Talk:Dedekind cut

Early comments
Hmmm. I don't see this as a stub. There's more to be added certainly, but the basic facts are here. Andrewa 10:30 16 Jul 2003 (UTC)

I've just added some definitions for comparison, arithmetic operations, and evaluations of supremum and infimum. These definitions are not all from the same source (though presumably all "public knowledge", "well-known" or "sufficiently obvious" ); they may not be entirely consistent with each other and/or with the definitions that were stated (and presently remain unaltered) at the beginning of the article. Please comment, and correct as appropriate. (I've put the same request on Peer review.) Best regards, Frank W ~@) R 02:39, 8 May 2004 (UTC)

Suggestion

 * Start off by defining Dedekind cut in the the simplest case: as a dedeckind cut in the rationals.


 * insert a picture


 * Leave generalizations for later in the article. CSTAR 19:55, 9 Jul 2004 (UTC)


 * (William M. Connolley 21:40, 9 Jul 2004 (UTC)) Probably a good idea. Most of us don't knwo what posets are, Charles.
 * Most may not, but they are pretty ordinary objects in math (e.g. they are studied routinely in lower-division discrete math classes), and so cuts in posets aren't extraordinarily "esoteric" (a relative term), so it's hardly out of place to give the full poset presentation here. Revolver 15:37, 15 Dec 2004 (UTC)

Dodgy (would "imprecise" be more NPOV?) language
(William M. Connolley 21:40, 9 Jul 2004 (UTC)) In several places the article seems to take for granted the existence of the reals. For example, it talks about being able "to do arithmetic on cuts, just like you can do on the reals" (I paraphrase). But (as far as I can understand this) that's the wrong way round. The reals *are* dedekind cuts (or some other definition starting from the rationals). Arithmetic on reals is *defined* by the corresponding actions on cuts. Just as arithmetic on rationals is defined by the corresponding operations on sets of pairs.

Not entirely. One can introduce the reals in a number of ways (e.g. Cauchy sequences of rationals). The main result is that a Dedekind cut of the reals must be at some real x: so, (-&infin;,x) and [x,+&infin;) or the other one with x in the left interval. So, there is no 'gap' remaining to isolate within the reals.

I'd be quite happy to move all the general order theory down to the end; and get the Dedekind-story at the top.

Charles Matthews 22:12, 9 Jul 2004 (UTC)


 * (William M. Connolley 22:34, 9 Jul 2004 (UTC)) Err, far be it from me to disagree, but I shall. You too are speaking as if the reals have some independent existence. No matter how you introduce the reals (Cauchy or whatever) their arithmetic properties are defined by the properties of the rationals in the definition; which of course end up being the properties you expect or things would have gone badly wrong...

Actually that's not the approach always adopted. One can just write down axioms for the real numbers qua complete field. People hope there is essentially one model; there might be no model (sceptical, finitist approach), or too many models to handle (let in the logicians with the continuum hypothesis etc.). It doesn't matter so much in practice - one just uses the axioms one has responsibly, and it would be 'bad luck' if there were no model at all. This, IIRC, is how Dieudonne's Foundations of Modern Analysis starts out. Charles Matthews 09:38, 10 Jul 2004 (UTC)


 * WMC, all of the ways Charles mentioned above to construct the reals do not assume any properties of existence of the reals. When he says, "the main result is that a Dedekind cut of the reals must be at some real x", he is speaking about Dedekind cuts of reals. Remember, the reals form a poset, too! So, once you have constructed them, it's perfectly consistent to form Dedekind cuts of reals, and say things about them. Revolver 15:37, 15 Dec 2004 (UTC)

Proposal for simplification
(William M. Connolley 20:05, 11 Jul 2004 (UTC)) I'd like to propose switching the definition to that used at http://planetmath.org/encyclopedia/DedekindCuts.html. This is essentailly that instead of calling (A,B) a cut, we call B the cut. This has the virtue of removing a whole pile of stuff relating to "the other half" which is never interesting. It allows ordering by simple subsetting, which seems a huge virtue. And its logically identical (at least in the context of the reals - CM may know otherwise for more esoteric bases). I think we should also point out somewhere that one can *define* the reals as the cuts (whilst also, of course, noting that there are other ways).


