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.