Q:

Dedekind's cut and axioms

What is the importance of 3rd axiom of dedekind's cut?
a Dedekind cut is a partition of a totally ordered set into two non-empty parts (A and B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element.(From Wikipedia)
what is importance of statement "A contains no greatest element"??
Please explain in intuitive way.
Also my reasoning is that if you don't know what is greatest number in A how can you calculate Least Upper Bound for A which is required for completeness of R. 

A:

Dedekind cuts are used for creating reals from rational numbers, that is, axiomatically, the reals are THE Dedekind cuts of the rationals. Without the condition, however, every rational would have two representations as a Dedekind cut: one where it is added to the lower class, and another in which it is added to the upper class. Hence the condition. 

