# Dedekind Cuts in Construction of the real line [closed]

Is each Dedekind cut a unique real number? or when we apply the process(Dedekind cut), do we get a bunch of real numbers instead of a unique one.

If we get a unique real number, is the unique real number then plotted as a line segment between rationals on the number line? Or is it plotted as a single point (just like 0 and 1)?

If not so, Is the bunch of numbers that we get infinite?

## closed as unclear what you're asking by Andrés E. Caicedo, hardmath, dustin, user147263, TravisJan 22 '15 at 3:12

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• Depending on your definition of real number, the answer to the first question may be "Yes, by definition" – Hagen von Eitzen Jan 21 '15 at 10:20
• Lets say it is defined as - a number on the real number line – novice Jan 21 '15 at 10:21
• What is the standard definition btw? – novice Jan 21 '15 at 10:34
• Some authors use Dedekind cuts to represent only non-negative real numbers, so that these sets are known to be bounded below. The negative real numbers can then be represented algebraically. – hardmath Jan 22 '15 at 2:14
• Could you please indicate what is unclear so I may edit the question properly – novice Jan 22 '15 at 8:13

## 1 Answer

There are several ways to construct (or if you prefer, define) the real numbers. The two most familiar are as Dedekind cuts in $\Bbb Q$, the ordered set of rational numbers, and as equivalence classes of Cauchy sequences of rational numbers with respect to a certain equivalence relation. If one constructs them using Dedekind cuts, then each Dedekind cut is by definition a unique real number. If one constructs them in some other way, it’s no longer the case that each real number is a Dedekind cut in $\Bbb Q$, but it is still true that there is a nice bijection between the set of Dedekind cuts in $\Bbb Q$ and the reals as constructed.

Saying that the reals are the numbers on the real number line is not a definition: in order to make it one, you’d need to define the real number line independently of the notion of real number. As it stands, you’re pretty much just saying that a real number is a member of the set of real numbers.

• Yes I knew it was a circular definition, which is why I asked what it is the standard definition. Thanks. – novice Jan 22 '15 at 1:55
• @novice: You're welcome. There really isn't one. The mathematically important fact is that all of the constructions produce isomorphic objects: up to isomorphism there is only one Dedekind-complete ordered field. – Brian M. Scott Jan 22 '15 at 2:03
• So it's by definition a unique real number - But it is used to plugin the gap between two rational points and is plotted as a line segment between them? – novice Jan 22 '15 at 2:07
• @novice: No, it's a single point, not a segment. No two rational numbers are adjacent: the average of any two rational numbers is a rational number between them. The gaps in $\Bbb Q$ are a bit more subtle than that. For example, if $R$ is the set of positive rational numbers $x$ such that $x^2>2$, and $L$ is the rest of the rationals, then every member of $L$ is less than each member of $R$, but there is no rational number between $L$ and $R$. In $\Bbb R$ the number $\sqrt 2$ fills that gap. – Brian M. Scott Jan 22 '15 at 2:25
• That gap can only be filled by a line segment right? How can a point fill that gap. – novice Jan 22 '15 at 2:34