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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.
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reference to establish that this was
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.
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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.
2533:"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."
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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.
446:. 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).
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848:(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.
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267:. 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
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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,"
718:. 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
2510:"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."
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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."
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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
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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
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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
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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
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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
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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
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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;
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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
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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
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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.
840:" (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 = (−∞, 0], then A-up = = A. NOTICE:
920:" 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."
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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:
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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.
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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
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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.
339:*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.
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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.
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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.
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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.
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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.
1996:, 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
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agreed 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
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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.
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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?
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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.
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There is no room in the Dedekind Cut article for Dedekind's quote, which precisely explains the cut. But, on Knowledge, there is plenty of room, for example, to include such things as a reference in
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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?
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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
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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
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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
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number, which we regard as completely defined by this cut ... . From now on, therefore, to every definite cut there corresponds a definite rational or irrational number ....
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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
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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. —
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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.
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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.
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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. --
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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
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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:
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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).
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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.
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1333:'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
2099:. 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.
1907:, 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)?
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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
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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
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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?
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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
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I am not interested. In my opinion, neither D. Cuts nor Cauchy sequences define the "real" numbers because irrational numbers don't exist.
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there is considered to be a real number, either rational or irrational, at every point on the number line continuum, with no discontinuity
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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.
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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)
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The above suggestion "you might want to add that the lower cut is a Cauchy sequence" is completely wrong. It is most definitely
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what's the difference between making a rational or an irrational cut? Anyways the two sets are open towards the cutting value.
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421:. Remember, the reals form a poset, too! So, once you have constructed them, it's perfectly consistent to form Dedekind cuts
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assume any properties of existence of the reals. When he says, "the main result is that a Dedekind cut of the reals must be
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Elements of the theory of functions By Konrad Knopp. It doesn't do the proof in detail but then neither does this article.
2488:. 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.
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Of course, if we ignore "The cut itself is in neither set", then we have a clear definition. Equivalently for some
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I'd be quite happy to move all the general order theory down to the end; and get the Dedekind-story at the top.
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2091:. According to the definition of a partition, this means every rational number is a member of exactly one of
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1110:), or has a greatest element? It wouldn't be a Dedekind cut, would it? The simplest example I can think of is
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if we change the order of the mappings, we get intervals going to ∞ rather than −∞, but they are still closed
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If the phrase "Dedekind cut" applies to distinct notions or versions (e.g. "original", "simplified", and/or "
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If you are responsible for this article, you might want to add that the lower cut is a Cauchy sequence.
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versions of "Dedekind cut". (See also my note from 02:39, 8 May 2004 (UTC), above). Referencing between
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perhaps not necessarily on different pages, but at least in distinct sections, for separate reference;
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Actually that's not the approach always adopted. One can just write down axioms for the real numbers
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Hmmm. I don't see this as a stub. There's more to be added certainly, but the basic facts are here.
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I have problems with that part, too. Re your last question: similarly to a mistake I just made at
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Start off by defining Dedekind cut in the the simplest case: as a dedeckind cut in the rationals.
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for that content in the latter page, and it must not be deleted as long as the latter page exists.
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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then these new rationals work just like the fractions and that cut mentioned "is" the rational
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which of course end up being the properties you expect or things would have gone badly wrong...
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21:40, 9 Jul 2004 (UTC)) Probably a good idea. Most of us don't knwo what posets are, Charles.
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1521:. We ought to be clear that to in modern mathematics this is only one possible approach. --
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Whenever, then, we have to do with a cut produced by no rational number, we create a new, an
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doesn't see this. The readers of the article are the losers here. Let the article remain as
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should have to be correspondingly precise, and edits on either page (ideally) synchronized.
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I've just added some definitions for comparison, arithmetic operations, and evaluations of
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More generally, a Dedekind cut is a partition of a totally ordered set into two non-empty
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Arguably the business of Dedekind cuts of the rationals giving one the reals should be at
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This appears to be an anachronism; Zermelo didn't even formulate AC until 1904, whereas
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Downward closed sets are the lower sets; order ideals are directed lower sets; they are
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I have deleted the following paragraph from the article, for the reasons stated above:
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20:05, 11 Jul 2004 (UTC)) I'd like to propose switching the definition to that used at
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Again, WMC, I think you're confusing the order theory def with the particular case of
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http://www.spacetimeandtheuniverse.com/math/4507-0-999-equal-one-317.html#post21409
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in the left interval. So, there is no 'gap' remaining to isolate within the reals.
