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?
