Intuition between Dedekind cut construction of real numbers I understand the construction of real numbers via Dedekind cuts I think, the set of Dedekind cuts over $Q$ are real numbers.  However, I don't see how it gives us any more understand or power than the trivial definition of real numbers $=$ set of rationals $+$ set of irrationals (not rationals).  With the set of irrationals we have already shown is nonempty by $\sqrt2$ as an example.  A number is either rational or it is not.
So I'm not sure what all the effort to define an algebra over Dedekind cuts and mapping to real numbers really buys.
 A: The point of Dedekind cuts is to rigorously define the irrational numbers (and consequently all real numbers) starting from the rational numbers alone. The definition "reals = rationals + irrationals" presupposes that we already have a good definition of "irrational number," but the whole point is that we don't yet at this stage in the game. The "big theorem" of Dedekind cuts is:

The set of Dedekind cuts (with the appropriate operations) forms a complete ordered field, and there is exactly one complete ordered field up to isomorphism.

Intuitively this is best thought of as saying that the Dedekind cut construction accurately captures our pre-formal intuitions about the real numbers. Note that before this construction it's not even clear that any complete ordered field exists! Basically, the chain of ideas is:


*

*We start out with $\mathbb{Q}$ as our "well-understood" object. (Of course, we can separately ask how $\mathbb{Q}$ is constructed - for now though  we're taking it for granted.)

*We next lay down some basic properties we want "the real number system" to have. At this point "the real number system" is merely an intuition; we don't know that anything fitting it actually exists. One of these basic properties is that the real numbers should form a complete ordered field; another, perhaps less fundamental but still important, is that it should be "unique" in some appropriate sense.

*The Dedekind cut construction lets us rigorously construct something with those desired properties, and we call that thing "the real number system."

OK, now some comments:


*

*Dedekind cuts aren't the only way to construct the reals (that is, to demonstrate the existence and appropriate uniqueness of a complete ordered field). The other most common approach is via Cauchy sequences. If you pick a different approach to start with, then Dedekind cuts are thought of as yielding an alternative construction which we subsequently prove is equivalent and may be useful as a technical tool.

*It's not quite true that the Dedekind cut construction rests only on the rational numbers. It also requires some basic set theory. The role of set theory in the construction of the real numbers is quite interesting, and one of the things that sparked modern mathematical logic, but that's a topic for another time.

*Finally, one might reasonably have different intuitions about what "the real number system" ought to be. This is absolutely something we can explore. The main competition to the standard approach is via nonstandard analysis. Very briefly, in nonstandard analysis we take the stance that infinitesimals are desirable (and so completeness is not) and that uniqueness in any particular sense isn't especially important (there is no such thing as "the" hyperreal numbers, although any two hyperreal number systems are similar in a very strong sense). Interestingly, we can show that results provable in one context can be translated to the other!
A: The point of having a construction of the real numbers is that, before they are constructed, we don't know that they exist!
It seems that you have the idea that the number line has already been defined, and divided into the rational numbers and the irrational numbers. While this is a reasonable idea to have most of the time when working with math, when we are dealing with the foundations of math we have to build up our knowledge from scratch.
So, without having some construction of the real numbers (such as Dedekind cuts, or equivalence classes of Cauchy sequences), we can't refer in a meaningful way to "the set of irrational numbers", because we don't yet have a "background set" of numbers that can be filtered by (ir)rationality.
Even proving that $\sqrt2$ (for example) is irrational depends first on knowing that $\sqrt2$ exists!—in other words, that there is a "number" (in some background set) whose square equals $2$. The existence of such a number is impossible to establish without first having the set of real numbers and some of its properties (such as the intermediate value theorem).
So in summary: when working with the foundations of math, we have to construct (or otherwise establish the existence of) the set of real numbers, before doing any of the math we probably saw earlier in life but which depends rigorously on the existence of the reals.
A: The problem that Dedekind is addressing is the following: how can we define the set of real numbers in terms of a concept we already feel comfortable with (the rational numbers). We like the rational numbers because you can always write them down (given enough space) and can do arithmetic on them mechanically. Irrational numbers pose a problem: we can draw a plot of $x^2$ and see that it ought to cross $2$ somewhere, but how do we describe where? We could just start writing out digits
$$\sqrt{2}=1.4142135\ldots$$
and, while we could compute those digits as far as we like (via bisection, for instance), just writing out the digits is never going to suffice to pin down which number we meant - so what does it even mean to say that $\sqrt{2}$ is a number if we can't write it down?
Well, we could talk about its algebraic properties - for instance, $\sqrt{2}$ is a solution to $x^2 = 2$. That's fine, but $-\sqrt{2}$ is also a solution, so how then do we tell those two values apart, if they have the same algebraic properties*? Worse still, what's the difference between $\sqrt{2}$ and $\sqrt{-1}$, where we sort of think one is on the real line and the other one appears not to be - yet nothing stops us from fitting together an algebraic picture including $\sqrt{-1}$ (since, of course, we could work in the complex plane - but that's not what we want to do right now!)
Well, we "believe" in $\sqrt{2}$ because we can sort of figure out where it would have to live on a number line - we suspect $1<\sqrt{2} < 2$ for instance since $1^2 < 2 < 2^2$. In fact, if we suspend our disbelief about whether $\sqrt{2}$ is even a thing for a moment, we realize that we can compare any rational number to $\sqrt{2}$ just by saying that $p > \sqrt{2}$ if and only if $p > 0$ and $p^2 > 2$ - and that statement is purely a statement about rational numbers, but tells us quite well where we would expect to find $\sqrt{2}$. Otherwise said, we imagine that the real numbers "fill in" all the gaps between rational numbers, and thus imagine that a real number is something to which you can ask the following question:

Are you greater than, less than, or equal to this rational number?

And whose answers to that question are somehow consistent. That's what Dedekind cuts formalize: they say that since we can ask this question of $\sqrt{2}$, there is a meaningful real number - defined exactly by the cut we described on $\mathbb Q$ - in that area we described. One can then define arithmetic on such cuts and find out, thankfully, that $(\sqrt{2})^2 = 2$ - which justifies calling that cut $\sqrt{2}$.
Once you have these definitions, you can start to say "we think that $\sqrt{x}$ exists for any non-negative $x$" and then you can construct Dedekind cuts that really are square roots of $x$! Even better, you can start defining and reasoning about concepts such as integrals and limits and prove theorems about this structure that let you reason without dealing with all the specifics of cuts***. However, a priori, this view of mathematics doesn't feel that $\sqrt{2}$ is meaningful: we only feel that rational numbers are meaningful** - the rest we must define ourselves!
(*Indeed, any polynomial with rational coefficients that has $\sqrt{2}$ as a root also has $-\sqrt{2}$ as a root, so we can't use algebraic properties to tell these numbers apart! You get an interesting view of mathematics if you take "we'll define numbers by their algebraic properties" as your goal - you get field theory and Galois theory that way, but you notably don't get the real numbers from the rational numbers that way and you can't really talk about order in this)
(**Of course, some people go further back - maybe we only like natural numbers or sets or something. It doesn't really matter where you start - there's still a meaningful relationship between rational and real numbers)
(***For instance: if you wanted to define $\pi$, it's easier to say define it as a limit of various sequences of partial sums or as an integral than to say "A rational $p$ is less than $\pi$ if and only if there is some natural $k$ such that $p<\sum_{i=0}^k\frac{8}{(4n+1)(4n+3)}$" which is a correct definition of $\pi$ via Dedekind cut (via the Leibniz formula for $\pi$), but is not at all intuitive and is not easy to compute!)
