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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.

Willie Wong
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Chandresh Sharma
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  • See http://math.stackexchange.com/questions/95271/why-does-the-dedekind-cut-work-well-enough-to-define-the-reals and http://math.stackexchange.com/questions/427257/motivation-behind-dedekinds-cut-set and http://math.stackexchange.com/questions/427448/a-basic-question-on-intuition-of-rational-cut-set-in-the-construction-of-real-nu – Asaf Karagila Aug 05 '13 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. – Silent Oct 07 '17 at 12:44

3 Answers3

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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.

Pete L. Clark
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Péter Komjáth
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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.

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cactus314
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  • 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, – Chandresh Sharma Aug 05 '13 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. – cactus314 Aug 05 '13 at 19:55
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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.

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7hi4g0
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