Talk:Construction of the real numbers

message: Note, that if you define (A_x,B_x)\, as the cut corresponding to  2-\sqrt2\,, and (A_y,B_y)\, as corresponding to 2+\sqrt2\,, and apply the rules as written, then neither (A_{\rm sum},B_{\rm sum})\,, nor (A_{\rm prod},B_{\rm prod})\, is a Dedekind cut.  The former is not even a partition, since 4 \notin A_{\rm sum} \cup B_{\rm sum}; and the latter is a partition but not a correct cut, since 2 is the greatest element of A_{\rm prod}\,.
 
The older constructions were based only on the set of positive rational numbers.  The cuts may be defined for the whole set \mathbb{Q}\,; 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)
replies: text: I suggest we only do it for positive rationals, as that will give an idea of what is going on, (and anyone who can understand it for positives, can work out the extension.. its just tedious..). any ideas? --TM-77 14:23, 28 October 2006 (UTC)
thread_title:  The Dedekind cut arithmetics '''is''' erroneous 

message: 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)
:: Doing that is the easy part. The hard part is to prove that the newly created field is complete, that is any Cauchy sequence converges. Oleg Alexandrov 15:14, 24 Apr 2005 (UTC)
replies: text:  maybe you guys can tell me how to properly define the sum of 0.44444444 and 0.66666666 (I mean the infinite decimals, of course)....in such a way that your definition applies to any two decimal expansions...--345Kai 20:40, 3 October 2006 (UTC)
replies: text:  0.444... + 0.666... = (0.4 + 0.6) + (0.04 + 0.06) + (0.004 + 0.006) + ... = 1 + .1 + .01 + ... = 1.111... --Ian Maxwell 04:15, 14 October 2006 (UTC)
thread_title:  Construction by decimal expansions 

message: 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 q \times q = 2.
We would expect that the Dedekind cut 
: A = \{ x\in \textbf{Q} : x < 0 \lor (x \geq 0 \land x \times x  < 2) \}
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 x \times x >  2, or must exclude some positive x such that x \times x <  2 (recalling again that there is no rational x such that x \times x =  2).
However, given the definition in the main article, we get A \times A =
: { \{ a_\mathrm{prod} \in \textbf{Q} : a_\mathrm{prod} \, {\not \in} \, \{ b_\mathrm{prod} \in \textbf{Q} : b_\mathrm{prod} = b_x \times b_y \land b_x \in A \land b_y \in A \} \}} 
It should be apparent that, by this definition, the value of A \times A  includes the rational number 2.  Since by definition the embedding of 2 in the real numbers is \{x: x < 2\} 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 A \times A  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)
replies: text: Wikibooks simply suggests that the left product set consisting of the products of left set elements (instead of the right product set consisting of the products of right set elements, as here). But as noted on its talk page, that doesn't really work either, because minus times minus is plus and all that. Neither the definition given here nor the one given there actually makes the cuts closed under multiplication. I'm looking forward to the refactoring. -- Jao (talk) 20:14, 29 August 2008 (UTC)
thread_title:  Definition of multiplication of Dedekind cuts seems wrong 

message: 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 A as the representative of any given Dedekind cut (A, B), since A completely determines B.  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 r is any subset of the set \textbf{Q} of rational numbers that fulfils the following conditions:
# r is not empty
# r \neq \textbf{Q}
# r is closed downwards.  In other words,  for all x, y \in \textbf{Q} such that x < y, if y \in r then x \in r
# r contains no greatest element.  In other words, there is no x \in r such that for all y \in r, y \leq x
* We define a total ordering on the set  \textbf{R}  of real numbers as follows: x < y 
\Leftrightarrow x \subset y
* We embed the rational numbers into the reals by identifying the rational number q with the set of all smaller rational numbers  \{ x \in \textbf{Q} : x < q \} .  Since the rational numbers are dense, such a set can have no greatest element and thus fulfils the conditions for being a real number laid out above.
* Addition. A + B := \{a + b: a \in A \land b \in B\}
* Subtraction. A - B := \{a - b: a \in A \land b \notin B\}
* Negation is a special case of subtraction: -B := \{a - b: a < 0 \land b \notin B\}
* Defining multiplication is less straightforward.  
** if A, B \geq 0 then  A \times B := \{ a \times b : a \geq 0 \land a \in A \land b \geq 0 \land b \in B \} \cup \{ x : x < 0 \}
** if either A or B is negative, we use the identities  A \times B = -(A \times -B) = -(-A \times B) = (-A \times -B) to convert A and/or B to positive numbers and then apply the definition above.
* We define division in a similar manner:
** if  B > 0  then  A / B := \{ a / b : a \in A \land b \notin B \}
** If B is negative, we use the identity  A / B = -(A / -B)  to convert B to a positive value and then apply the definition above.
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)
replies: text: I haven't analyzed it in detail, but I think it looks safe. Do you have a source? Otherwise, it might look like OR. -- Jao (talk) 11:14, 30 August 2008 (UTC)
replies: text: I've been scouring Google Books in a very frustrating search.  Most textbooks just give a basic definition of cuts and possibly addition, before going on to say something like "multiplication is more complicated, and is left as an exercise for the reader" :)  However, the best ref I've found so far has been .  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)
replies: text: Alright, but don't hesitate to go ahead and update the article, and give that reference for the multiplication. At any rate, a correct definition that could be accused of being partly OR is quite a few orders of magnitude better than a clearly flawed and completely unsourced definition. -- Jao (talk) 16:59, 30 August 2008 (UTC)
replies: text: Done.  Feel free to add any more references, etc.  Cheers. Grover cleveland (talk) 19:08, 30 August 2008 (UTC)
replies: text: Looks great! -- Jao (talk) 19:09, 30 August 2008 (UTC)
thread_title: Proposed replacement for definition of Dedekind cuts

