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Archive 1 |
Apart from the difficulties that such a requirement creates, it is contrary to intuition. For example the sequence {0,1,0,1,0,...} may be greater than the sequence {2,0,2,0,2,0,....} (depending upon which sets of indices belong to the ultrafilter used). It would be much better to restrict the set of pairs of comparable sequences and define it in a way that does not contradict intuition. Leocat 15:10, 24 January 2007 (UTC)
It is stated that "(In fact, there are many such U, but it turns out that it doesn't matter which one we take.)". Can anyone provide some explanation as to why this is so? (Are all free ultrafilters on N isomorphic in some sense, or is there some other reason why the choice of U doesn't matter?) - Chinju 19:41, 6 Oct 2004 (UTC)
(In fact, this seems to conflict with the sentence at the top: "The use of the definite article 'the' in the phrase 'the hyperreal numbers' is somewhat misleading in that there is not a unique ordered field that is referred to, in most treatments.") - Chinju 19:50, 6 Oct 2004 (UTC)
A look at the provided external link seems to indicate that the ultrafilter chosen doesn't matter (the constructed hyperreals are isomorphic) if the continuum hypothesis is assumed, but can matter if the continuum hypothesis's negation is assumed. If no one argues otherwise, or that I've misinterpreted the reference, I'll modify the article accordingly. - Chinju 19:56, 6 Oct 2004 (UTC) --
I don't think epsilon-delta definitions are really that unintuitive, and they have the advantage of working entirely within the reals, where infinitesimals truly don't exist. But they're still very cumbersome and tend to give miraculous results that should be obvious with a better set up. The only formal construction I've seen of differentials is as members of a cotangent space, which is no help at all. I've heard of hyperreals but never seen a treatment - any chance you could augment the above with a formal construction and/or axiomatization for us less enlightened? Thanks!
There is an on-line article with a short description of such a beast:
http://www.math.vt.edu/people/elengyel/thesis/thesis.html I've read it but am not confident enough to wikipedify it. By the way, I've been searching for some more information on-line on this subject and guess where
http://www.google.com sends you.... that's right, to Wikipedia. :-) --
Jan Hidders
Differential forms living in cotangent spaces are not the same thing as
infinitesimals even though the notation may or may not be identical.
Sorry, my mistake, I thought you meant a description of the formal construction of hyperreal numbers. --
Jan Hidders
How can you tell if a sequence is a valid hyperreal? Here's a pair of sequences for which which is greater depends on which ultrafilter you use: (1/2, 1/4, 1/4, 1/6, 1/6, ...) and (1/3, 1/3, 1/5, 1/5, 1/7, ...). -
PierreAbbat
Because it is so easy to construct ordered fields that contain infinitesimals but will not serve the purposes of non-standard analysis, it would be a good idea to mention the special properties of this one that enable it to do so, i.e., the transfer principle. Maybe I'll add something on this if I get around to it. Michael Hardy 16:29, 2 Sep 2003 (UTC)
I would like to change the displayed formulas in this article to LaTex, unless there is a compelling reason not to. User:CSTAR
I think the construction of the hyperreals is wrongly attributed to Lindstrom. Probably Zakon is the originator of this idea.
Could we put the section on Hyperreal fields after the more elementary exposition? Yes fine we know if you mod a ring by a maximal ideal we get a field, but lets try to keep it elementary, at the beginning at least. It's OK, in my view, if you put the hyperreal fields section after and say this is a generalization. CSTAR 06:24, 23 Oct 2004 (UTC)
The most recent edits to this article have added a long paragraph on the process of extension of number "systems" (including fields, rings and semirings); the text of this paragraph belongs somewhere in a wikipedia article, preferably in another article specifically about extensions of number systems. That would be a useful article.
In addition, I disagree strongly with the claim (emphasis mine)
What is the accident of history? Is it really an accident of history that 1st order logic was developed after calculus? We don't need this kind of historical revisionism in explaining the development of mathematics. I suggest that the paragraph in question be removed or be thoroughly re-edited. CSTAR 15:34, 15 Mar 2005 (UTC)
The most recent edits adding a new section An intuitive approach to the ultrapower construction contain some confusing statements : For example, it seems to define an "infinitesimal sequence" as one containing a subsequence converging to zero (The specific quote is "...the true infinitesimals are the classes of sequences that contain a sequence converging to zero").This is clearly inadequate, since the sum of such sequences may not be infinitesimal. Moreover, the style seems to me to be unsuitable for an article. For example, the opening sentence of the section: "Here is the simplest approach to infinitesimals that I could think of. " -- CSTAR 14:12, 16 September 2005 (UTC)
I say "the classes of sequences that contain a sequence...," I never say "the sequences that contain a subsequence..." It is the classes that contain, not the sequences that contain. Please, don't misquote me. As for the style, the article originated as an e-mail to a friend, and I am sorry if it lacks in pompous and authoritarian air that some people expect from the articles in an encyclopedia. In any case, this place is free, anybody is welcome to make improvements.