 * Arguably the business of Dedekind cuts of the rationals giving one the reals should be at construction of real numbers, which currently has no detail to speak of. Before changing the definition, and remembering that WP is not supposed to innovate on terminology, one should hesitate a moment, anyway. The point of the definition in terms of order theory is that it is a monotone mapping to the two-element ordered set {0,1} with 0 < 1. Well, that is no more and no less complicated than propositions taking values in the Booleans {yes, no}; and one can equally replace those by the set on which the proposition takes value 'no'. In practice one needs to be fluent in saying things either way. So we should have all that on this page, in some form; as I said, I think the axiomatics can be elsewhere (cf. the approach on the real number page, in fact). Charles Matthews 07:53, 12 Jul 2004 (UTC)

OK, I have now moved the detailed definitions for operations on reals to construction of real numbers. I think that will make it easier to shuffle this page into some better logical order. Charles Matthews 14:59, 12 Jul 2004 (UTC)

William M. Connolley: I'd like to propose switching the definition
 * If the phrase "Dedekind cut" applies to distinct notions or versions (e.g. "original", "simplified", and/or "instead of calling (A,B) a cut, we call B the cut", etc. etc.) then I'd suggest that they should all be covered (eventually);
 * perhaps not necessarily on different pages, but at least in distinct sections, for separate reference;
 * and surely not omitting version(s) "due to Dedekind" and/or "particular representative of a Schnitt (German)"
 * Frank W ~@) R 03:47, 14 Jul 2004 (UTC)

Charles Matthews: I have now moved the detailed definitions for operations on reals to construction of real numbers
 * On one hand that's a good move because that's where the detailed definitions for arithmetic, set, and order operations on Dedekind cuts would seem most relevant. (But why did the move mangle the careful formatting? ...)
 * On the other hand however, while (any and all versions of) "Dedekind cut" can be defined without requiring the detailed definitions for operations, it may well be that different such definitions go with distinct versions of "Dedekind cut". (See also my note from 02:39, 8 May 2004 (UTC), above). Referencing between construction of real numbers and Dedekind cut should have to be correspondingly precise, and edits on either page (ideally) synchronized.
 * Frank W ~@) R 03:47, 14 Jul 2004 (UTC)

The formatting - I just made it standard for this site - it was extraordinarily hard to edit as it was. I don't think we need various different versions of 'Dedekind cut', unless and until they fulfil a clear need. Charles Matthews 07:32, 14 Jul 2004 (UTC)


 * Again, WMC, I think you're confusing the order theory def with the particular case of Q. One of the reasons the case with Q is so important is that one can transfer and expand the field operations on Q into Dedekind cuts of rationals. In other words, the "distinct version of Dedekind cut" you're referring to about the rationals is able to happen not just because we have the poset Q, it's the poset and field Q, altogether, the ordered field. That is precisely why Charles thinks the drawn-out details of this particular construction from Q to R belong at a separate page -- because the field operations are so essential, and field operations aren't part of the theory of general Dedekind cuts in posets. Revolver 15:37, 15 Dec 2004 (UTC)

Definition
There is a problem with the definition: "In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rational numbers into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. Dedekind cuts are one method of construction of the real numbers."

It states "non-empty 'parts' A and B" and then goes on to talk about these "parts" as if they are sets, which indeed they are. I will correct this so that it states "sets" and not parts, because only sets have elements by definition. 197.79.9.234 (talk) 08:27, 19 May 2015 (UTC)


 * Yes, thanks IP. It was implicit in the wikilinking of "partition", but it's obviously better to use "sets". I've wikilinked "sets". "Subsets" perhaps? c1cada (talk) 12:03, 19 May 2015 (UTC)


 * There are still more errors:

More generally, 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.

If you are responsible for this article, you might want to add that the lower cut is a Cauchy sequence. 197.79.0.247 (talk) 14:31, 21 May 2015 (UTC)


 * Not responsible for the article . My only contribution here was the last paragraph of the lede. I'm not aware that the definition is seriously in error. I was brought up on Whittaker and Watson myself and the treatment here looks much the same as theirs. I encourage you simply to correct any errors where you see them. As for Cauchy sequence stuff that's probably best dealt with in Construction of the real numbers where there is a section on Dedekind cuts stressing your remarks above. What I would like to see in this article is a discussion of the historical context of Dedekind's construction, including reservations expressed at the time about its perceived geometrical bias, and then perhaps a really careful description of the construction, following perhaps Hardy's celebrated account. Incidentally, as far as getting your concerns looked at by editors here, it would be best to start a new section at the end of the Talk page. Otherwise they are quite likely not to be noticed. c1cada (talk) 15:59, 21 May 2015 (UTC)


