Talk:Dedekind cut

message: 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)
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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 Wikipedia:Peer review.) Best regards, <TT>F</TT>rank  <TT>W ~@) R</TT> 02:39, 8 May 2004 (UTC)
thread_title: Early comments

message: * 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)
replies: text: (William M. Connolley 21:40, 9 Jul 2004 (UTC)
replies: text: 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)
thread_title: Suggestion

message: (William M. Connolley 21:40, 9 Jul 2004 (UTC)
replies: text: (William M. Connolley 22:34, 9 Jul 2004 (UTC)

text: 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)
thread_title:  Dodgy (would "imprecise" be more NPOV?) language 

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

text: 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)
thread_title:  Definition 

message: From the article:
:"[...] is a Dedekind cut we could call ( −∞, 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)
replies: text: There is no B for which {(−∞,a],B} is a Dedekind cut - the first set cannot have a greatest element. --Fibonacci 03:06, 10 December 2005 (UTC)
thread_title:  Ordering Dedekind cuts 

message: 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)
thread_title:  axiom of choice 

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title: [[User:Fredrik]]'s Deletion

message: 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)
replies: text: I wonder why User:SCZenz writes: "… define (or 'create' if you will) … ." 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, "… we create a new, an irrational number … ." Beman translated this from the German " … erschaffen wir eine neue, eine irrationale Zahl … ." 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)
thread_title:  Second paragraph POV (heh) 

message: 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)
replies: text: 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)
thread_title:  Dedekind-MacNeille completion 

message: ...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)
thread_title:  Dedekind completions in posets 

message: 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)
thread_title:  Anachronism 

message: 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:
* 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)
— Smjg (talk) 14:21, 13 August 2011 (UTC)
replies: text: ℚ is the standard symbol for the set of all rational numbers.  See Set (mathematics)#Special sets. — Smjg (talk) 13:08, 15 November 2011 (UTC)
thread_title:  Can the cut be rational? 

message: 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)
thread_title:  Relation to Euclidean Geometry needs references/clarification/expansion? 

message: 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)
replies: text: 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)
thread_title:  Problems with the section on the construction of the reals 

message: 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)
thread_title:  Transcendentals 

message: 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)
thread_title:  How can B have the smallest element if A can't be empty?