Sir, I don't see anything eschatological about our discussion. You just misunderstood the sentence that was clear enough in my opinion and I have clarified it for you. I have read what you wrote, sir. I only used the word "subsequence" in reminding the Bolzano-Weierstrass, the expression "infinitesimal sequence" is not used at all. Your quotation is accurate, and I take it back that you misquoted, but you took this sentence out of context, have totally twisted and mangled what I wrote, probably in order to discredit it, probably because you didn't like my informal style. The sentence that you did not like is not a formal definition, it is a part of an explanation, and the next sentence says: "Let us explain where these classes come from." It is you who is not reading carefully and picking at minutia. My article is not a bunch of clinically dry definitions forced into a linearly ordered sequence. I am trying to show how one can arrive at these definitions, starting from some natural assumptions and simple observations. Definitions come first only in lousy textbooks. As Michael Atiyah says, "don't give them a definition, give them an example."
Hi there. Thank you for your contributions to hyperreal number. And a request. Would you mind making an account? It is rather hard for us to chase an ever changing identity whenever you change IP address. Also, it may confuse you with vandals, distracting people from the task of watching for the bad guys. It will be better for you too, as you can track your contributions better. What do you think?
On other matters, one should put a comma after "i.e.". According to the manual of style, one better even use "that is" instead. Thanks. Oleg Alexandrov 16:11, 17 September 2005 (UTC)
Hi Michael. I was not offended, but I could have. It seems you don't know that well what tzar means, and the history of that part of the world I come from. But never mind, I lived enough in US that I know what you mean.
Again, thank you for your contributions. I will not interfere with what you write as I have plenty of other things to take care of. One piece of advice. Your comments both to CSTAR and to me were rather combative. I think you need to mellow out a bit if you plan to continue contributing to this encyclopedia. Wikipedia is a place where for better or worse everybody steps on each other's foot all the time, and interpersonal skills are very important (that is, it is not enough that you have a good point, you also need to know how to drive it home without irritating yourself or others around you). In the future, if you have comments and questions, you can use my talk page. Oleg Alexandrov 15:31, 20 September 2005 (UTC)
It seems to me that the section titled "An intuitive approach to the ultrapower construction" is almost completely redundant with the preexisting section titled "The ultrapower construction." IMO, either the "intuitive approach" section should be deleted, or the two sections should be merged.-- Bcrowell 05:07, 4 January 2006 (UTC)
At the bottom, there is a couple links that it says to "compare with". These are: Surreal numbers, Superreal numbers, and Real closed fields. I think that it would be very useful if there was a section comparing the three for the reader. It is very hard to tell the diference between three sets that all contain reals, and all contiain infintesimals and infinities. Fresheneesz 23:52, 10 February 2006 (UTC)
Hello
I have a comment regarding what I believe to be an incorrect statement in the "Hyperreal number" article. In "The ultrapower construction", near the end of the section, one finds the following sentence (referring to the hyperreal field constructed in that section):
As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R.
I do not believe this is the case. More precisely, I don't think that the hyperreal field constructed there is isomorphic to R. One way to see this, for example, would be to notice that the hyperreal field contains an element (any of the so-called "infinitely large numbers", for example) x such that for every positive integer n, x-n is a square. The existence of an isomorphism between the hyperreal field and R would imply (since an isomorphism would, of course, map 1 into 1 and n into n) the existence of an element of R with the same property. This is absurd.
Am I missing something obvious here?
P.S.
I cannot remember where, but I have seen this comment on Wikipedia before (the statement that two real closed fields of continuum cardinality are isomorphic, but, perhaps, not isomorphic as ordered fields). As noted above, this is false. Moreover, the distinction between an isomorphism of fields and an isomorphism of ordered fields doesn't make sense for real closed fields: an isomorphism maps squares into squares, and in a real closed field the non-negative elements are precisely the squares; this means that any field isomorphism will automatically be increasing and hence an order isomorphism as well.