 * I am not interested. In my opinion, neither D. Cuts nor Cauchy sequences define the "real" numbers because irrational numbers don't exist. http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409 197.79.0.85 (talk) 01:45, 23 May 2015 (UTC)
 * Well, yes. You might like to read Hardy's classic account in "A Course of Pure Mathematics" ( first year study in his time at Cambridge around 1910 and still very readable) where he discusses exactly that point of view. c1cada (talk) 03:10, 23 May 2015 (UTC)


 * Have read it and it's nonsense. 197.79.3.185 (talk) 05:26, 14 June 2015 (UTC)

Now I'm getting confused on the definition. Taken at face value, it is true, the standard way of defining addition is not closed. This is a problem. It's okay if we restrict the lower subset not to have an upper bound, e.g. This is how I usually see the definition in analysis books. Now, I looked up in an order theory book, and they disagreed with the definition given here. Basically, their definition was in terms of Galois connections, where the Galois mappings are given (in words) by "take the set of all things bigger than or equal to all things in A" (denote A-up) and similarly for smaller than. This is a Galois connection on the power set, and a Dedekind cut is defined to be a "Galois subset", i.e. a subset so that after you apply each mapping one by the other, you get the same thing. Example: A = (&minus;&infin;, 0], then A-up = [0, &infin;) and (A-up)-down = (&minus;&infin;, 0] = A. NOTICE: if we change the order of the mappings, we get intervals going to &infin; rather than &minus;&infin;, but they are still closed. If we think of "identifying" these intervals [a, &infin;) with their complements, we get what we think of as open intervals down to &minus;&infin;, but these complements aren't Galois! (in either order!) I think this is where the confusion about the definition lies. You have to fix the order of the mappings, it seems. By using the "closed-down", "closed-up" mappings, you get both defintions (corr. to different orders of mappings) thrown together instead of just one. If you restrict to just one order of the Galois mappings, then suddenly addition is closed. This leads me to think maybe we have the definition of a Dedekind cut slightly wrong altogether. Revolver 16:28, 15 Dec 2004 (UTC)


 * It's not okay to restrict the lower subset not to have an upper bound, because that is required in the definition. What you see and what you understand from "analysis books" is open to varying interpretations. 197.79.0.247 (talk) 14:33, 21 May 2015 (UTC)

Why is there no mention of the upper set B in the definition? Dutugamunu (talk) 14:24, 14 May 2019 (UTC)

The above suggestion "you might want to add that the lower cut is a Cauchy sequence" is completely wrong. It is most definitely not a Cauchy sequence, especially because it is not a sequence. — Preceding unsigned comment added by 2601:200:C082:2EA0:7D50:88F7:2D96:18EC (talk) 19:33, 20 January 2024 (UTC)

Ordering Dedekind cuts
From the article:
 * "[...] is a Dedekind cut we could call ( &#8722;&#8734;, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S."

If we do identify (-inf, a) with a, then we are missing all the cuts (-inf, a] -- so the set of Dedekind cuts is bigger than S even if S "enjoys" the l-u-b property. --SirJective 84.151.225.29 15:25, 20 May 2005 (UTC)


 * There is no B for which {(&minus;&infin;,a],B} is a Dedekind cut - the first set cannot have a greatest element. --Fibonacci 03:06, 10 December 2005 (UTC)

Definition changed
I've changed the definition here and at Construction of real numbers to incorporate the condition that the low/left set contain no greatest element. This is a requirement made by Walter Rudin, MathWorld, PlanetMath, any other source I have ever seen, and most importantly, it is needed to make the construction of R to actually work. Otherwise, as that article pointed out itself, the cuts would not be closed under addition. -- Jao 13:34, Jun 24, 2005 (UTC)

Embedding of a set in the set of its Dedekind cuts

 * "If a is a member of S then the set


 * $$\{ \{ x\in S: x < a \}, \{ x\in S: x \ge a \} \}$$


 * is a Dedekind cut we could call ( &minus;&infin;, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S."

