Real Analysis – Why Can a Real Number Be Defined as a Dedekind Cut?

Question

I don't know if my textbook is written poorly or I'm dumb. But I can't bring myself to understand the following definition.

A real number is a cut, which parts the rational numbers into two classes. Let $\mathbb{R}$ be the set of cuts. A cut is a set of rational numbers $A \subset \mathbb{Q}$ with the following properties:

i) $A \neq \emptyset$ and $A \neq \mathbb{Q}$.

ii) if $p \in A$ and $q < p$ then $q \in A$.

iii) if $p \in A$, there exists some $r \in A$ so that $p < r$ (i.e. $A$ doesn't contain the "biggest" number).

That's a literal translation from my textbook (which is written in Slovenian). All seems fine and I can get my head around all of the postulations except for one. The definition states in the beginning "A real number is a cut…", but then it also states "A cut is a set of rational numbers…" So a real number is 'a set of rational numbers'?!

It's not my bad translation, I swear, I'm quite good at English. Either the textbook is written in such a convoluted manner that I can't properly understand the wording the author chose or I'm overlooking something big. Could you please clarify and explain the definition in full detail?

Answer 1

As I said in my comment, you are in good company---in fact, the company of Dedekind himself! In a letter to Heinrich Weber, Dedekind says the following:

(...) I would advise that by [natural] number one understand not the class itself (...) but something new (corresponding to this class) which the mind creates. (...) This is precisely the same question that you raise at the end of your letter in connection with my theory of irrationals, where you say that the irrational number is nothing other than the cut itself, while I prefer to create something new (different from the cut) that corresponds to the cut and of which I prefer to say it brings forth, creates the cut. (Ewald, From Kant to Hilbert, vol. 2, p. 835)

So Dedekind himself preferred not to identify the real number with the cut, merely saying that the mind somehow creates the real number which then corresponds to the cut. This is, however, a little bit obscure, so it's not surprising that most mathematicians (such as Weber!) decided to ignore Dedekind's suggestion and simply identify the real number with the cut. The reasoning behind this identification is roughly the following.

We know that any Dedekind-complete ordered field is isomorphic to the field of the real numbers. In particular, this means that any construction or theorem carried out in the real numbers could be reproduced inside an arbitrary Dedekind-ordered field, and vice-versa, by simply using the isomorphism as a "translation" between the fields. Hence, it doesn't matter what the real numbers actually are; for mathematical purposes, even supposing that there is such a thing as the real numbers, anything that we wanted to do with them could also be accomplished in an arbitrary Dedekind-complete ordered field.

Thus, if we could show that the cuts themselves satisfy the axioms for being a Dedekind-complete ordered field, then we could dispense with the real numbers altogether and simply work with the cuts themselves. And, in fact, we can show that this is the case! One need only to show that, given two cuts, $X$ and $Y$, it's possible to define operations on them corresponding to the usual operations on the real numbers, such as addition and multiplication, and that after doing so these operations will satisfy the field axioms. It's not difficult to see that the obvious operations will yield the desired result (exercise!), though it is somewhat laborious. If you are interested in seeing a detailed verification, I recommend reading, say, Appendix A of Yiannis Moshovakis excellent book Notes on Set Theory, which contains a very thorough discussion of the matter.