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I just got through reading the first chapter of principles of mathematical analysis by Walter Rudin, the first chapter goes on and on about Dedekind cuts and then starts defining properties of them, talking about how there closed under certain mathematical operations and such, I just don't understand how defining the notion of a Dedkind cut is at all useful or interesting, and when I say useful I don't mean practical. I mean in the sense that it could contribute to other areas of mathematics, not including itself, I kind of had the same experience when I started studying parts of linear algebra, I know that notation and rigour is important but some of the parts seem like over kill in terms of rigor and the results don't seem very meaningful to me. I have only read the first chapter and don't know if I should continue, I would appreciate any advice.

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Rudin puts the proof in the Appendix for a reason. If you don't find it interesting, don't read it. (I don't say this to be rude - when I first got to this section, I skipped a lot of it.) –  wj32 Nov 2 '12 at 5:54
    
My professor mentioned that it is included for the intellectually curious. I don't think any of my friends have had a professor that rigorously covered Dedkind cuts. The usual reason is because it contributes to a deeper understand of mathematics, but you can get through a first course in analysis without ever knowing exactly how the Dedkind cuts work. –  emka Nov 2 '12 at 6:00
    
You can question the value of dedekind cuts, but it's difficult to question the value of linear algebra, which is used constantly in science and engineering (and so much of advanced math). –  littleO Nov 2 '12 at 6:04
    
Knowing the idea behind Dedekind cuts should be part of any well rounded mathematical education. The details in verifying all the algebraic rules are messy and not very interesting, so just try to get the big picture. –  Michael Greinecker Nov 2 '12 at 6:26
    
Dedekind cuts is one approach (among others) to construct $\mathbb{R}$. Often this book is the first rigorous book that a student encounters and I would guess Rudin wished to be self-contained. –  AD. Nov 2 '12 at 6:28
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1 Answer 1

If you don't like Dedekind cuts, which of course you are entitled to, what do you mean when you write $\sqrt2$?

Regarding the book, you don't say why you are reading it, so it is hard to give you advice on what to do. A very natural question for anyone interested in math is whether the objects exist, and how. Going back to my question above, you could say "$\sqrt2$ is a number that when squared gives 2". Very nice, but how do you know such an object exists? Defining an object by one of its properties doesn't make it to exist. For instance, I can say "let $m$ be the real number that is bigger than all other real numbers", or "let $s$ be a number such that $1/s=0$".

The point of that chapter 1 is to show that the real numbers can actually be constructed out of the rationals in such a way that we get a field with the properties that we usually expect from the reals. After all, if the reals didn't exist, then Calculus would be just hot air.

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I empathise with not liking Dedekind cuts. By $\sqrt{2}$ one can mean an equivalence of Cauchy sequences in $\mathbb Q$ converging to $\sqrt{2}$, to address the question with which you start your answer. –  Matt N. Nov 2 '12 at 6:31
    
I think it is strange that so many analysis books show how to construct the reals from the rationals, but do not show that there is up to isomorphism only one complete ordered field. I certainly do not mean a subset of $9$ when I write $\sqrt{2}$. –  Michael Greinecker Nov 2 '12 at 6:34
    
I thought the purpose of "constructing" the real numbers (using Dedekind cuts or equivalence classes of Cauchy sequences or whatever) was just to prove that: if the axioms for $\mathbb{R}$ are inconsistent, then the axioms for $\mathbb{Z}$ are also inconsistent. (And if $\mathbb{Z}$ can be constructed from some simpler set of axioms, then that set of axioms is also inconsistent.) This gives us reassurance that the axioims for the reals are consistent. I don't think constructing the reals gives us new insight into what a real number "really is". –  littleO Nov 2 '12 at 6:37
    
@MattN.: I personally don't like Dedekind cuts. But it is also true that when I have tried to teach the "equivalence classes of Cauchy sequences" approach I found it messier than I expected. –  Martin Argerami Nov 2 '12 at 13:29
    
@MichaelGreinecker: I think that uniqueness is very important, too. I don't think of $\sqrt2$ as a subset of the rationals, but neither I think of $e^{i\theta}$ as a pair of reals, or $-2$ as a pair of naturals, or $2$ as an equivalence class of sets. Which doesn't mean that they are not concrete methods of constructing such objects. –  Martin Argerami Nov 2 '12 at 13:33
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