Ehm... not quite. What if the left set of that "cut" is empty (so that (A,B) is not a partition of S), or has a greatest element? It wouldn't be a Dedekind cut, would it? The simplest example I can think of is S = {a,b}, with a<b - the set of its Dedekind cuts is empty, so there can be absolutely no embedding of S in it. --Fibonacci 03:13, 10 December 2005 (UTC)

Update: I have deleted the following paragraph from the article, for the reasons stated above:


 * If a is a member of S then the set


 * $$\{ \{ x\in S: x < a \}, \{ x\in S: x \ge a \} \}$$


 * is a Dedekind cut we could call ( &minus;&infin;, a ); by identifying a with it, the linearly ordered set S is embedded in the set of all Dedekind cuts of S. If the linearly ordered set S does not enjoy the least-upper-bound property, then the set of Dedekind cuts will be strictly bigger than S; conversely, if S has the least-upper-bound property, the set of its Dedekind cuts is order isomorphic to S, by identifying each cut (A,B) with the supremum of A.

axiom of choice
I didn't explain the axiom of choice thing last time, sorry. It was worded poorly. Dedekind was able to prove the completeness of the real numbers without using the axiom of choice by using cuts. This is the significance of Dedekind cuts in Mathematics. The construction using Cauchy sequences requires the proof to use the AoC. Benandorsqueaks 03:33, 17 December 2005 (UTC)

User:Fredrik's Deletion
I am really surprised at User:Fredrik's deletion. Here we had a clear, concise explanation of a Dedekind Cut, verified by a key quotation by Richard Dedekind himself. You couldn't ask for a more apropos account of the meaning of Dedekind's cut. The quotation was the one place in Dedekind's essay that the famous mathematician clearly and distinctly showed, in a few words, how the cut resolved the continuum problem by placing a rational or irrational number at every point on the number line. User:Fredrik doesn't see this. The readers of the article are the losers here. Let the article remain as User:Fredrik wants it, without an enlightening and concise explanation which was accompanied by a very pertinent and important quotation by the renowned mathematician. I reproduce it below for the benefit of curious readers who may judge for themselves the value of the paragraph and quotation:

The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves. Wherever a cut occurs and it is not on a real rational number, an irrational number (which is also a real number) is created by the mathematician. Through the use of this device, there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity. Lestrade 00:38, 27 August 2006 (UTC)Lestrade


 * There is no room in the Dedekind Cut article for Dedekind's quote, which precisely explains the cut. But, on Wikipedia, there is plenty of room, for example, to include such things as a reference in The Simpsons in a Henry Kissinger article. Lestrade 00:04, 1 October 2006 (UTC)Lestrade


 * Personally, I don't see the point of saying that the irrational numbers are "created" by mathematicians. For that matter, all numbers are "created" by mathematicians.  There's nothing more concrete about the number 1 than there is about the number pi.  The whole introduction seems to be rather lacking in understanding of modern math.  Gandalf (talk) 17:43, 24 November 2008 (UTC)

This is a typical lack of understanding of Dedekind's own actual words. What does being "concrete" have to do with positing the existence of a number? Who cares? Ours is an ignorant time and we are proud of our ignorance and our youthful superiority to the old thinkers. Would Gandalf favor us with a clear explanation of his opinion that "The whole introduction seems to be rather lacking in understanding of modern math" or is it so obvious that we should know without being told?Lestrade (talk) 20:26, 24 November 2008 (UTC)Lestrade


 * I beg your pardon. The comment was in haste.  My complaint is mainly against the second paragraph, not the entire introduction.  The sentence "The Dedekind cut resolves the contradiction between the continuous nature of the number line continuum and the discrete nature of the numbers themselves." does not make sense mathematically.  What is the difference between the continuum of numbers and the "numbers themselves"?  The set of numbers being discussed is ambiguous.  If the sentence is referring to the set of rational numbers, it is incorrect because that is not a discrete set as is claimed.  If the set being discussed is the set of integers (which is a discrete set) then the formulation of the reals via Dedekind cuts is incorrect.  I think the paragraph should be deleted due to its lack of mathematical foundation.  Gandalf (talk) 15:26, 26 November 2008 (UTC)

