When defining $\mathbb{R}$ from the rationals they use Dedekind cuts and there are basically two definitions:

Version 1:

A Dedekind cut is any set $A\subseteq \mathbb{Q}$ with the following three properties:

(I) $A$ is not empty, and $A\neq \mathbb{Q}$.

(II) If $p\in A, q\in \mathbb{Q}$, and $q< p$, then $q\in A$.

(III) If $p\in A$, then $p<r$ for some $r\in A$

Version 2:

A Dedekind cut is an ordered pair of subsets of $\mathbb{Q}$, $(A,B)$, satisfying:

(I) $\{A,B\}$ is a partition of $\mathbb{Q}$

(II) $p<q$ for all $p\in A$, $q\in B$

(III) $A$ does not have a greatest element

The main difference between them is that the first one names the set $A$ as a cut while in the second a cut is the pair $(A,B)$.

Now, in Geometry (From Greenberg's book) there is what I think is the analogous definition for the second version:

Dedekind cut in Geometry: A Dedekind cut of a line $l$ is a partition of $l$ into two non-empty subsets $A$ and $B$ such that no point of either subset is between two points of the other.

Question: I haven't seen any place that uses the first version of the Dedekind cuts for Geometry, and I wonder why. Is there a reason? What is the analogous (standard?) definition of Dedekid cut in Geometry for the first version?.


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    $\begingroup$ The second definition doesn't really work, because according to it there are two different cuts that intuitively represent each rational, depending on whether you put that rational in $A$ or in $B$. $\endgroup$ – Henning Makholm Nov 6 '13 at 13:31
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    $\begingroup$ Well, you can use betweenness to induce a total order on the line, and just carbon copy the definition. Also, to correct the mistake @HenningMakholm hints to, people usually assume $A$ to be open (that is, it has no maximal element. $\endgroup$ – Arthur Nov 6 '13 at 13:33
  • $\begingroup$ @Arthur Thanks. I just did the correction. I think your idea of inducing a total order works. It would be nice though if it could be done just like in Greenberg's definition, without appealing to extra definitions. $\endgroup$ – Daniela Diaz Nov 6 '13 at 13:52
  • $\begingroup$ @DanielaDiaz: But now your geometric definition is not analogous to your Version 2, because the mistake was not fixed there. (And without having a preferred direction on the line in question, it seems to be hard do fix elegantly). (Also, where did the requirement that $A$ and $B$ are nonempty go?) $\endgroup$ – Henning Makholm Nov 6 '13 at 13:55
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    $\begingroup$ @DanielaDiaz: Geometically, "unbounded" can be phrased as "not contained inside any circle". $\endgroup$ – Henning Makholm Nov 6 '13 at 16:25

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