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would be 'bad luck' if there were no model at all. This, IIRC, is how Dieudonne's
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I suppose the question is whether "The cut itself is in neither set" means that:
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I'm not sure whether the article contradicts itself or is just a bit confusing.
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nature of the numbers themselves. Wherever a cut occurs and it is not on a real
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is so important is that one can transfer and expand the field operations on
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writes: "… define (or 'create' if you will) … ." Chapter 4 of Dedekind's
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OK, I have now moved the detailed definitions for operations on reals to
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Relation to Euclidean Geometry needs references/clarification/expansion?
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Ehm... not quite. What if the left set of that "cut" is empty (so that (
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945:} is a Dedekind cut - the first set cannot have a greatest element. --
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property, then the set of Dedekind cuts will be strictly bigger than
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I have now moved the detailed definitions for operations on reals to
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and therefore A is not closed downwards, and it shouldn't be a cut.
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has the least-upper-bound property, the set of its Dedekind cuts is
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WMC, all of the ways Charles mentioned above to construct the reals
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a rational or irrational real number and is not the number itself
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Why does the Dedekind-MacNeille completion consist of the sets (
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ref that would help via Google Books. This might help a bit:
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Why is there no mention of the upper set B in the definition?
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a Cauchy sequence, especially because it is not a sequence.
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the cut is just the notional boundary between the values in
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The original and most important cases are Dedekind cuts for
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out itself, the cuts would not be closed under addition. --
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Problems with the section on the construction of the reals
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since it is not a fraction. Now if A has a largest member
2079:
The first paragraph states first that a Dedekind cut is a
2628:
How can B have the smallest element if A can't be empty?
1351:
The Dedekind cut resolves the contradiction between the
1237:{\displaystyle \{\{x\in S:x<a\},\{x\in S:x\geq a\}\}}
1065:{\displaystyle \{\{x\in S:x<a\},\{x\in S:x\geq a\}\}}
1600:
In the meanwhile I fonund the answer myself: the sets (
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by the mathematician. Through the use of this device,
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444:
http://planetmath.org/encyclopedia/DedekindCuts.html
148:, a collaborative effort to improve the coverage of
2648:have the smallest element of the rational numbers?
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1698:, so both give essentially the same completion. --
1690:
1657:
1236:
1064:
973:Embedding of a set in the set of its Dedekind cuts
1665:is order reversing and bijective, its inverse is
495:instead of calling (A,B) a cut, we call B the cut
497:", etc. etc.) then I'd suggest that they should
330:Dodgy (would "imprecise" be more NPOV?) language
302:Leave generalizations for later in the article.
2172:{\displaystyle A=\mathbb {Q} \cap (-\infty ,x)}
1388:
1262:is embedded in the set of all Dedekind cuts of
1090:is embedded in the set of all Dedekind cuts of
2218:{\displaystyle B=\mathbb {Q} \cap [x,\infty )}
8:
2666:Addition and multiplication of Dedekind cuts
1825:. It argues that (A,B) is a cut, because if
1549:"define" he would have written "definieren."
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488:I'd like to propose switching the definition
507:and surely not omitting version(s) "due to
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1454:lacking in understanding of modern math.
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511:" and/or "particular representative of a
417:", he is speaking about Dedekind cuts of
960:I've changed the definition here and at
214:Text and/or other creative content from
96:
66:
1745:I have removed the following snippet:
1250:is a Dedekind cut we could call ( −∞,
1078:is a Dedekind cut we could call ( −∞,
898:2601:200:C082:2EA0:7D50:88F7:2D96:18EC
638:of construction of the real numbers."
7:
2313:which are (of course) expressions
1312:requires the proof to use the AoC.
846:but these complements aren't Galois!
711:
142:This article is within the scope of
1795:I don't understand the example for
593:. One of the reasons the case with
85:It is of interest to the following
2209:
2157:
1758:. Dedekind used cuts to prove the
1258:with it, the linearly ordered set
1086:with it, the linearly ordered set
831:in an order theory book, and they
232:on 21 May 2011. The former page's
14:
2696:Mid-priority mathematics articles
2636:are smaller than all elements of
2123:{\displaystyle x\in \mathbb {R} }
1541:Continuity and Irrational Numbers
1401:Continuity and Irrational Numbers
162:Knowledge:WikiProject Mathematics
1998:construction of the real numbers
805:Have read it and it's nonsense.
