\begin{definition}[Definition:Dedekind Cut/Definition 2]
Let $\struct {S, \preceq}$ be a totally ordered set.
A '''Dedekind cut''' of $\struct {S, \preceq}$ is an ordered pair $\tuple {L, R}$ such that:
:$(1): \quad \set {L, R}$ is a partition of $S$.
:$(2): \quad L$ does not have a greatest element.
:$(3): \quad \forall x \in L: \forall y \in R: x \prec y$.
\end{definition}