192.129.3.135
21:00, 29 November 2006 (UTC) Alexandru Chirvasitu
I do not believe that using hyperreal numbers miraculously makes every derivative continuous, as it is stated in the article. In standard analysis it is of fundamental importance whether x approaches a, or a approaches x. The definition given in the article obliterates this distinction. Obviously e.g. the derivative of the function f(x) = sin(1/x^2)*x^2 for non-zero x, 0 at 0, is not continuous at 0. Leocat 08:49, 24 January 2007 (UTC)
Is "S-differentiable" supposed to mean "differentiable according to standard analysis"? If so, then my example contradicts your statement. Leocat 12:34, 25 January 2007 (UTC)
If h is a non-zero infinitesimal, then L = 0 satisfies the condition of being the value of the S-derivative of my f at 0. Actually I do not see why you state that S-differentiability is different from standard differentiability. Leocat 17:40, 25 January 2007 (UTC)
Let g(x)=|x|. Is g S-differentiable at values y infinitely close to 0? What is the definition of uniform differentiability? Is L in the definition of S-differentiability a real number? Leocat 20:45, 25 January 2007 (UTC)
Since L is to be a hyperreal number, I wonder about examples when L is not a real number. Are there standard functions, whose S-derivatives at standard points take on e.g. infinite hyperreal values? Leocat 22:04, 25 January 2007 (UTC)
I was going to add this bit about the hyperreals not forming a complete set, but I wanted to make sure my leg-work was correct; it's been a while.
A large weakness of the hyperreal field in analysis, however, is that the set of all hyperreal numbers is not
complete in the sense of a
metric space, since the existence of the hyperreals gives rise to hyperhyperreals, and thus it is impossible for a Cauchy sequence of hyperreals to converge to a unique point (since we can just define an infinitesimal number next to our real number to which the Cauchy sequence also converges).
134.39.100.70
19:47, 6 March 2007 (UTC)
Here is something that I have been thinking about:
This is fact according to the article:
0 = infinitesimal
x = infinite
1/x = infinitesimal
1/x = 0
0*x = 1
1/0 = x
That makes this possible:
(1/x)/(1/x) = 1
0^(-1) = 1
0/0 = 1
But if 0/0 = 1, then why does my calculators give me results as "divide by zero error", "undefined", etc?
—The preceding
unsigned comment was added by
161.52.141.39 (
talk) 08:00, 9 March 2007 (UTC).
Re: This is fact according to the article:
0 = infinitesimal
No, the article doesn't say this (unless it's been edited very recently to make that claim).--
CSTAR
14:51, 9 March 2007 (UTC)
I didn't get this:
Uh? How can a field F be isomorphic as a field to R but not isomorphic as an ordered field? If f: F -> R is a field isomorphism, then if x in F and x > 0 and f(x) < 0, then take y such that f(x) = -y^2, apply f^(-1) and get x = -(f^(-1)(y))^2 < 0, absurd. So, f must preserve order too. Albmont 12:45, 30 April 2007 (UTC)
Is this a valid construction of the hyperreals?
This way we might get the hyperreals even without the axiom of choice. Albmont 00:49, 4 May 2007 (UTC)
This is similar to, but not quite the same as, the polynomial ratio construction I added in Jan, 2007, and removed in Feb. because it was unpublished original research. When and if that work is published in a citable journal, I intend to re-contribute it. Basically, I am asserting that the non-Archemedian field H of real polynomial ratio (" rational function") sequences, the nth terms of which are those functions of the natural number n, where any statement about such ratios holds in H if it holds in R for all but a finite number of values of n, also models the transfer of first-order statements between H and R, and that this model, with the cofinite filter common to all nontrivial ultrafilters, does not require the axiom of choice, and should be much simpler to grasp than the ultrapower construction. Alan R. Fisher 05:50, 13 May 2007 (UTC)
Can anyone explain in more colloquial terms the comparison of two Hyperreals? I don't really have the background to understand all this ultrafilter stuff. Here's my understanding; am I correct, almost correct, or completely wrong? What's the correct description?
Two Hyperreals A and B are represented by their terms A1, A2, A3, etc and B1, B2, B3, etc. A≤B if and only if there exist an infinite number of terms n such that An≤Bn. All comparisons are defined as logical combinations of the ≤ operator. For example, A>B if and only if B≤A and not A≤B.
