# Pugh: Dedekind cuts and gaps in the real number line

I'm reading Charles C. Pugh's Real Mathematical Analysis, 2nd Ed. On page 12 he defines a Dedekind cut like so:

Definition. A cut in $$\mathbb{Q}$$ is a pair of subsets $$A, B$$ of $$\mathbb{Q}$$ such that
(a) $$A \cup B = \mathbb{Q}$$, $$A \neq \emptyset$$, $$B \neq \emptyset$$, $$A \cap B = \emptyset$$.
(b) If $$a \in A$$ and $$b \in B$$ then $$a < b$$.
(c) $$A$$ contains no largest element.

Also:

Definition. A real number is a cut in $$\mathbb{Q}$$.

On page 17, Pugh shows that $$\mathbb{R}$$ has no gaps:

Next, suppose we try the same cut construction in $$\mathbb{R}$$ that we did in $$\mathbb{Q}$$. Are there gaps in $$\mathbb{R}$$ that can be detected by cutting $$\mathbb{R}$$ with scissors? The natural definition of a cut in $$\mathbb{R}$$ is a division $$\mathcal{A}|\mathcal{B}$$, where $$\mathcal{A}$$ and $$\mathcal{B}$$ are disjoint, nonempty subcollections of $$\mathbb{R}$$ with $$\mathcal{A} \cup \mathcal{B} = \mathbb{R}$$, and $$a < b$$ for all $$a \in \mathcal{A}$$ and $$b \in \mathcal{B}$$. Further, $$\mathcal{A}$$ contains no largest element. Each $$b \in \mathcal{B}$$ is an upper bound for $$\mathcal{A}$$. Therefore $$y = \text{l.u.b.}(\mathcal{A})$$ exists and $$a \leq y \leq b$$ for all $$a \in \mathcal{A}$$ and $$b \in \mathcal{B}$$. By trichotomy, $$\mathcal{A} | \mathcal{B} = \{ x \in \mathbb{R} : x < y \} | \{ x \in \mathbb{R} : x \geq y \} .$$ In other words, $$\mathbb{R}$$ has no gaps. Every cut in $$\mathbb{R}$$ occurs exactly at a real number.

I can't convince myself that this works, since $$\mathcal{A}|\mathcal{B}$$ is a cut in $$\mathbb{R}$$ — but he's defined a real number as a cut in $$\mathbb{Q}$$. My mind is constructing a number system $$\mathbb{S}$$ where each $$s \in \mathbb{S}$$ is associated with some cut in $$\mathbb{R}$$, and I'm tempted to keep filling in the gaps of number systems ad infinitum.

What am I misunderstanding?

The point is that if you fill the gaps in $$\mathbb{R}$$, you get nothing new. The set of cuts in $$\mathbb{R}$$ is in bijection with $$\mathbb{R}$$, by mapping each element $$y\in\mathbb{R}$$ to the cut $$\{x\in\mathbb{R}:x (note that this map is also how you identify $$\mathbb{Q}$$ as a subset of the real numbers, but in the case of $$\mathbb{Q}$$ it is not surjective). Moreover, this bijection preserves the order relation, so the ordered set of cuts in $$\mathbb{R}$$ is order-isomorphic to $$\mathbb{R}$$.
• I think I understand what you're saying. My $\mathbb{S}$ does not exist because we have carefully designed $\mathbb{R}$ so that the least-upper-bound property holds, so by consequence of definition it "fills in it's own gaps". Is that a reasonable explanation? Aug 29, 2020 at 1:55
• More or less. Another way to put it is that your $\mathbb{S}$ exists, but it is the same as $\mathbb{R}$. Aug 29, 2020 at 1:55