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The definition of Dedekind cuts on this page seems incorrect. According to the page, a partiton of an ordered field, , is a Dedekind cut, where and are non-empty sets, such that is closed downwards and is closed upwards. Then, and intersect at a point - ie, they do not form a partition.
The page discusses a non-closed addition, where is not a Dedekind cut. But of course, neither nor is a Dedekind cut by this page's definition, which requires both sets that make the cut to be closed.
The definition I'm familiar with defines A as an open set, extended downwards. B as closed and extended upwards. The least upper bound of A is the number defined by the cut. See [2] for a reference to one such formulation.
Other references define them as a non-empty, downwards extended set, that is bounded-above, and doesn't contain a greatest element. The the least-upper bound of this set defines the number at the cut. See [3] for a reference to one such formulation.
Please see Talk:Dedekind cut Revolver 16:30, 15 Dec 2004 (UTC)
Note, that if you define as the cut corresponding to , and as corresponding to , and apply the rules as written, then neither , nor is a Dedekind cut. The former is not even a partition, since ; and the latter is a partition but not a correct cut, since 2 is the greatest element of .
The older constructions were based only on the set of positive rational numbers. The cuts may be defined for the whole set ; but in that case the rules for the arithmetic get immensely more complex than the thing exhibited at the moment. (The same goes for any other ordered field.) The rules for arithmetics now are not completely adapted to this, even in spite of the 'cheating' at multiplication by making a restriction to nonnegative cuts. My personal suggestion is to make another approach. I've made a longer suggestion for this at de:Diskussion:Dedekindscher_Schnitt, which I prefer not to repeat here. JoergenB 01:16, 10 October 2006 (UTC)
That is the right way to do it. I might look up Landau Foundatioons of Analysis and adapt if still needed. --Gentlemath (talk) 22:59, 28 March 2009 (UTC)
We can take the infinite decimal expansion to be the definition of a real number, considering expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally.
I don't know whether such a definition would work; it seems reasonable. I think it should be pointed out that we could use (any base? any integer as a base?). In any case surely it doesn't need to have anything to do with the number 10. Brianjd | Why restrict HTML? | 11:46, 2005 Apr 24 (UTC)
Yet it can be done. The first to do so was [[Weierstrass] about the same time as Cauchy and Dedekind. Ultimately it is neither more nor less tedious than the other methods. --Gentlemath (talk) 22:50, 28 March 2009 (UTC)
We want there to be a positive real number x such that x . x = 2.
In the following discussion I'm going to refer only to the left element of the Dedekind cut for conciseness.
Let's first recall that there is no rational number q such that .
We would expect that the Dedekind cut
should represent the positive square root of 2. It should certainly be clear this could be the only possible definition of positive root 2, since any other positive Dedekind cut must either contain some positive x such that , or must exclude some positive x such that (recalling again that there is no rational x such that ).
However, given the definition in the main article, we get
It should be apparent that, by this definition, the value of includes the rational number 2. Since by definition the embedding of 2 in the real numbers is and thus excludes 2, we are forced to conclude that the positive square root of 2 is not a real number!
In fact, we should see that the value of contains an upper bound, namely 2, and thus does not even fit the defintion of a real number!
I'm going to work on an alternative presentation of this section, with what I regard as a correct definition of multiplication -- I'll present it in talk before making any changes. Cheers! Grover cleveland (talk) 18:58, 29 August 2008 (UTC)
A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.
For convenience we may take the lower set as the representative of any given Dedekind cut , since completely determines . By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number is any subset of the set of rational numbers that fulfils the following conditions:
We could add an example, such as the construction of root 2. Let me know what you guys think. Cheers. Grover cleveland (talk) 20:39, 29 August 2008 (UTC)
((cite book))
: |pages=
has extra text (help). Google Books link here. It gives a definition of multiplication that agrees with the one above: however it doesn't define division and gives a somewhat different (though equivalent) def of subtraction. However, it does have a really cool diagram of a pair of scissors cutting the number line :) The search continues.... Grover cleveland (talk) 15:40, 30 August 2008 (UTC)Maybe the right thing to do would be to follow Landau Foundations of Analysis. I think he defines + * / and < for the positive integers and then makes the right definitions to allow signs. No free online version, alas. --Gentlemath (talk) 22:56, 28 March 2009 (UTC)
It is enough to define addition and multiplication for the Dedekind cuts. To me the simplest route seems to be the following:
If one is ready to omit the details, the following definition is even more effective. Here we denote the set of Dedekind cuts by and the embedding by .
