Rudin's Construction of Real Numbers

At the end of chapter 1 of Principles of Mathematical Analysis, Rudin provides a proof of the construction of real numbers. The first step in the proof is to define members of $\mathbb{R}$ that are subsets of $\mathbb {Q}$ known as cuts. Rudin gives the following definition of a cut:

1. $\alpha$ is not empty, and $\alpha \neq \mathbb{Q}$.
2. If $p \in \alpha, q \in \mathbb {Q}$, and $q < p$, then $q \in \alpha$
3. If $p \in \alpha$, then $p < r$ for some $r \in \alpha$.

The letters $p, q, r, \ldots$ will always denote rational numbers, and $\alpha, \beta, \gamma, \ldots$ will denote cuts.

My question is the following: Doesn't the definition of the cut provided lead to the conclusion that each cut contains all rational numbers?

In other words, that $\alpha = \mathbb {Q}$ for all $\alpha$?

The third property of cuts provided shows that there is no maximal element in a cut. The second property implies that given an element in a cut, all rational numbers less than that element are in the cut. Doesn't this clearly imply that cuts include all the rationals?

I was hoping someone could clear this up. Thanks.

• The set of all strictly negative rationals ought to be a cut, and it doesn't equal $\mathbb Q$. How do you think it fails to satisfy the definition? Feb 4 '13 at 21:30
• Not every linear ordering shares the property of $\mathbb{N}$ that every nonempty subset with no greatest element is unbounded. Feb 4 '13 at 21:34
• @user1709828: It's clearly relevant and appropriate to tell the asker that this construction in generally credited to Dedekind (under the particular name "Dedekind cuts"). What made your initial comment sound off was that it looked like the OP was in error for not already knowing this. Feb 4 '13 at 21:37
• One way to read (3) is: "There is no strict maximum element of a cut." Feb 4 '13 at 21:40
• I think you just have to realize that a nonempty set of numbers can be bounded above and still have no maximum. Feb 4 '13 at 21:48

No, of course not. For example, the set of negative rational numbers are a cut.

• But it appears to me that this is not a cut according to the three properties given to me in the question. For example, if $p = 0$ in the third property, there is no rational number $r$ that is greater than $p$ and within the set of negative rational numbers. Feb 4 '13 at 21:34
• $p=0$ is not a negative rational number, @user60994 Feb 4 '13 at 21:36
• @user60994: $0$ is not a negative number, so it is not in Chris’s cut. Feb 4 '13 at 21:36
• Chris, that should be "is a cut." A set is singular, it is not "are a cut." Feb 4 '13 at 21:37
• @BrianM.Scott That's not a sport idiom that I have read before, but sports makes for bad examples. In the states, most team names are plural, so you mights write "The Cubs are pitching badly," but write "The team is pitching badly." As a rule, does the sentence "The set are a cut" read like English? My teachers always said to eliminate auxiliarly phrases to find the correct plural/singular choice. Feb 4 '13 at 21:54

Think of it this way: A cut is a set of the form: $$\mathbb Q\cap (-\infty,r)$$ where $r$ is any real number.

The reason they haven't been defined this way, is because you haven't defined the real numbers yet.

Funny, I thought this too when I was learning about cuts.

It actually seems reasonable: if there is no largest element, and all the rational numbers less than an element in the cut are included, then the cut should contain all rational numbers.

The thing is, the definition doesn't define how the numbers get larger. Rather, it just says that there is no largest member.

Then, we could pick any open interval and change it so that it matches the conditions. For example, if I had the open interval from 0 to 1, then you can see that this interval has no largest element. That's the idea, that cuts are just open intervals with some special properties.

At least, that's how I think of it.