Second paragraph POV (heh)
The second paragraph seems to me to imply the use of Dedkind cuts is the unique way to define (or "create" if you will) the irrational numbers, thus neglecting the method of cauchy sequences. We ought to be clear that to in modern mathematics this is only one possible approach. -- SCZenz 15:16, 8 March 2007 (UTC)


 * I wonder why User:SCZenz writes: "&hellip; define (or 'create' if you will) &hellip; ." Chapter 4 of Dedekind's Continuity and Irrational Numbers is entitled, in German, "Schöpfung der irrationalen Zahlen." As translated by W. W. Beman, in the English translation published by Open Court and Dover, this definitely means, "Creation of Irrational Numbers." In this chapter, Dedekind wrote, "&hellip; we create a new, an irrational number &hellip; ." Beman translated this from the German " &hellip; erschaffen wir eine neue, eine irrationale Zahl &hellip; ." There is no mention of "define." There is only Dedekind's use of the words meaning creation and creating. We can't arbitrarily change Dedekind's words and freely substitute "define" for "create" whenever we feel like doing so. By adding the phrase, "if you will," User:SCZenz seems to imply that his word "define" is the preferred word, but, the word "create" will be tolerated if necessary. This is not the case. We have Dedekind's own exact words as a reference. He, the original author, used the words meaning creation and creating. If he wanted to use the word meaning "define" he would have written "definieren."Lestrade 18:38, 8 March 2007 (UTC)Lestrade

Dedekind-MacNeille completion
Why does the Dedekind-MacNeille completion consist of the sets (Au)l = A and not of the sets (Al)u = A? Or is this the same? --NeoUrfahraner (talk) 15:35, 23 April 2008 (UTC)
 * In the meanwhile I fonund the answer myself: the sets (Au)l = A are exactly the sets A=Bl, and the sets (Bl)u = B are exactly the sets B=Au, the map $$A\mapsto A^u$$ is order reversing and bijective, its inverse is $$B\mapsto B^l$$, so both give essentially the same completion. --NeoUrfahraner (talk) 08:28, 25 April 2008 (UTC)

Dedekind completions in posets
...One completion of S is the set of its downwardly closed subsets (also called order ideals)...  Downward closed sets are the lower sets; order ideals are directed lower sets; they are not the same. 81.210.248.125 (talk) 05:19, 14 May 2008 (UTC)chrystomath

Anachronism
I have removed the following snippet:
 * The original and most important cases are Dedekind cuts for rational numbers and real numbers. Dedekind used cuts to prove the completeness of the reals without using the axiom of choice (proving the existence of a complete ordered field to be independent of said axiom). 

This appears to be an anachronism; Zermelo didn't even formulate AC until 1904, whereas Essays on the Theory of Numbers was published in '01. There may be a valid underlying point here (I suppose if you wanted to prove completeness in terms of Cauchy sequences, you'd be tempted to use AC to pick a representative from each equivalence class of sequences), but it's not really explained. --Trovatore (talk) 06:18, 4 October 2008 (UTC)

sqrt(2) example
I don't understand the example for $$\sqrt(2)$$. It argues that (A,B) is a cut, because if $$x > 0 \wedge x \notin B$$ then there exist $$y > x \wedge y \notin B$$, but how does this prove that A is closed downward, B closed upwards or that A contains no greatest element (like the cut is defined in the introduction)?

Also $$-3 < -1 \in A$$, but $$-3 \notin A$$ and therefore A is not closed downwards, and it shouldn't be a cut. 78.49.23.196 (talk) 12:13, 28 May 2010 (UTC)
 * I have problems with that part, too. Re your last question: similarly to a mistake I just made at Talk:Maximum spacing estimation, you may not have noticed that it says "positive", so your example using negative numbers doesn't apply.  However, the couple of sentences after "neither claim is immediate" need work.  Similar sentences appear in the construction of the real numbers article. Both versions seem not quite right to me.  For one thing, you have to define multiplication of cuts before you can prove things involving multiplication. Both articles have the sentence "However, neither claim is immediate", but seem to be talking about different claims!  I tried to find a ref that would help via Google Books.  This might help a bit:  Elements of the theory of functions By Konrad Knopp. It doesn't do the proof in detail but then neither does this article. ☺ Coppertwig (talk) 20:52, 10 October 2010 (UTC)