720:Construction of the real numbers
712:
165:Template:WikiProject Mathematics
129:
119:
98:
67:
2083:of the rationals into two sets
1994:Talk:Maximum spacing estimation
1773:Essays on the Theory of Numbers
182:This article has been rated as
2623:15:19, 20 September 2018 (UTC)
2367:relatively prime integers and
2292:Set (mathematics)#Special sets
2212:
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2151:
1812:
1804:
1714:Dedekind completions in posets
1691:{\displaystyle B\mapsto B^{l}}
1675:
1658:{\displaystyle A\mapsto A^{u}}
1642:
1266:. If the linearly ordered set
458:of the definition in terms of
395:Foundations of Modern Analysis
1:
2558:17:59, 18 December 2012 (UTC)
2526:17:45, 18 December 2012 (UTC)
2304:13:08, 15 November 2011 (UTC)
2062:16:49, 28 November 2010 (UTC)
1941:{\displaystyle -3<-1\in A}
1818:{\displaystyle {\sqrt {(}}2)}
1565:Dedekind-MacNeille completion
1507:15:26, 26 November 2008 (UTC)
1490:20:26, 24 November 2008 (UTC)
1464:17:43, 24 November 2008 (UTC)
1317:03:33, 17 December 2005 (UTC)
1139:03:13, 10 December 2005 (UTC)
950:03:06, 10 December 2005 (UTC)
425:, and say things about them.
226:Dedekind–MacNeille completion
156:and see a list of open tasks.
2691:C-Class mathematics articles
2686:Old requests for peer review
2389:the cut is not in either set
2285:21:35, 7 November 2011 (UTC)
2040:14:01, 14 October 2010 (UTC)
2017:20:52, 10 October 2010 (UTC)
962:Construction of real numbers
906:19:33, 20 January 2024 (UTC)
722:where there is a section on
710:responsible for the article
675:There are still more errors:
558:construction of real numbers
538:construction of real numbers
477:construction of real numbers
452:construction of real numbers
2632:Since, all elements in set
2269:14:21, 13 August 2011 (UTC)
1786:06:18, 4 October 2008 (UTC)
1436:00:04, 1 October 2006 (UTC)
1413:00:38, 27 August 2006 (UTC)
1290:, by identifying each cut (
605:, it's the poset and field
434:Proposal for simplification
362:,+∞) or the other one with
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2661:09:59, 23 April 2020 (UTC)
2498:03:50, 25 April 2013 (UTC)
1970:{\displaystyle -3\notin A}
1905:x\wedge y\notin B}" /: -->
1864:0\wedge x\notin B}" /: -->
1708:08:28, 25 April 2008 (UTC)
1595:15:35, 23 April 2008 (UTC)
1513:Second paragraph POV (heh)
969:13:34, Jun 24, 2005 (UTC)
2644:cant be empty, how could
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2602:22:48, 13 June 2015 (UTC)
2586:22:36, 13 June 2015 (UTC)
1760:completeness of the reals
1585:? Or is this the same? --
1554:18:38, 8 March 2007 (UTC)
1526:15:16, 8 March 2007 (UTC)
1329:I am really surprised at
852:16:28, 15 Dec 2004 (UTC)
815:05:26, 14 June 2015 (UTC)
585:07:32, 14 Jul 2004 (UTC)
552:such definitions go with
483:14:59, 12 Jul 2004 (UTC)
401:09:38, 10 Jul 2004 (UTC)
224:was copied or moved into
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18:
2229:has a least element iff
2072:Can the cut be rational?