message: It is enough to define addition and multiplication for the Dedekind cuts. To me the simplest route seems to be the following:
* Define A+B := \{a+b \ | \ a\in A , b\in B\}. This already determines the additive inverse -A = \{ a-b | a<0 , b\notin B \}.
* Define A/B := \{a/b \ | \ a\in A , b\notin B \} for A,B>0. (I think this is wrong in the article.)
* Define (-A)(-B) := AB := A/(1/B) and A(-B) := (-A)B := -(AB) for A,B>0 and AB:=0 if either is zero.
If one is ready to omit the details, the following definition is even more effective. Here we denote the set of Dedekind cuts by \mathcal D and the embedding \mathbb Q \hookrightarrow \mathcal D by
q\mapsto q_{\mathcal D}. 
* Define a topology for the set \mathcal D by introducing the open intervals ]a,b[\subset \mathcal D in the obvious way.
* Define the addition and multiplication as the unique continuous extensions of the corresponding operators *:\mathbb Q\times \mathbb Q \to \mathbb Q into  *:\mathcal D\times \mathcal D \to \mathcal D.
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)
thread_title:  Streamlining arithmetics 

message: 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)
thread_title:  Construction from the group of integers 

message: 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)
replies: text: Sounds like a flawless argument. I guess nobody just thought of it before. -- Jao (talk) 19:09, 18 September 2008 (UTC)
replies: text: I am copying my edit from WPM: True, but it can also be read as construction of "real numbers", it being understood that everybody has heard of the famous "real numbers".  It can also be read as an editorial contraction, headline-style.  I think this should be a question of wiki style guidelines rather than a semantic issue. Katzmik (talk) 05:44, 19 September 2008 (UTC)
replies: text: Relevant guideline policy is here: WP:NAME.  I can't immediately see anything which prescribes leaving out the definite article in the middle of a noun phrase. However, this sub-page, WP:COMMONNAME, may have some bearing – are people more (or less) likely to type "of the real numbers" than "of real numbers" when referring to or looking for this material? H.G. 06:34, 19 September 2008 (UTC)
replies: text: An answer is provided by traffic statistics.  construction of real numbers is hit on the average 40 times daily, whereas construction of the real numbers is hit about once.  This does not mean the former title is preferable necessarily.  The discrepancy may be due to the fact that the pages linking to it all link to the former rather than the latter.  At any rate, if we are worried about the stray bird who types in the definite article in his query, the worries are unfounded.  Katzmik (talk) 06:40, 19 September 2008 (UTC)

text: P.S. The results of all the olympiads indicate that the russians are the ones who are interested in, and good at, math, and they all tend to leave out the definite article (Russian does not have one). :) Katzmik (talk) 06:42, 19 September 2008 (UTC)
replies: text: Do the traffic statistics distinguish between page visits by people who've typed something into the search box and page visits made by people who've clicked on an internal link?  H.G. 07:30, 19 September 2008 (UTC)
thread_title:  Proposed move 

message: 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
:Absolutely correct. Or in simpler terms, the construction of R filled all "gaps", so there are no more "gaps" to fill. -- Jao (talk) 12:53, 12 January 2009 (UTC)
thread_title:  dedekind cut the real numbers? 

message: 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)
replies: text: Not sure what the problem is.  Take A = 1, B = 2.  Then A/B = {x/y : x < 1 && y >= 2}.  This has no greatest element, since (1/2) is not in A/B but all smaller rationals are.  Can you explain the worry?  Grover cleveland (talk) 14:33, 29 March 2009 (UTC)
thread_title:  Dedekind cuts still not ok! 

