I apologise for this question to be quite long... it looks wordy since I'm trying to express the process of my thinking.
I'm recently using the appendix at the end of chapter 1 in Rudin's Principle of Mathematical Analysis to study the construction of $\mathbb{R}$ from $\mathbb{Q}$.
I fall into a somewhat circular argument $\color{red}{(\ast)}$ when I try to visualize cuts as finite lengths on an infinite straight line (by finite length I mean the segment from a reference point "0" to some point on the line).
I'm trying to visualize cuts as lengths on the line because I think "length" is the intuitive concept that is central to the story: people in the old days first found out that there exist some certain lengths that could not be represented by the elements of $\mathbb{Q}$ (namely the diagonals of squares), then this became the reason for cuts to come.
So, the first question I asked myself is:
How could one be sure that, after defining the cuts, every finite length on the line is indeed included in $\mathbb{R}$?
My answer to this is: I can now pick any one point on the line, denote it as $x$, and I can associate a set $r$ which consists of all the rational points to the left of the point $x$ so that $r$ is a cut. And I'm now representing $x$ by $r \in \mathbb{R}$.
Then, I asked myself the second question:
How could I know that such a cut $r$ is actually unique to my length $x$? That is, how could I ensure that my cut $r$ will not represent $x$ anymore even if I move my point $x$ to the right on the line for just a tiny tiny bit?
For this question, let's denote the original $x$ as $x_0$ and the $x$ after moving as $x_1$. My idea goes like: if we can always find a rational number between $x_0$ and $x_1$ (no matter how close they are), then $x_1$ must be represented by a different cut (let's denote it as $s$) because by the definition of cut, $r$ is now a proper subset of $s$.
[Please note also that I did not justify my premise above using the property that $\mathbb{Q}$ is dense in $\mathbb{R}$, because I still don't know if $x_0$ and $x_1$ are two distinct reals at the moment]
So, at this point, I made two important assumptions:
Any two points on the line must either be the same point or be different. So, as long as I move the point, it becomes a different one to the original.
The settings for the lengths on the line make them isomorphic to the cuts. i.e. assume that some proper, possibly customary, addition, multiplication and order which operates on the lengths are defined, and are preserved by some bijection between the set of all lengths and the the set of all cuts.
Using those two assumptions, we now have the two different points on the line representing by two different cuts.
For assumption 1, I think it is rather like an axiom.
And Finally this leads to the humble question I wrote in the title:
How do we prove assumption 2? That is, how can we construct an isomorphism between the set of all real numbers defined by (Dedekind's) cuts and the set of all finite lengths (including zero length) on an infinite straight line?
For now, there are two angles that I tried to solve this question in general:
As it is a fact that any two ordered fields with the least-upper-bound property are isomorphic, I was trying to prove that the set of all lengths on the line forms an ordered field with the least-upper-bound property under some good definitions. But I'm stucking somewhere in the middle.
Assume that there exist two points on the line $x_0$ and $x_1$ as described above where $x_1$ is to the right of $x_0$, and there exists some cut $r$ that catches them both. Then, the length between $x_0$ and $x_1$ must be finite (due to assumption 1), and I don't see a reason why any rational cuts wouldn't be surrounded by such "blank zone" or why those blank zones would have various lengths (I don't think my naive understanding to the concept of length would let me say more on these). By mathematical induction, there are infinitely many rationals between any two different rationals, and thus we would have some finite constant multiplied by infinity equals some finite quantity, which seems logically incorrect. But to me, such "proof by contradiction" is far not rigorous.
By the way, I have a feeling that constructing $\mathbb{R}$ from $\mathbb{Q}$ by means of Cauchy sequences might be a better approach to this question, I haven't read such construction yet because I believe that cuts should be equally good.
$\color{red}{(\ast)}$ : I think what is (probably) being circular here is as follows (and this is what really confuses me):
From the definition of cut, it is clear that any two distinct cuts cannot represent the same length on the line. But now I want to prove that any two distinct lengths on the line cannot be represented by the same cut. Symbolically, I want to prove that if the two lengths on the line are denoted as $x_0$ and $x_1$ where $x_0 \neq x_1$, and their associated cuts are denoted as $r_0$ and $r_1$ respectively, then $r_0 \neq r_1$. By the definition of the order defined on cuts, to prove $r_0 \neq r_1$ I need to show that one is the proper subset of another. This means that I need to find a rational in between. However, to conclude that there exists a rational between two cuts it requires the two cuts not to be equal in the first place.
Could anyone please tell me how to get rid of this circularity?
There may be ambiguities in multiple places in this question as I am still being confused about it right now... so anyone who notices why I'm confused is more than welcomed to write an answer even if the answer is unrelated to my arguments above :)
Thank you for your patience :)