1987:12:13, 28 May 2010 (UTC)
1870:x\wedge y\notin B}": -->
1829:0\wedge x\notin B}": -->
1735:05:19, 14 May 2008 (UTC)
1321:
1106:) is not a partition of
932:15:25, 20 May 2005 (UTC)
882:14:24, 14 May 2019 (UTC)
867:14:33, 21 May 2015 (UTC)
781:03:10, 23 May 2015 (UTC)
766:01:45, 23 May 2015 (UTC)
737:15:59, 21 May 2015 (UTC)
698:14:31, 21 May 2015 (UTC)
667:12:03, 19 May 2015 (UTC)
652:08:27, 19 May 2015 (UTC)
625:15:37, 15 Dec 2004 (UTC)
576:03:47, 14 Jul 2004 (UTC)
530:03:47, 14 Jul 2004 (UTC)
501:be covered (eventually);
471:07:53, 12 Jul 2004 (UTC)
429:15:37, 15 Dec 2004 (UTC)
375:22:12, 9 Jul 2004 (UTC)
323:15:37, 15 Dec 2004 (UTC)
255:10:30 16 Jul 2003 (UTC)
188:project's priority scale
1298:) with the supremum of
306:19:55, 9 Jul 2004 (UTC)
282:02:39, 8 May 2004 (UTC)
145:WikiProject Mathematics
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912:Ordering Dedekind cuts
486:William M. Connolley:
75:This article is rated
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1898:{\displaystyle y: -->
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269:Knowledge:Peer review
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313:William M. Connolley
168:mathematics articles
2360:{\displaystyle a,b}
2334:{\displaystyle a/b}
1270:does not enjoy the
238:provide attribution
2653:TheFibonacciEffect
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1762:without using the
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1399:Richard Dedekind,
1254:); by identifying
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1082:); by identifying
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956:Definition changed
941:for which {(−∞,a],
916:From the article:
609:, altogether, the
534:Charles Matthews:
137:Mathematics portal
81:content assessment
23:Article milestones
2588:
2576:comment added by
2561:
2544:comment added by
2481:{\displaystyle r}
2458:{\displaystyle r}
2438:{\displaystyle r}
2380:{\displaystyle b}
2225:. Consequently,
1866:then there exist
1807:
1374:(which is also a
1372:irrational number
1278:; conversely, if
1272:least-upper-bound
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896:comment added by
271:.) Best regards,
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1616:, and the sets (
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583:Charles Matthews
481:Charles Matthews
469:Charles Matthews
467:page, in fact).
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373:Charles Matthews
297:insert a picture
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1791:sqrt(2) example
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1429:Henry Kissinger
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1314:Benandorsqueaks
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1307:axiom of choice
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1261:
1257:
1253:
1249:
1248:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1201:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1164:
1163:
1162:
1161:
1157:
1153:
1149:
1148:
1147:
1145:
1141:
1140:
1137:
1133:
1129:
1125:
1121:
1117:
1113:
1109:
1105:
1101:
1093:
1089:
1085:
1081:
1077:
1076:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1029:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
992:
991:
990:
989:
985:
981:
977:
976:
972:
970:
968:
963:
955:
951:
948:
944:
940:
936:
935:
934:
933:
930:
929:84.151.225.29
927:
919:
918:
917:
911:
909:
907:
903:
899:
895:
889:
884:
883:
879:
875:
868:
864:
860:
855:
854:
853:
851:
847:
843:
839:
834:
816:
812:
808:
804:
803:
802:
801:
800:
799:
798:
797:
796:
795:
794:
793:
782:
778:
774:
769:
768:
767:
763:
759:
756:
752:
751:
750:
749:
748:
747:
746:
745:
738:
734:
