The real numbers $\mathbb{R}$ as Dedekind cuts

The real numbers are defined to be the set of all Dedekind cuts of $\mathbb{Q}$ (Dedekind cut): $\mathbb{R} := \{\, A \subseteq \mathbb{Q} : A \text{ is a Dedekind cut} \,\}.$ Elements of $\mathbb{R}$ are written $A, B, C, \dots$; each is a subset of $\mathbb{Q}$ satisfying (C1)–(C3).

The rationals embed into $\mathbb{R}$ by the rational embedding: for $q \in \mathbb{Q}$ set $q^{*} := \{\, r \in \mathbb{Q} : r < q \,\},$ the cut of all rationals strictly below $q$. Each $q^{*}$ is a Dedekind cut (Each rational cut $q^{*}$ is a Dedekind cut ↗), and $q \mapsto q^{*}$ sends $\mathbb{Q}$ into $\mathbb{R}$. The images of $0$ and $1$ are written $0^{*}$ and $1^{*}$; being cuts they lie in $\mathbb{R}$ and serve as its additive and multiplicative identities.

A cut $A$ is exactly the set of rationals lying below a putative real point; $\mathbb{R}$ is thus built by naming each point through the downward gap of rationals it determines. Where $\mathbb{Q}$ has a genuine rational point $q$, the cut $q^{*}$ recovers it, but the construction also admits cuts $A$ with no largest excluded rational and no rational boundary at all, such as $\{\, q : q < 0 \text{ or } q^{2} < 2 \,\}$. These are precisely the missing limits of $\mathbb{Q}$: the cut convention manufactures a real number wherever $\mathbb{Q}$ leaves a hole, which is why $\mathbb{R}$ is complete while $\mathbb{Q}$ is not.

The order on $\mathbb{R}$ is set inclusion, $A \le B :\Leftrightarrow A \subseteq B$ (Order on the Dedekind reals). That $q \mapsto q^{*}$ is an order-preserving ring embedding, and that its image is dense, is recorded in The rational cuts embed densely in $\mathbb{R}$, preserving sums, products, $0$, $1$ and the order; the arithmetic and order structure making $\mathbb{R}$ a complete ordered field is developed in The Dedekind reals form a totally ordered field and Dedekind completeness: the least-upper-bound property.