Obviously, this can lead to problems. For example, one can build a scenario that defies the transitive property of equality:
A = (2, -2, 2, -2, 2, -2...), B = (0, 0, 0, 0, 0, 0, 0...), C = (1, 1, 1, 1, 1, 1, 1...)
A≤B and B≤A, so A=B.
A≤C and C≤A, so A=C.
Therefore, by transitive property, B=C, an obvious contradiction, since the statement C≤B is false. This may be prevented by not allowing such sequences as A, but then one has to figure out how to make the proper distinction between a non-converging sequence like A, and an infinite sequence, like (1, 2, 3, 4...).
Additionally, Hyperreals that converge can be exactly equal to what they converge onto (not an infinitesimal smaller or larger). Though not a contradiction, this seems to be slightly against the philosophy of Hyperreals. For example (1, 1-1/3, 1-1/3+1/5, 1-1/3+1/5-1/7...) = π/4 exactly by my definition of equality.
Eagerly waiting on the edge of my seat for answers, -- 69.91.95.139 ( talk) 15:43, 5 April 2008 (UTC)
I deleted some material from the transfer principle section, as per discussion at talk:transfer principle. Katzmik ( talk) 10:03, 15 September 2008 (UTC)
I found the following comment in the article and I find it remarkable:
Is this true? Can it be documented? Is this referring to IST? Katzmik ( talk) 12:05, 15 September 2008 (UTC)
I think we should have a simpler summary of the properties to give the lay reader a feel for what's going on. Perhaps even more advanced readers (like me!) would find it helpful to get a feel for all the properties before getting stuck into the formalities. Then, an example of the use of hyperreals in proving an everyday calculus theorem (would the Quotient_rule be a candidate?) might possibly help too. We should also highlight the fact they are totally ordered, as opposed to something 'two-dimensional' like complex numbers, as I think this might help newbies to understand.
Update 2008-11-08: I've just retracted a more detailed comment I had made on this page, visible at [1]. There were two comments by me, and I'm losing confidence in the accuracy of that maths that I had written. I appreciate that it may be bad form to just delete comments, but nobody had replied to me, so I think this tidy up will help a little.
Aaron McDaid ( talk - contribs) 11:31, 8 November 2008 (UTC)
I think this article should be merged with non-standard analysis, for reasons given on that article's talk page.-- 76.167.77.165 ( talk) 16:44, 8 March 2009 (UTC)
User Alansohn reverted some of my edits. I've re-reverted and left a message on his talk page asking him to discuss it here.-- 76.167.77.165 ( talk) 17:30, 8 March 2009 (UTC)
Is the topology induced on hyperreals by open intervals of hyperreal length homeomorphic to real numbers? Why/why not? -- 129.116.47.169 ( talk) 20:53, 8 December 2009 (UTC)
The first paragraph of the article gives the impression that Newton and Leibniz used the hyperreal numbers, which is not supported by any evidence I'm aware of. They did use infinitesimals, and the sentence in question should be moved to the page Infinitesimal. In particular, the Hyperreals lack the least upper bound property, which is why they can be a consistent field with infinitesimal elements. It is not at all clear that Newton or Leibniz were aware of this requirement; indeed, the assumption that the real numbers had both infinitesimal elements and the least upper bound property is why their calculus is generally regarded as inconsistent (though groundbreaking and incredibly useful!). AshtonBenson ( talk) 19:47, 11 December 2009 (UTC)
is not a field extension of but an ultrapower.It's not a subfield , it's the same field but seen differently in two differents models. Field extension , it's for Galois theory , not for non-standard analysis.-- Titi2 ( talk) 11:07, 3 March 2010 (UTC)
There is a comment by Titi dating from yesterday that is a bit hard to find as it is buried among other editors' comments. The suggestion, again, is to replace a reference to a field extension, by an ultrapower. Now the introduction is certainly not set in stone and can be changed. The immediate difficulty with the suggested change is that a hyperreal field is not necessarily an ultrapower, as has been mentioned already. Tkuvho ( talk) 21:08, 10 April 2010 (UTC)
Here is a google translation of the French wiki: "In mathematics The hyperreal numbers are an extension of real numbers usual, to give a rigorous meaning to the concepts of infinitely small or infinitely large, technically, are often used Ultra Power to build this extension." Tkuvho ( talk) 21:39, 10 April 2010 (UTC)
![]() | This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 |
Apart from the difficulties that such a requirement creates, it is contrary to intuition. For example the sequence {0,1,0,1,0,...} may be greater than the sequence {2,0,2,0,2,0,....} (depending upon which sets of indices belong to the ultrafilter used). It would be much better to restrict the set of pairs of comparable sequences and define it in a way that does not contradict intuition. Leocat 15:10, 24 January 2007 (UTC)