In order to justify this definition one must be able to show that the continuous extensions exist and are unique, but I think that the main idea in the definition is transparent enough to pay off this drawback.Lapasotka (talk) 23:14, 19 October 2009 (UTC)
A recent discussion "A new (?) construction of R" (2008-09-05) on sci.math mentions a related definition of a positive real given in 1946 by A. N. Kolmogorov. — Loadmaster (talk) 18:01, 5 September 2008 (UTC)
I noticed an edit in which someone changed a link to construction of the real numbers to a link to construction of real numbers, with no "the". The title with the definite article is a redirect. Omitting "the" seems weird. It is as if individual real numbers were to be constructed, rather than the system as a whole. Are there opinions on moving the page to construction of the real numbers? Michael Hardy (talk) 18:56, 18 September 2008 (UTC)
For what it's worth (and it's part of what I had in mind when mentioning WP:COMMONNAME), I'd expect to see this written as "construction of the real numbers", as we tend to invoke the definite article when referring to them as a group (in the non-mathematical sense). A quick gander at Google Books and Google Scholar also suggests a preference for using "the". I'm willing to bet that the proliferation of links to construction of real numbers has more to do with some editors' preference for avoiding redirects rather than common practice in real life. H.G. 08:42, 19 September 2008 (UTC)
ok i'm just a beginner in analysis. after finishing the first chapter in rudin's principle of mathematical analysis, my first thought is, wat if we run throught the dedekind construction on the set of real numbers? i think because rudin goes on sumthin about isomorphism between Q, the rationals, and Q*, the set of rational cuts, and since each r* is the supremum of all q* such that q* is a subset of r* , we wouldn't be doing any good if we apply this construction to the reals because they already got the least upper bound property, i.e. this new set is isomorphism to the original set of reals R. if sum1 can help a poor dumbass out pls do
Proof that the details will get you in the end! I'm going to need to look up Landau and get this right!
Note that in the definition of 1/B there would be a greatest element if B is rational. One could define A/B for A,B>0 as the set of all a/b with a in A and b in the complement of B ***but not the miminimum element of that complement if there is one***. As usual, the thorniest details are for rationals when we already know the result we want.
I think that the Landau definition for 1/B with B>0 is the cut whose lower set is essentially {all x>0 | xB<1 }--Gentlemath (talk) 23:58, 28 March 2009 (UTC)
I agree now. I'm never sure if it is legit to erase ones own talk contributions. But at least -3/10 was a problem before.--Gentlemath (talk) 17:42, 29 March 2009 (UTC)
The last part of the construction using Cauchy sequences seems very handwaving. It is a bit confusing that the two constructed sequences and are equivalent (i.e. in the same class representing a real number), yet one of them is said to not be an upper bound, while the other one is. —Preceding unsigned comment added by 80.216.180.6 (talk) 16:20, 16 May 2009 (UTC)
Does anyone know if one can construct R along the lines of the hyperreal construction, but using the Frechet filter instead of the ultrafilter? This would eliminate the need for choice. Tkuvho (talk) 11:31, 21 January 2010 (UTC)
one more reference if someone wants to follow them up:
http://www.ams.org/bull/1909-16-02/S0002-9904-1909-01864-6/S0002-9904-1909-01864-6.pdf this is a review of the reference I gave and describes its contents
also: —Preceding unsigned comment added by Gentlemath (talk • contribs) 23:32, 14 March 2010 (UTC)
There is also http://eom.springer.de/r/r080060.htm which may actually make too strong a case. --Gentlemath (talk) 00:52, 15 March 2010 (UTC)
Dear Tkuvho
It has been seen by me that my remarks added to the topics on the connection between construction of real number system and the physical operation of measurement have been undone and removed. I think that it is a fault in that the physical operation of measurement served as a great motivation for the construction of real number system and should never be omitted because of its shear importance. If the remarks is not satisfied with, I am inclined to think that it is better improved rather than simply removed. Sorry, I am new to wikipedia and do not know much about the policy and courtesy here. I am going to undo the deletion, no implication of insulting because I do appreciate your great contribution to wikipedia. For any suggestions, maybe we could contact via e-mail: tschijnmotschau@gmail.com. Thank youTschijnmotschau (talk) 08:55, 10 December 2010 (UTC)
I am really sorry for undoing the change without a discussion, I am
new to wikipedia and is not very proficient in the regulation here. I
am sorry. But I cannot agree with the idea that the rational numbers
is sufficient for physical measurement. If it is indeed the case,
there might have never been the introduction of real numbers into
mathematics. Real numbers can be viewed as the completion (also in the
sense of the convergence of Cauchy sequence) of the set of rational
numbers for it to be able to be used for measurement of any physical
quantity. Physically, if the measurement of some object, say, length
of a line segment, is equivalent to another set of identical objects
which is chosen as the unit for measurement, then the cardinality of
the set of units can be taken as a measure for the physical
quantity. For example, when measuring the length, if a line segment is
equivalent to another set of three identical line segments, the length
of the line segment can be assigned the integer value of three
units. But it is not always possible to find a set of integral number
of units which gives equivalent physical measurement. So rational