730:
725:
724:Dedekind cuts
721:
709:
706:
705:
704:
703:
702:
701:
700:
699:
695:
691:
686:
684:
674:
673:
672:
671:
668:
664:
660:
656:
655:
654:
653:
649:
645:
639:
632:
630:
624:
620:
616:
612:
611:ordered field
608:
604:
600:
596:
592:
588:
587:
586:
584:
575:
572:
568:
566:
563:
559:
555:
551:
547:
543:
542:
541:
540:
539:
529:
526:
522:
520:
517:
515:
510:
506:
503:
500:
496:
492:
491:
490:
489:
484:
482:
478:
470:
466:
461:
457:
453:
449:
448:
447:
445:
441:
433:
428:
424:
420:
416:
412:
408:
404:
403:
402:
400:
396:
391:
382:
378:
377:
376:
374:
370:
367:
365:
361:
357:
353:
349:
345:
340:
337:
329:
322:
317:
316:
314:
310:
309:
305:
301:
300:
296:
295:
291:
290:
286:
284:
283:
280:
277:
273:
270:
266:
262:
256:
254:
246:
239:
235:
231:
227:
222:
217:
213:
206:
205:
189:
185:
179:
176:
175:
172:
155:
151:
147:
146:
138:
132:
127:
125:
122:
118:
117:
113:
107:
104:
101:
97:
92:
88:
82:
74:
70:
65:
64:
52:
50:
49:
45:
42:
38:
37:
33:
30:
27:
26:
21:
17:
2672:
2669:
2650:
2645:
2641:
2637:
2633:
2631:
2611:
2578:73.147.18.72
2572:— Preceding
2568:
2540:— Preceding
2535:
2532:
2529:
2516:
2512:
2509:
2506:
2465:
2388:
2310:
2308:
2273:
2258:
2249:
2245:
2241:
2235:
2230:
2226:
2101:
2096:
2092:
2088:
2084:
2080:
2078:
2075:
2033:
1979:78.49.23.196
1909:
1794:
1772:
1770:
1756:real numbers
1749:
1744:
1737:chrystomath
1722:
1718:
1717:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1582:
1578:
1574:
1570:
1568:
1540:
1516:
1474:
1425:The Simpsons
1408:
1403:, Section IV
1400:
1391:
1389:
1383:
1379:
1350:
1344:User:Fredrik
1340:User:Fredrik
1334:
1331:User:Fredrik
1328:
1323:User:Fredrik
1310:
1299:
1295:
1291:
1287:
1279:
1275:
1267:
1263:
1259:
1255:
1251:
1158:then the set
1155:
1151:
1143:
1142:
1131:
1127:
1123:
1119:
1115:
1111:
1107:
1103:
1099:
1097:
1091:
1087:
1083:
1079:
986:then the set
983:
979:
959:
942:
938:
937:There is no
923:
915:
892:— Preceding
887:
885:
871:
859:197.79.0.247
845:
841:
837:
832:
829:
807:197.79.3.185
707:
690:197.79.0.247
687:
682:
680:
644:197.79.9.234
640:
636:
628:
618:
614:
610:
606:
602:
598:
594:
590:
579:
562:Dedekind cut
553:
549:
535:
533:
513:
498:
494:
487:
485:
474:
460:order theory
455:
437:
422:
418:
414:
410:
406:
397:starts out.
394:
389:
387:
371:
368:
363:
359:
355:
351:
347:
343:
341:
333:
258:
250:
221:Dedekind cut
216:this version
184:Mid-priority
183:
143:
109:Mid‑priority
87:WikiProjects
46:
1741:Anachronism
1725:the same.
1546:User:SCZenz
1537:User:SCZenz
1376:real number
1357:number line
1325:'s Deletion
758:197.79.0.85
465:real number
159:Mathematics
150:mathematics
106:Mathematics
48:Peer review
2680:Categories
2490:Gentlemath
2130:, we have
2054:Coppertwig
2009:Coppertwig
1632:, the map
1392:irrational
1353:continuous
926:SirJective
874:Dutugamunu
633:Definition
413:some real
354:: so, (-∞,
350:some real
287:Suggestion
2466:rationals
2311:fractions
2081:partition
1778:Trovatore
1492:Lestrade
1431:article.
1415:Lestrade
1360:continuum
1136:Fibonacci
1134:in it. --
947:Fibonacci
550:different
230:this edit
2574:unsigned
2554:contribs
2542:unsigned
2030:dixit. (
1556:Lestrade
1551:Lestrade
1482:Lestrade
1438:Lestrade
1433:Lestrade
1410:Lestrade
1364:discrete
1362:and the
1122:}, with
894:unsigned
850:Revolver
623:Revolver
554:distinct
516:(German)
509:Dedekind
427:Revolver
423:of reals
346:must be
321:Revolver
261:supremum
53:Reviewed
1499:Gandalf
1478:Gandalf
1456:Gandalf
1380:created
1335:apropos
1144:Update:
573:W ~@) R
527:W ~@) R
514:Schnitt
358:) and [
278:W ~@) R
265:infimum
253:Andrewa
234:history
186:on the
77:C-class
31:Process
2277:RHB100
2028:Rursus
1948:, but
1523:SCZenz
773:c1cada
729:c1cada
659:c1cada
571:rank
525:rank
407:do not
276:rank
83:scale.