message: The last part of the construction using Cauchy sequences seems very handwaving. It is a bit confusing that the two constructed sequences l_n and u_n 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)
thread_title:  Cauchy sequences  

message: 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)
replies: text: Not really, as far as I can see. QN modulo the cofinite filter is a rather unpleasant object (it has uncountably many square roots of 1, for example), and you'd need some way of restricting to well-behaved points. All the ways I've thought of (restricting to finite elements that are order-comparable with every rational, or restricting to points that are limits of rational sequences) more-or-less boil down to restricting to Cauchy sequences, and if you're going to do that, you might as well forget the filter and just use the Cauchy sequence construction from the beginning. By the way, article talk pages are for discussions about what should and shouldn't be in the article. Questions such as this one about actual mathematics relating to articles are better posted to WP:RD/Math, which is watched by far more people. Algebraist 18:45, 21 January 2010 (UTC)
thread_title:  schmieden-laugwitz type construction using a Frechet filter 

message: 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)
replies: text: The first reference is a review of a book from 1908 by someone other than Weierstrass.  It does not claim that W. defined reals by decimal expansions.  Briefly, W. defines a number as a series of positive rationals.  The second reference is inaccurate, and does not provide any sources for the claim made (that W. defined reals by decimal expansions).  With all due respect to Springer, this is simply incorrect.  The sentence following Stevin in the current version is therefore inaccurate.  Tkuvho (talk) 10:54, 16 March 2010 (UTC)
thread_title:  Did Weierstrass construct the reals by means of decimals? 

message: 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)
replies: text: As far as I can see, almost homomorphisms are not even closed under pointwise integer multiplication.—Emil J. 13:21, 14 December 2010 (UTC)
replies: text: Good point.  Suffices to look at the identity homomorphism.  Still, you could multiply f(n) and g(n), divide by n, and take integer part.  That presumably gives the product in R, doesn't it?  Tkuvho (talk) 14:00, 14 December 2010 (UTC)
replies: text: Why would you do it in such a complicated way when you can just compose the two functions? And what does it have to do with the article? The construction is described using composition in the sources, therefore we should report it as such, we are not supposed to report your OR. Leaving aside the fact that the general construction can be usefully applied to other abelian groups, such as Zn, that have no multiplication (see the appendix to Arthan's paper).—Emil J. 14:24, 14 December 2010 (UTC)
replies: text: OK, thanks.  Tkuvho (talk) 15:08, 14 December 2010 (UTC)
thread_title:  group or ring? 

message: 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)
replies: text: I cannot find the statement you quote in the article. What I can find in the article is the correct statement that A ≤ B iff A is equivalent to B (defined earlier to mean \lim_{n\to\infty} a_n-b_n=0) or there is N such that an ≤ bn for all n > N. This means that A < B iff A is not equivalent to B and there is an N such that an < bn for all n > N. In your example, A and B are equivalent.—Emil J. 13:51, 11 November 2012 (UTC)
replies: text: Ah, so it does. I suppose I misread it. Thanks. 174.140.112.18 (talk) 04:01, 12 November 2012 (UTC)
thread_title:  Is the information on the ordering of Cauchy sequences correct? 

message: 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)
thread_title: #Construction from Cauchy sequences

message: 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)
thread_title:  the opening paragraph 

message: Substituted at 01:55, 5 May 2016 (UTC)
thread_title: Assessment comment

message: Here it is for reference:
It has been known since Simon StevinKarin Usadi Katz and Mikhail G. Katz (2011) Stevin Numbers and Reality. Foundations of Science.  Online First.   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.V. Dantscher,   "Vorlesungen über die Weierstrass'sche Theorie der irrationalen Zahlen" , Teubner  (1908)
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:
*According to the source, these notions weren't new with Stevin, although he did seem to be an important advocate.
*It wasn't "known" since the time of Stevin, since the concept of "real number" would continue to develop over the next few centuries.
*The bit about Weierstrass feels tacked on and makes no sense in context.
*The second paragraph is highly dubious and sounds like personal opinion.
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)
thread_title:  Removed section "Stevin's Construction" 

message: 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)
thread_title:  Citation needed 

message: 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)
replies: text: Looking at the McTutor biography of these two authors (accessible form their WP page), it appears that both had important contributions which were independent from those Weierstraß and Cantor. So they must be mentioned here, and I'll revert your edit. However, I agree that the respective contributions must be clarified. D.Lazard (talk) 14:38, 6 March 2019 (UTC)
replies: text: Regarding Meray, I see that he indeed apparently had the same idea as Cantor. But for Otto Stolz, it says nothing about him constructing the real numbers. (Of course he had important contributions to mathematics in general, otherwise he wouldn't have a McTutor page.) I have searched online but cannot find any reference to Stolz working on the real numbers (he obviously worked on real analysis though). KarlFrei (talk) 08:04, 7 March 2019 (UTC)
thread_title:  Otto Stolz? 

message: 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)
thread_title:  "Separates" definition could maybe use clarification. 

message: 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 \mathbb{R}, 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 \mathbb {R}" 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)
replies: text: I have added a definition of the notation D.Lazard (talk) 14:05, 22 January 2023 (UTC)
thread_title:  A suggestion