It is stated that "(In fact, there are many such U, but it turns out that it doesn't matter which one we take.)". Can anyone provide some explanation as to why this is so? (Are all free ultrafilters on N isomorphic in some sense, or is there some other reason why the choice of U doesn't matter?) - Chinju 19:41, 6 Oct 2004 (UTC)
(In fact, this seems to conflict with the sentence at the top: "The use of the definite article 'the' in the phrase 'the hyperreal numbers' is somewhat misleading in that there is not a unique ordered field that is referred to, in most treatments.") - Chinju 19:50, 6 Oct 2004 (UTC)
A look at the provided external link seems to indicate that the ultrafilter chosen doesn't matter (the constructed hyperreals are isomorphic) if the continuum hypothesis is assumed, but can matter if the continuum hypothesis's negation is assumed. If no one argues otherwise, or that I've misinterpreted the reference, I'll modify the article accordingly. - Chinju 19:56, 6 Oct 2004 (UTC) --
I don't think epsilon-delta definitions are really that unintuitive, and they have the advantage of working entirely within the reals, where infinitesimals truly don't exist. But they're still very cumbersome and tend to give miraculous results that should be obvious with a better set up. The only formal construction I've seen of differentials is as members of a cotangent space, which is no help at all. I've heard of hyperreals but never seen a treatment - any chance you could augment the above with a formal construction and/or axiomatization for us less enlightened? Thanks!
There is an on-line article with a short description of such a beast:
http://www.math.vt.edu/people/elengyel/thesis/thesis.html I've read it but am not confident enough to wikipedify it. By the way, I've been searching for some more information on-line on this subject and guess where
http://www.google.com sends you.... that's right, to Wikipedia. :-) --
Jan Hidders
Differential forms living in cotangent spaces are not the same thing as
infinitesimals even though the notation may or may not be identical.
Sorry, my mistake, I thought you meant a description of the formal construction of hyperreal numbers. --
Jan Hidders
How can you tell if a sequence is a valid hyperreal? Here's a pair of sequences for which which is greater depends on which ultrafilter you use: (1/2, 1/4, 1/4, 1/6, 1/6, ...) and (1/3, 1/3, 1/5, 1/5, 1/7, ...). -
PierreAbbat
Because it is so easy to construct ordered fields that contain infinitesimals but will not serve the purposes of non-standard analysis, it would be a good idea to mention the special properties of this one that enable it to do so, i.e., the transfer principle. Maybe I'll add something on this if I get around to it. Michael Hardy 16:29, 2 Sep 2003 (UTC)
I would like to change the displayed formulas in this article to LaTex, unless there is a compelling reason not to. User:CSTAR
I think the construction of the hyperreals is wrongly attributed to Lindstrom. Probably Zakon is the originator of this idea.
Could we put the section on Hyperreal fields after the more elementary exposition? Yes fine we know if you mod a ring by a maximal ideal we get a field, but lets try to keep it elementary, at the beginning at least. It's OK, in my view, if you put the hyperreal fields section after and say this is a generalization. CSTAR 06:24, 23 Oct 2004 (UTC)
The most recent edits to this article have added a long paragraph on the process of extension of number "systems" (including fields, rings and semirings); the text of this paragraph belongs somewhere in a wikipedia article, preferably in another article specifically about extensions of number systems. That would be a useful article.
In addition, I disagree strongly with the claim (emphasis mine)
What is the accident of history? Is it really an accident of history that 1st order logic was developed after calculus? We don't need this kind of historical revisionism in explaining the development of mathematics. I suggest that the paragraph in question be removed or be thoroughly re-edited. CSTAR 15:34, 15 Mar 2005 (UTC)
The most recent edits adding a new section An intuitive approach to the ultrapower construction contain some confusing statements : For example, it seems to define an "infinitesimal sequence" as one containing a subsequence converging to zero (The specific quote is "...the true infinitesimals are the classes of sequences that contain a sequence converging to zero").This is clearly inadequate, since the sum of such sequences may not be infinitesimal. Moreover, the style seems to me to be unsuitable for an article. For example, the opening sentence of the section: "Here is the simplest approach to infinitesimals that I could think of. " -- CSTAR 14:12, 16 September 2005 (UTC)
I say "the classes of sequences that contain a sequence...," I never say "the sequences that contain a subsequence..." It is the classes that contain, not the sequences that contain. Please, don't misquote me. As for the style, the article originated as an e-mail to a friend, and I am sorry if it lacks in pompous and authoritarian air that some people expect from the articles in an encyclopedia. In any case, this place is free, anybody is welcome to make improvements.
