is too symmetrical; the above statement could equally well be seen as the claim that there's a hole at $-\sqrt{2}$. Okay, lets go ahead and destroy that symmetry.

There is a hole at $\sqrt{2},$ which means that the downward closure of $\{x \in \mathbb{Q} \mid x^2 < 2\}$ has exactly the same elements as the downward closure of $\{x \in \mathbb{Q} \mid x^2 \leq 2\}.$

Aha! So maybe we should define that $\sqrt{2}$ is the downward closure of $\{x \in \mathbb{Q} \mid x^2 < 2\}.$ But wait, why not the downward closure of the non-strict version? Okay, lets just disregard that possibility for the moment.

So, a real number is the downward closure of a set of rational numbers that is bounded above, but has no greatest element.

Wait. How about (Eureka!):

A real number is a downward-closed set of rational numbers that is bounded above, but has no greatest element.