 * You, guys are lucky, because it seems you understand something. I don't. What't the meaning of this? Any fool can make an open-bordered set by not including the surface (in Dedekind's case a zero-dimensional surface constituted by the "cut" value), but so what? Wasn't that known before Dedekind? And what's the difference between making a rational or an irrational cut? Anyways the two sets are open towards the cutting value. Rursus dixit. ( m bork3 !) 14:01, 14 October 2010 (UTC)
 * Set A is open: it contains no greatest element. Set B may be open or closed: if it's at a rational number, then it contains a least element, which is that number.  If it's a cut of the rationals and is at an irrational number, then it contains no least element (among the rationals) and is open, like A.
 * OK, any fool can divide a set into two sets. The brilliant thing here is to define the cut as a significant mathematical object and start building a new number system out of it – that is, to start from the rationals and create a model for the set of real numbers. Defining how to multiply cuts is part of that process. ☺ Coppertwig (talk) 16:49, 28 November 2010 (UTC)

Can the cut be rational?
I'm not sure whether the article contradicts itself or is just a bit confusing.

The first paragraph states first that a Dedekind cut is a partition of the rationals into two sets A and B. According to the definition of a partition, this means every rational number is a member of exactly one of A and B. However, it goes on to say "The cut itself is in neither set", thereby contradicting the prior statement in the case where the cut is a rational number.

Of course, if we ignore "The cut itself is in neither set", then we have a clear definition. Equivalently for some $$x \in \mathbb{R}$$, we have $$A = \mathbb{Q} \cap (-\infty, x)$$ and $$B = \mathbb{Q} \cap [x, \infty)$$. Consequently, B has a least element iff x is rational.

I suppose the question is whether "The cut itself is in neither set" means that: — Smjg (talk) 14:21, 13 August 2011 (UTC)
 * the cut is just the notional boundary between the values in A and the values in B, and it merely corresponds to a rational or irrational real number and is not the number itself
 * any cut on a rational number isn't a Dedekind cut (though I'm not sure that the least upper bound property can be satisfied that way)

I agree with your first two statements above. The definition is different from that in "Advanced Calculus" by Angus Taylor. I think the definition is wrong; but I don't know what you mean by Q in your definition of A and B. RHB100 (talk) 21:35, 7 November 2011 (UTC)
 * ℚ is the standard symbol for the set of all rational numbers. See Set (mathematics). — Smjg (talk) 13:08, 15 November 2011 (UTC)

I'm just looking at this and who knows if anyone cares any more, but I'd say that we could avoid the word rational for just a little bit and talk about fractions which are (of course) expressions  $$ a/b $$ with $$a,b$$ relatively prime integers and $$b$$ positive. A (fraction) cut is an ordered pair (A,B) of (infinite) sets of fractions (meeting certain conditions) so it is not a fraction itself. Hence the cut is not in either set since it is not a fraction. Now if A has a largest member $$r=a/b$$ (which is a fraction since all the members are fractions) then that one fraction $$r$$ tells us everything there is to know about the cut (A,B). Now it gets confusing because we define addition and multiplication of cuts. If we call the cuts with a greatest element $$r$$ rationals then these new rationals work just like the fractions and that cut mentioned "is" the rational $$r$$. But the cut is not really the same thing as the largest fraction in the left-hand member of the cut. I hope that helps. Gentlemath (talk) 03:50, 25 April 2013 (UTC)

Relation to Euclidean Geometry needs references/clarification/expansion?
I wonder if the following should provided with references, and the discussion expanded just slightly (although probably in a later section, not in the introduction).

"Dedekind used the German word Schnitt (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind's construction of the number line ensures that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other."

This suggests a motivation for defining the real numbers, coming from a desire for a two-dimensional space to have the properties one might want, e.g. to satisfy Euclid's postulates. It does seem plausible to me that this was one of Dedekind's motivations, but would like to see a fairly specific reference to establish that this was [an important part of] Dedekind's motivation for the "Schnitt". (I am not familiar with the primary sources so I'm not the person to do this.) The relation of "the number line" (why not say the real numbers) to Euclidean geometry might be made more explicit here, or at least a reference to a rigorous treatment of the proposition claimed ("the construction ensures...") would be very desirable. If that takes up too much space in the introduction, perhaps putting it in a later section would be appropriate. I don't think a detailed treatment is warranted; just something a little bit clearer, e.g. some reference to Dedekind's construction of the reals allowing R^2 to be a model of Euclid's postulates in the two-dimensional case, or something along these lines. If anyone thinks they can do this concisely, and with awareness of the extent to which it reflects Dedekind's motivation, I for one think it would be a valuable improvement to the article.

