# Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?

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?

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.