Sir, I don't see anything eschatological about our discussion. You just misunderstood the sentence that was clear enough in my opinion and I have clarified it for you. I have read what you wrote, sir. I only used the word "subsequence" in reminding the Bolzano-Weierstrass, the expression "infinitesimal sequence" is not used at all. Your quotation is accurate, and I take it back that you misquoted, but you took this sentence out of context, have totally twisted and mangled what I wrote, probably in order to discredit it, probably because you didn't like my informal style. The sentence that you did not like is not a formal definition, it is a part of an explanation, and the next sentence says: "Let us explain where these classes come from." It is you who is not reading carefully and picking at minutia. My article is not a bunch of clinically dry definitions forced into a linearly ordered sequence. I am trying to show how one can arrive at these definitions, starting from some natural assumptions and simple observations. Definitions come first only in lousy textbooks. As Michael Atiyah says, "don't give them a definition, give them an example."
Hi there. Thank you for your contributions to hyperreal number. And a request. Would you mind making an account? It is rather hard for us to chase an ever changing identity whenever you change IP address. Also, it may confuse you with vandals, distracting people from the task of watching for the bad guys. It will be better for you too, as you can track your contributions better. What do you think?
On other matters, one should put a comma after "i.e.". According to the manual of style, one better even use "that is" instead. Thanks. Oleg Alexandrov 16:11, 17 September 2005 (UTC)
Hi Michael. I was not offended, but I could have. It seems you don't know that well what tzar means, and the history of that part of the world I come from. But never mind, I lived enough in US that I know what you mean.
Again, thank you for your contributions. I will not interfere with what you write as I have plenty of other things to take care of. One piece of advice. Your comments both to CSTAR and to me were rather combative. I think you need to mellow out a bit if you plan to continue contributing to this encyclopedia. Wikipedia is a place where for better or worse everybody steps on each other's foot all the time, and interpersonal skills are very important (that is, it is not enough that you have a good point, you also need to know how to drive it home without irritating yourself or others around you). In the future, if you have comments and questions, you can use my talk page. Oleg Alexandrov 15:31, 20 September 2005 (UTC)
It seems to me that the section titled "An intuitive approach to the ultrapower construction" is almost completely redundant with the preexisting section titled "The ultrapower construction." IMO, either the "intuitive approach" section should be deleted, or the two sections should be merged.-- Bcrowell 05:07, 4 January 2006 (UTC)
At the bottom, there is a couple links that it says to "compare with". These are: Surreal numbers, Superreal numbers, and Real closed fields. I think that it would be very useful if there was a section comparing the three for the reader. It is very hard to tell the diference between three sets that all contain reals, and all contiain infintesimals and infinities. Fresheneesz 23:52, 10 February 2006 (UTC)
Hello
I have a comment regarding what I believe to be an incorrect statement in the "Hyperreal number" article. In "The ultrapower construction", near the end of the section, one finds the following sentence (referring to the hyperreal field constructed in that section):
As a real closed field with cardinality the continuum, it is isomorphic as a field to R but is not isomorphic as an ordered field to R.
I do not believe this is the case. More precisely, I don't think that the hyperreal field constructed there is isomorphic to R. One way to see this, for example, would be to notice that the hyperreal field contains an element (any of the so-called "infinitely large numbers", for example) x such that for every positive integer n, x-n is a square. The existence of an isomorphism between the hyperreal field and R would imply (since an isomorphism would, of course, map 1 into 1 and n into n) the existence of an element of R with the same property. This is absurd.
Am I missing something obvious here?
P.S.
I cannot remember where, but I have seen this comment on Wikipedia before (the statement that two real closed fields of continuum cardinality are isomorphic, but, perhaps, not isomorphic as ordered fields). As noted above, this is false. Moreover, the distinction between an isomorphism of fields and an isomorphism of ordered fields doesn't make sense for real closed fields: an isomorphism maps squares into squares, and in a real closed field the non-negative elements are precisely the squares; this means that any field isomorphism will automatically be increasing and hence an order isomorphism as well.