MorphismOfDoom (talk) 17:45, 18 December 2012 (UTC)

If it seems consistent with the mathematical facts, I think the following wording might improve the passage from the article quoted at the top of my previous comment:

"Dedekind used the German word Schnitt (cut) in a visual sense rooted in Euclidean geometry. When two straight lines cross, one is said to cut the other. Dedekind's construction of the real numbers allows us to use R^2 (the Cartesian product of R with itself) as a model of two-dimensional Euclidean space. In particular, and unlike the Cartesian product of the rationals with themselves, it satisfies Euclid's postulate that two crossing lines always have one point in common because each of them defines a Dedekind cut on the other."

Again I'm not that aware of the history... in some sense the fact that the Cartesian product of the rationals with themselves does not satisfy Euclid's postulates has been known at least since the Greeks, so Descartes was surely aware of it... so it may not be that Dedekind really solved a problem with the Cartesian plane... on the other hand, I guess the point may be that the construction preceded or came simultaneously with the first rigorous axiomatization of the reals? One could argue that, if the rationals were correctly (even if somewhat informally) axiomatized, then Dedekind's construction provides, in some sense, the first rigorous mathematical theory of the real numbers, and therefore of Euclidean two-space, so that it really does solve the problem in a way the Greeks and Descartes, even though they recognized the desirability of talking about irrational numbers, and even did so to a great extent. Anyone care to comment? This seems close to the essence of Dedekind's achievement, so it is central to the article. — Preceding unsigned comment added by MorphismOfDoom (talk • contribs) 17:59, 18 December 2012 (UTC)

Problems with the section on the construction of the reals
The example of the cut representing the square root of 2 is fine. But there's nothing unclear about it actually being a cut. Also, any discussion of it representing the square root of 2 is inappropriate unless some information is added about how the algebraic structure of the rationals is extended to the set of cuts. Otherwise, the discussion trying to show that this really is the square root of 2 doesn't really make any sense. — Preceding unsigned comment added by 73.147.18.72 (talk) 22:36, 13 June 2015 (UTC)
 * The information about the algebraic structure extension should be added anyway, regardless of this issue — the section on constructing the reals is incomplete without it. —David Eppstein (talk) 22:48, 13 June 2015 (UTC)

Transcendentals
The prototypical examples of dedekind cuts are always rational numbers and simple square roots like sqrt(2). I can easily see how dedekind cuts give rise to algebraic numbers using some kind of integer polynomial in the set construction bounding our rationals, such as saying x^2 < 2 as per the case of sqrt(2). I also understand the argument that the "reals" constructed with dedekind cuts have the least upper bound property. What I fail to see, however, is why the "reals" contain the transcendentals. Why cannot the reals complete with the least upper bound property contain only algebraics? That is to say, I dont see how dedekind cuts give rise to transcendentals. I dont see how to express the bounds on the rationals in the lower cut to give rise to, say, pi, for example. It seems to me that constructing the transcendentals with dedekind cuts requires reference to other transcendentals... or best case scenario, non-rational algebraics. I feel when it comes to transcendentals there is a circular reasoning problem going on here. Can someone show me an example of a non-trivial transcendental being defined with a dedekind cut? Or even for that matter a particular algebraic root of a fifth degree polynomial? I have yet to see such a thing in my research. Alternatively, I *can* construct transcendentals if I bound the rationals in the lower cut with an infinite polynomial or infinite series, but then at that point we'd might as well employ the Cauchy construction of the reals instead. 50.35.103.217 (talk) 15:19, 20 September 2018 (UTC)

How can B have the smallest element if A can't be empty?
Since, all elements in set A are smaller than all elements of B and A cant be empty, how could Bhave the smallest element of the rational numbers?

--TheFibonacciEffect (talk) 09:59, 23 April 2020 (UTC)

Addition and multiplication of Dedekind cuts
It would be good to include in this article a discussion of how addition and multiplication of Dedekind cuts are defined. Currently these operations are left unmentioned.

(Addition is pretty easy, but multiplication of cuts can be a bit awkward.)