# 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.

• Aug 5, 2013 at 17:27
• More striking importance of third axiom is pointed out in Rudin's Principles of Mathematical Analysis' chapter 1, exercise 20: If we omit third axiom, then there are cuts for which additive inverse does not exist. Oct 7, 2017 at 12:44

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.

• I wonder whether you would be willing having a look at this question: Bartoszyński's results on measure and category and their importance. It was partially inspired by your comments on Wikipedia. (Sorry for posting comment unrelated to the post, but I wanted to find some place where to ping you. Feel free to ping me here or in chat to let me know that you have seen this comment and I can remove it.) Thanks! Jan 6, 2017 at 8:41

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.

• my question is that if we make a cut for sqrt(2) then how we are using third axiom to say that it is a cut i.e. what is use of third axiom that there should be no greatest element in the set which contains elements below sqrt(2). Thanks, Aug 5, 2013 at 17:17
• @ChandreshSharma There is no largest rational element in the cut. The real number $\sqrt{2}$ is defined as the "number" between the upper and lower cut. Aug 5, 2013 at 19:55

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.