There is a property of Dedekind cuts that states that there exists no maximum $a_0\in A$ for all cuts $A\in\mathbb{R}$.
My question is: Why do Dedekind cuts need this property?
What would happen if Dedekind cuts didn’t have this property?
With these questions I aim to better understand the motivation behind the definition of Dedekind cuts and the construction of $\mathbb{R}$.