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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.

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    The no greatest is so that for any rational $r$, there will be a unique associated "real." It's just a technical device, we could equally well insist that the upper set have no minimum. The intuition for cuts presumably comes from the standard experience of approximation by terminating decimals. Those are too special, not intrinsic enough. "Approximating" by all rationals less than $x$ is structurally nicer. In particular, if we use decimal approximation there is a technical nightmare in verifying the basic properties of addition, multiplication. With cuts it is smooth. – André Nicolas Jun 23 '13 at 02:37
  • The no greatest is so that for any rational r, there will be a unique associated "real".- this is not clear to me –  Jun 23 '13 at 02:48
  • Let $x$ be the (informal number line) real number $2$. If we allow a cut to have a largest element, $x$ would have two cut representations: (1) The cut determined by all rationals $\le 2$ and (2) the cut determined by all rationals $\lt 2$. By the way, I don't know what version of cut you are using. For Dedekind, it was a pair $(L,U)$, but more modern presentations usually just use $L$. – André Nicolas Jun 23 '13 at 03:08
  • Suppose I am given the set of rational numbers less than $e$ and a rational number in that set, say, $p$, how to find another number in that set which is larger than $p$ ? –  Jun 23 '13 at 05:50
  • Add up terms of the series $1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}$ until you get past $p$. – André Nicolas Jun 23 '13 at 06:13
  • yes. So, the intuition for cuts presumably comes from the standard experience of approximation by terminating decimals. For example, we can approximate $\sqrt{2}$ by the sequence $1, 1.4, 1.41, 1.414, 1.4142, \dots$. Now, why a cut set for $\sqrt{2}$ requires "all" rationals less than $\sqrt{2}$. Intuitively, only these rationals $1, 1.4, 1.41, 1.414, 1.4142, \dots$ uniquely identifies $\sqrt{2}$. Are not they sufficient ? –  Jun 23 '13 at 06:48
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    I mentioned sort of why in my first comment. – André Nicolas Jun 23 '13 at 06:59
  • Is there a reason that my answer got a lot of flak and some down votes, and then some other folks said basically the same things later? – dfeuer Jun 23 '13 at 07:21
  • @AndréNicolas: sorry, i missed that –  Jun 23 '13 at 08:31
  • @AndréNicolas :actually, I need more clarification on it. I am posting this as a separate post. –  Jun 23 '13 at 10:44
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    See the question. – Tony Piccolo Jun 25 '13 at 06:56
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    You had better find English translation of Dedekind's pamphlet "Stetigkeit und Irrationale Zahlen" online. Or better read its exposition in chapter 1 of G H Hardy's "A Course of Pure Mathematics". The treatment of Dedekind cuts in modern textbooks is full of mathhematical symbols without any appreciation of this simple idea of real numbers. – Paramanand Singh Oct 03 '13 at 03:09

3 Answers3

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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.

goblin GONE
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Hope this helps:

To understand why the cuts had no greatest element you need to understand why we need the real numbers in the first place.

If you consider the problem of finding $\sqrt{2}$, you will find that it is possible to come up with a set of rational numbers which are all less than the $\sqrt{2}$.

We can easily define such a set.

$S = \lbrace q \mid q^2 \lt 2 \text{ or } q \lt 0, q \in \mathbb{Q} \rbrace$

The set $S$ is a cut of the rational numbers which we can identify with $\sqrt{2}$. In Dedikends construction this set actually is $\sqrt{2}$.

The idea is that there are gaps in the rational numbers where we expect there to be a value. The easiest way to identify these gaps is by describing them with something like a cut.

Then we define the collection of cuts to be our new number system. The reason for this is that every rational has a unique cut and every gap has a unique cut. Therefore we don't lose the rationals by working in this new system, but do manage to fill in the gaps.

It may be helpful if you study the construction of the reals using a Cauchy completion. That is I think a little more intuitive.

Spencer
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One way to approach mathematics is to see it as building up more and more complex structures on top of simple foundations. One commonly used foundation (but not the only one) is set theory of one sort or another. Once you have set theory, you can build on it to form the natural numbers, the the integers, then the rational numbers, and then the reals. Dedekind cuts are one of the simpler ways to construct reals.

Added after question edit:

It's important that every real number be represented by exactly one Dedekind cut. Without the "no greatest element" bit, every rational number could be represented by two different cuts.

dfeuer
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