34:Result
2341:with
1910:Also
1878:: -->
1837:: -->
1427:in a
1378:) is
1370:, an
683:parts
456:point
419:reals
344:reals
304:CSTAR
228:with
2657:talk
2640:and
2619:talk
2598:talk
2582:talk
2550:talk
2522:talk
2494:talk
2300:talk
2296:Smjg
2294:. —
2281:talk
2265:talk
2261:Smjg
2179:and
2095:and
2087:and
2058:talk
2034:bork
2013:talk
1983:talk
1924:<
1782:talk
1754:and
1731:talk
1704:talk
1620:) =
1604:) =
1591:talk
1581:) =
1573:) =
1503:talk
1486:talk
1460:talk
1193:<
1126:<
1021:<
978:"If
902:talk
878:talk
863:talk
811:talk
777:talk
762:talk
733:talk
694:talk
663:talk
648:talk
560:and
545:...)
263:and
28:Date
2038:!)
1723:not
1286:to
1150:If
1114:= {
967:Jao
888:not
833:dis
708:Not
617:to
499:all
390:qua
218:of
178:Mid
2682::
2659:)
2651:--
2621:)
2600:)
2584:)
2556:)
2552:•
2524:)
2496:)
2302:)
2283:)
2267:)
2259:—
2210:∞
2198:∩
2158:∞
2155:−
2149:∩
2113:∈
2060:)
2045:A.
2015:)
1985:)
1962:∉
1956:−
1933:∈
1927:−
1918:−
1890:∉
1884:∧
1849:∉
1843:∧
1784:)
1776:--
1733:)
1706:)
1676:↦
1643:↦
1593:)
1505:)
1488:)
1462:)
1397:—
1386:.
1223:≥
1211:∈
1181:∈
1094:."
1051:≥
1039:∈
1009:∈
904:)
880:)
865:)
813:)
779:)
764:)
735:)
696:)
665:)
650:)
411:at
348:at
2655:(
2646:B
2642:A
2638:B
2634:A
2617:(
2596:(
2580:(
2548:(
2520:(
2492:(
2476:r
2453:r
2433:r
2413:b
2409:/
2405:a
2402:=
2399:r
2375:b
2355:b
2352:,
2349:a
2329:b
2325:/
2321:a
2298:(
2279:(
2263:(
2246:B
2242:A
2231:x
2227:B
2213:)
2207:,
2204:x
2201:[
2194:Q
2190:=
2187:B
2167:)
2164:x
2161:,
2152:(
2145:Q
2141:=
2138:A
2117:R
2110:x
2097:B
2093:A
2089:B
2085:A
2056:(
2051:☺
2011:(
2006:☺
1981:(
1965:A
1959:3
1936:A
1930:1
1921:3
1893:B
1887:y
1881:x
1875:y
1852:B
1846:x
1840:0
1834:x
1813:)
1810:2
1805:(
1780:(
1729:(
1702:(
1684:l
1680:B
1673:B
1651:u
1647:A
1640:A
1630:A
1628:=
1626:B
1622:B
1618:B
1614:B
1612:=
1610:A
1606:A
1602:A
1589:(
1583:A
1579:A
1575:A
1571:A
1501:(
1484:(
1458:(
1302:.
1300:A
1296:B
1294:,
1292:A
1288:S
1280:S
1276:S
1268:S
1264:S
1260:S
1256:a
1252:a
1232:}
1229:}
1226:a
1220:x
1217::
1214:S
1208:x
1205:{
1202:,
1199:}
1196:a
1190:x
1187::
1184:S
1178:x
1175:{
1172:{
1156:S
1152:a
1132:S
1128:b
1124:a
1120:b
1118:,
1116:a
1112:S
1108:S
1104:B
1102:,
1100:A
1092:S
1088:S
1084:a
1080:a
1060:}
1057:}
1054:a
1048:x
1045::
1042:S
1036:x
1033:{
1030:,
1027:}
1024:a
1018:x
1015::
1012:S
1006:x
1003:{
1000:{
984:S
980:a
943:B
939:B
900:(
876:(
861:(
838:A
809:(
775:(
760:(
731:(
692:(
661:(
646:(
619:R
615:Q
607:Q
603:Q
599:Q
595:Q
591:Q
569:F
523:F
518:"
438:(
415:x
379:(
364:x
360:x
356:x
352:x
334:(
311:(
274:F
190:.
89::
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