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. 
A: For example, a dedekind cut for $\sqrt{2}$:
$$\frac{1}{1} < \frac{7}{5} <  \frac{41}{29} < \frac{239}{169} < \dots < \sqrt{2} < \dots < \frac{577}{408} < \frac{99}{70} < \frac{17}{12} < \frac{3}{2}  $$
The left half has no biggest element since $\sqrt{2} \notin \mathbb{Q}$.  
Here's an algorithm to approximate $\sqrt{2}$.  He says if $\frac{m}{n}$ is an estimate, then $\frac{m+2n}{m+n}$ is a better estimate.
A: From what I understand, the third axiom basically states that the cut is infinite. The elements in the cut A grow higher and higher towards the represented real number without ever equating or surpassing it.
Basically, for $C = \{q \mid q\in\mathbb{Q}, q < x\}$ representing the real $x$, then for every $q\in C$ there is a $q'\in C$ such that $q<q'$. Hence there is no greatest element in $C$. There is always a greater element.
Not sure if i got that right, but that is how I understood it.
Check this answer. It helped get my mind around the concept of a Dedekind cut.
