Definition:Dedekind Completion

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

The Dedekind completion of $\left({S, \preceq}\right)$ is defined as the unique Dedekind complete ordered set $\left({\widetilde{S}, \widetilde{\preceq}}\right)$, up to order isomorphism, that satisfies the following axioms:
 * $\left({1}\right): \quad$ There exists a subset $S^* \subseteq \widetilde{S}$ that is order isomorphic to $S$.
 * $\left({2}\right): \quad$ If a Dedekind complete ordered set $\left({S', \preceq'}\right)$ satisfies condition $\left({1}\right)$, then there exists a subset $\widetilde{S}^* \subseteq S'$ that is order isomorphic to $\widetilde{S}$.

Also see
This is not to be confused with the Dedekind–MacNeille completion.