192.129.3.135
21:00, 29 November 2006 (UTC) Alexandru Chirvasitu
I do not believe that using hyperreal numbers miraculously makes every derivative continuous, as it is stated in the article. In standard analysis it is of fundamental importance whether x approaches a, or a approaches x. The definition given in the article obliterates this distinction. Obviously e.g. the derivative of the function f(x) = sin(1/x^2)*x^2 for non-zero x, 0 at 0, is not continuous at 0. Leocat 08:49, 24 January 2007 (UTC)
Is "S-differentiable" supposed to mean "differentiable according to standard analysis"? If so, then my example contradicts your statement. Leocat 12:34, 25 January 2007 (UTC)
If h is a non-zero infinitesimal, then L = 0 satisfies the condition of being the value of the S-derivative of my f at 0. Actually I do not see why you state that S-differentiability is different from standard differentiability. Leocat 17:40, 25 January 2007 (UTC)
Let g(x)=|x|. Is g S-differentiable at values y infinitely close to 0? What is the definition of uniform differentiability? Is L in the definition of S-differentiability a real number? Leocat 20:45, 25 January 2007 (UTC)
Since L is to be a hyperreal number, I wonder about examples when L is not a real number. Are there standard functions, whose S-derivatives at standard points take on e.g. infinite hyperreal values? Leocat 22:04, 25 January 2007 (UTC)
I was going to add this bit about the hyperreals not forming a complete set, but I wanted to make sure my leg-work was correct; it's been a while.
A large weakness of the hyperreal field in analysis, however, is that the set of all hyperreal numbers is not
complete in the sense of a
metric space, since the existence of the hyperreals gives rise to hyperhyperreals, and thus it is impossible for a Cauchy sequence of hyperreals to converge to a unique point (since we can just define an infinitesimal number next to our real number to which the Cauchy sequence also converges).
134.39.100.70
19:47, 6 March 2007 (UTC)
Here is something that I have been thinking about:
This is fact according to the article:
0 = infinitesimal
x = infinite
1/x = infinitesimal
1/x = 0
0*x = 1
1/0 = x
That makes this possible:
(1/x)/(1/x) = 1
0^(-1) = 1
0/0 = 1
But if 0/0 = 1, then why does my calculators give me results as "divide by zero error", "undefined", etc?
—The preceding
unsigned comment was added by
161.52.141.39 (
talk) 08:00, 9 March 2007 (UTC).
Re: This is fact according to the article:
0 = infinitesimal
No, the article doesn't say this (unless it's been edited very recently to make that claim).--
CSTAR
14:51, 9 March 2007 (UTC)
I didn't get this:
Uh? How can a field F be isomorphic as a field to R but not isomorphic as an ordered field? If f: F -> R is a field isomorphism, then if x in F and x > 0 and f(x) < 0, then take y such that f(x) = -y^2, apply f^(-1) and get x = -(f^(-1)(y))^2 < 0, absurd. So, f must preserve order too. Albmont 12:45, 30 April 2007 (UTC)
Is this a valid construction of the hyperreals?
This way we might get the hyperreals even without the axiom of choice. Albmont 00:49, 4 May 2007 (UTC)
This is similar to, but not quite the same as, the polynomial ratio construction I added in Jan, 2007, and removed in Feb. because it was unpublished original research. When and if that work is published in a citable journal, I intend to re-contribute it. Basically, I am asserting that the non-Archemedian field H of real polynomial ratio (" rational function") sequences, the nth terms of which are those functions of the natural number n, where any statement about such ratios holds in H if it holds in R for all but a finite number of values of n, also models the transfer of first-order statements between H and R, and that this model, with the cofinite filter common to all nontrivial ultrafilters, does not require the axiom of choice, and should be much simpler to grasp than the ultrapower construction. Alan R. Fisher 05:50, 13 May 2007 (UTC)
Can anyone explain in more colloquial terms the comparison of two Hyperreals? I don't really have the background to understand all this ultrafilter stuff. Here's my understanding; am I correct, almost correct, or completely wrong? What's the correct description?
Two Hyperreals A and B are represented by their terms A1, A2, A3, etc and B1, B2, B3, etc. A≤B if and only if there exist an infinite number of terms n such that An≤Bn. All comparisons are defined as logical combinations of the ≤ operator. For example, A>B if and only if B≤A and not A≤B.
