**Definition** A *Dedekind cut* is any [[Set]] $C$ satisfying the following conditions: 1. $C \subset \mathbb{Q}$, but $C \neq \emptyset$ and $C\neq \mathbb{Q}$; 2. If $p \in C$ and $q<C$ then $q \in C$; 3. if $p \in C$ then $\exists q\in C$ such that $q \in C$. **Remark** A *cut* should be thought of as the set $(-\infty, b)\cap \mathbb{Q}$ for some 'real number' $r$. See [[The set of real numbers is the set of Dedekind cuts]].