numbers can be introduced to solve this problem to some extent. If
some set with n identical objects of something can give identical
measurement as the unit of measurement, then it is assigned the
measurement of 1/n units. Taking a line segment with 2/3 cm as an
example, no set of integral numbers of the unit of centimeter can
given identical result with the 2/3 cm segment when its length is
measured. But if the segment a set of three of which can give
identical measurement with the unit is found first, a set of two of
such segments can give equivalent measurement with the segment to be
measured, then the result of 2/3 cm can be reasonably assigned. In
this way, to measure a line segment with a unit, first the unit itself
might be tested to see whether or not a set of integral numbers of it
can give equivalent measurement as the unknown. If it cannot be so,
the 1/n of the unit can be used progressively to measure the unknown
until a set of integral number x of the 1/y unit segment can give
equivalent measurement as the unknown, then the unknown can be
assigned the measurement of x/y units. It is pointed out above that
the rational numbers would be sufficient for physical measurement, but
in fact, it is sufficient ONLY WHEN THE PROCESS DESCRIBED ABOVE CAN
TERMINATE AT SOME POINT OF x. But is it indeed the case? Obviously
not. It can well occur that when finer and finer partition of the unit
is used the unknown can still not be measured exactly. Mathematically,
it can be proved that the hypotenuse of an isosceles right-angled
triangle can never be equal in length to a set of any number of any
1/n partition of its sides, i.e. the irrationality of
. This leads, directly and quite intuitively, the Dedekind
construction of real numbers, and the Cantor's construction came in
another way but the essence is exactly the same. So I am inclined to
think that my remarks added on the article is pertinent and helpful,
hence should not be simply deleted. Thank you! Tschijnmotschau (talk) 17:54, 12 December 2010 (UTC)
Sorry for having failed to make my point clear. Rather than saying
that a real number can be approximated ever closer by rational
numbers, what I was going to say is that mere rational numbers are not
sufficient to represent the measurement of any physical quantities,
hence the need for the construction of real numbers. The rational
numbers can approximate ever closer to any point on a line, but it
cannot specify, i. e. overlap with any point on the line. Only after
the introduction of real numbers, the ideal physical measurement with
infinite accuracy can be represented mathematically. This is what I
want to say. Thank you!Tschijnmotschau (talk) 02:51, 13 December 2010 (UTC)
Sorry, I think that maybe what quantum mechanics is telling us is that we cannot make measurement of infinite accuracy without any disturbance to the state of the system, rather than that such measurement is in principle impossible. In principle, the measurement of, say, position of a particle would cause its state to collapse into a Dirac delta, a state in no conflict with the basic concepts of quantum mechanics. But this maybe is not important here. What is important is that some mathematical object is needed by physicists to represent the result of ANY POSSIBLE MEASUREMENT, including the conceptual measurement of infinite accuracy. And real numbers is a perfect construct that can make it possible. I am sorry for my improper phrase "ideal physical measurement", what I wanted to express is that the measurement is to be applied to some solid physical entity, rather than some pure mathematical structure. hence "physical", and such measurement is going to take place in imagination rather than really to be implemented in some experiment, hence "ideal". I am sorry for my improper wording. Tschijnmotschau (talk) 12:21, 16 December 2010 (UTC)
In the construction from the integers, one uses the composition in Maps(Z,Z) to define multiplication in the reals. It seems a bit misleading to claim that one uses "only" the addition. It seems almost simpler to use the multiplication in Z to define the product, instead of composition, but perhaps there are reasons not to do this. Any comments? Tkuvho (talk) 13:07, 14 December 2010 (UTC)
The article states that for some Cauchy sequences of rational numbers A = (a1, a2, ...), B = (b1, b2, ...) then A < B iff there exists N > 0 : for all n > N, an < bn. Is this correct? For example, take A := (0.9, 0.99, 0.999, 0.9999, ...), B := (1, 1.0, 1.00, 1.000, ...). For all n, an < bn, but they both converge to the same number (the number 1). Am I missing something? I am ardently not a mathematician, so I can't just say for sure that something's up, but this seems incorrect. 174.140.112.18 (talk) 06:13, 11 November 2012 (UTC)
Is L a rational number we already had before construction? Or is it a rational that belongs to R?--578985s (talk) 07:28, 14 October 2014 (UTC)
I would like to improve the opening paragraph so that it more correctly introduces the subject matter. This is my first wiki edit ever, so any comments would be greatly appreciated. Something along these lines (and later on I would like to improve the article to more systematically present the constructions and related to the three aspects presented below):
The axioms of the real number system, which were colloquially well-known well before any rigorous construction of any model of the real numbers was ever constructed, are well known to essentially uniquely characterize the reals. More precisely, it follows from the axiomatic formulation of the reals that any two models satisfying these axioms are isomorphic, and thus essentially the same. This however does not rule out the possibility that the axioms are contradictory. A construction of the real numbers is a rigorous presentation of mathematical objects which are also shown to form a complete ordered field, namely the conform to the axiomatic formulation of the reals.