Obviously, this can lead to problems. For example, one can build a scenario that defies the transitive property of equality:
A = (2, -2, 2, -2, 2, -2...), B = (0, 0, 0, 0, 0, 0, 0...), C = (1, 1, 1, 1, 1, 1, 1...)
A≤B and B≤A, so A=B.
A≤C and C≤A, so A=C.
Therefore, by transitive property, B=C, an obvious contradiction, since the statement C≤B is false. This may be prevented by not allowing such sequences as A, but then one has to figure out how to make the proper distinction between a non-converging sequence like A, and an infinite sequence, like (1, 2, 3, 4...).
Additionally, Hyperreals that converge can be exactly equal to what they converge onto (not an infinitesimal smaller or larger). Though not a contradiction, this seems to be slightly against the philosophy of Hyperreals. For example (1, 1-1/3, 1-1/3+1/5, 1-1/3+1/5-1/7...) = π/4 exactly by my definition of equality.
Eagerly waiting on the edge of my seat for answers, -- 69.91.95.139 ( talk) 15:43, 5 April 2008 (UTC)
I deleted some material from the transfer principle section, as per discussion at talk:transfer principle. Katzmik ( talk) 10:03, 15 September 2008 (UTC)
I found the following comment in the article and I find it remarkable:
Is this true? Can it be documented? Is this referring to IST? Katzmik ( talk) 12:05, 15 September 2008 (UTC)
I think we should have a simpler summary of the properties to give the lay reader a feel for what's going on. Perhaps even more advanced readers (like me!) would find it helpful to get a feel for all the properties before getting stuck into the formalities. Then, an example of the use of hyperreals in proving an everyday calculus theorem (would the Quotient_rule be a candidate?) might possibly help too. We should also highlight the fact they are totally ordered, as opposed to something 'two-dimensional' like complex numbers, as I think this might help newbies to understand.
Update 2008-11-08: I've just retracted a more detailed comment I had made on this page, visible at [1]. There were two comments by me, and I'm losing confidence in the accuracy of that maths that I had written. I appreciate that it may be bad form to just delete comments, but nobody had replied to me, so I think this tidy up will help a little.
Aaron McDaid ( talk - contribs) 11:31, 8 November 2008 (UTC)
I think this article should be merged with non-standard analysis, for reasons given on that article's talk page.-- 76.167.77.165 ( talk) 16:44, 8 March 2009 (UTC)
User Alansohn reverted some of my edits. I've re-reverted and left a message on his talk page asking him to discuss it here.-- 76.167.77.165 ( talk) 17:30, 8 March 2009 (UTC)
Is the topology induced on hyperreals by open intervals of hyperreal length homeomorphic to real numbers? Why/why not? -- 129.116.47.169 ( talk) 20:53, 8 December 2009 (UTC)
The first paragraph of the article gives the impression that Newton and Leibniz used the hyperreal numbers, which is not supported by any evidence I'm aware of. They did use infinitesimals, and the sentence in question should be moved to the page Infinitesimal. In particular, the Hyperreals lack the least upper bound property, which is why they can be a consistent field with infinitesimal elements. It is not at all clear that Newton or Leibniz were aware of this requirement; indeed, the assumption that the real numbers had both infinitesimal elements and the least upper bound property is why their calculus is generally regarded as inconsistent (though groundbreaking and incredibly useful!). AshtonBenson ( talk) 19:47, 11 December 2009 (UTC)
is not a field extension of but an ultrapower.It's not a subfield , it's the same field but seen differently in two differents models. Field extension , it's for Galois theory , not for non-standard analysis.-- Titi2 ( talk) 11:07, 3 March 2010 (UTC)
There is a comment by Titi dating from yesterday that is a bit hard to find as it is buried among other editors' comments. The suggestion, again, is to replace a reference to a field extension, by an ultrapower. Now the introduction is certainly not set in stone and can be changed. The immediate difficulty with the suggested change is that a hyperreal field is not necessarily an ultrapower, as has been mentioned already. Tkuvho ( talk) 21:08, 10 April 2010 (UTC)
Here is a google translation of the French wiki: "In mathematics The hyperreal numbers are an extension of real numbers usual, to give a rigorous meaning to the concepts of infinitely small or infinitely large, technically, are often used Ultra Power to build this extension." Tkuvho ( talk) 21:39, 10 April 2010 (UTC)