Constructions of the reals, and there are plenty of such constructions, may be classified based on three aspects, namely the assumed simpler model, the explicitness of the constructed objects, and the analyticity of the definition of the algebraic operations. In more detail, construction of the reals must use some agreed upon simpler entities, and most constructions appeal to the rational or the integer number systems. Then there is the issue of how tangible the constructed objects representing the real numbers are. This is of particular relevance to related efforts of implementing any particular construction on a computer, for instance for the purposes of automated proof verifiers when arguing about the the reals. Lastly, and somewhat vaguely, the amount of analytic machinery required in order to obtain the definition of the algebraic operations may vary greatly between constructions. A purist approach would demand that no analytic properties, i.e., limit-like arguments utilizing suprema or infima should be used at all when defining addition and multiplication, and such constructions exist. Other constructions may first develop certain of analytic tools as part of the construction required for the algebraic operations. — Preceding unsigned comment added by IttayWeiss (talk • contribs) 00:28, 21 February 2015 (UTC)
The comment(s) below were originally left at Talk:Construction of the real numbers/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.
== Construction from Cauchy sequences == In the article, the constructed sequences and are said to be trivially Cauchy. I don't think it is trivial. I'm a undergraduate math student, and fail to show it. I suppose the sequences are Cauchy because they are bounded from above/below and nondecreasing/nonincreasing respectively. Still, more justification is necessary. |
Last edited at 07:35, 20 March 2015 (UTC). Substituted at 01:55, 5 May 2016 (UTC)
Here it is for reference:
It has been known since Simon Stevin[1] that real numbers can be represented by decimals. We can take the infinite decimal expansion to be the definition of a real number, defining expansions like 0.9999... and 1.0000... to be equivalent, and define the arithmetical operations formally. This is equivalent to the constructions by Cauchy sequences or Dedekind cuts and incorporates an explicit modulus of convergence. Similarly, another radix can be used. Weierstrass attempted to construct the reals but did not entirely succeed. He pointed out that they need only be thought of as complete aggregates (sets) of units and unit fractions.[2]
This construction has the advantage that it is close to the way we are used to thinking of real numbers and suggests series expansions for functions. A standard approach to show that all models of a complete ordered field are isomorphic is to show that any model is isomorphic to this one because we can systematically build a decimal expansion for each element.
This whole section is kind of a mess:
There should really be some context and qualifications to all this before it's put back in. --Deacon Vorbis (talk) 15:23, 1 April 2017 (UTC)
References
Text says "We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks."
I could not find proof in any text book, and I doubt anyone can, other than those just saying it's fact. No text books provide proof. Prove me wrong, should be easy if it's common. — Preceding unsigned comment added by 71.17.109.113 (talk) 00:17, 29 May 2017 (UTC)
The text attributes one of the first three constructions of the real numbers to "Karl Weierstraß/Otto Stolz". I had never heard of Otto Stolz until now and it seems he was a student of Weierstraß. Is there any source for him being a co-inventor (if that is the right word)? Similarly, I thought until now that the construction using Cauchy sequences was solely due to Cantor. KarlFrei (talk) 18:34, 28 January 2019 (UTC)
In the informal description of one of the axioms, it states: "If a set of reals precedes another set of reals, then there exists at least one real number separating the two sets." While it is intuitive that "separates" could certainly refer to a number appearing between both sets, the definition also allows (e.g.) X = {x|x < 0.0} and Y = {0.0}, in which case the "separator" is the only element of Y. Suggesting that 0 somehow "separates" itself from other numbers is not intuitive, particularly if you need this sort of analogy to understand the axiom. Maybe "either A has a largest element, or B has a smallest element, or there exists a z which separates A from B"...? TricksterWolf (talk) 03:31, 17 December 2021 (UTC)
It would be a massive help to those who do not have an advanced degree in Mathematics -- or whose knowledge of the symbology of mathematics is rusty -- if the symbol was defined. (A quick look at R (disambiguation) shows it stands for the set of real numbers. Which means the passage "A model for the real number system consists of a set " is a circular statement.) In other words, avoid jargon so the rest of us understand the article, unless you are willing to explain it in non-technical words. -- llywrch (talk) 03:59, 22 January 2023 (UTC)