Talk:Hyperreal number

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[edit] Should all sequences be comparable?

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)

[edit] Junk at the top

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) --

There is a saturation condition which which guarantees that reduced ultrapowers of superstructures re isomorphic. I think this result may be in a Keisler article in a collection published in 1986 by London University Press (one of the blue book series). I don't have the reference available at the moment. I'm not familiar with the other result you mentioned (However, my reaction to something which is true mod continuum hypothesis is ... yuck)CSTAR 21:55, 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!

Heck, irrational numbers don't exist. I've never measured something with a ruler, or a scale, or a stopwatch, and gotten an infinite decimal expansion that never repeats :-) --Bcrowell 07:00, 16 Mar 2005 (UTC)



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

If you choose to define the hyperreals using the ultrapower method (which is not the only way to do it), then every sequence is a valid hyperreal; there aren't any invalid ones. In your example, you're right, the comparison depends on the ultrafilter. Since an ultrafilter can't be explicitly constructed, it doesn't actually matter. Nothing you calculate using hyperreals depends on the ability to construct them as specific sequences. --Bcrowell 07:00, 16 Mar 2005 (UTC)




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.

I think you're right. Zakon is certainly the originator of the superstructure approach to NSA, and I'm almost certain Lindstrom had nothing to do with either (although he did figure among the coauthors of a very important book on NSA). see nonstandard analysis. CSTAR 22:28, 27 Jul 2004 (UTC)

[edit] Hyperreal fields

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)

[edit] Editorializing on the history of mathematics and other oddities

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)

"Although the use of the infinitesimals predate the reals by some 170 years, by an accident of history, the tools for formalizing their treatment in terms of set theory and formal logic were developed earlier than the tools for doing the same with the infinitesimals and the other hyperreals (1870 versus 1960)."

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)

You're right. It was bad. I've deleted it. --Bcrowell 07:00, 16 Mar 2005 (UTC)

[edit] Recent edits

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.

If you actually read what I wrote, I did not misquote you: In fact I included in italics and in quotation marks what you actually said. Now granted you did say classes, but you haven't defned at this point in your exposition what classes you are referring to. I did precede your quote with a paraphrase of your quote, suggesting that this is what it seems to say.
If we are going to need an eschatological discussion of what is written down in the article, I think we might be better off with a pompous and authoritarian style.--CSTAR 16:49, 17 September 2005 (UTC)

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."

[edit] Note to the anonimous contributor

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)

I've just made an account. As for the grammer, feel free to edit, but I will appreciate if you do it without mutilating the meaning. michaelliv 03:28, 18 September 2005 (UTC)
And by the way, comrade commissar, using a comma after i.e. is not mandatory, it is a matter of whether i.e. is used as an introductory modifier, that is not always the case. At London University they recommend not to use periods either, ie, to write it just like this: ie. As for mathematics, I would rather take ie in a long chain of obvious implications or before a simple reminder, rather than combersome and totally uninformative "or, to put is the other way," "in other words," or even "that is," that merely dilutes the text and distracts the mind.
As for editing, I don't think it is fruitful to hack at the piece written by somebody else just to shoehorn it into your idea of how mathematics should be written or explained. After all, somebody may understand it better when it is written or explained the way you don't like. Let's show some tolerance and respect. The uniformity of style will not make the encyclopedia better, it will make it worse. Making some minor corrections or additions that do not distort the original ideas of the author is useful, but let's not be arrogant. If you think you can do it better, write your own piece, and let the readers have their own opinions. michaelliv 17:00, 18 September 2005 (UTC)
Wikipedia is a place where anybody can edit and specific disagreements need to be stated explicitely and dealt with on the talk pages. Ambiguous comments of the form "I don't think it is fruitful to hack at the piece written by somebody else" are not helpful. Oleg Alexandrov 20:22, 18 September 2005 (UTC)
Oh, please, read the rest of the sentence! Don't resort to the low debating tactics! And most of all, please, don't play a tzar here! 206.15.138.85 04:42, 19 September 2005 (UTC)
Tzar Oleg! If this weren't an ethnic slur, it might be funny. I suggest you find another outlet for your humor.--CSTAR 04:52, 19 September 2005 (UTC)
Man, since when did tzar become an ethnic slur? I've totally missed it! You must know better, Mr. Shadowy Deputy of Mathematical and Political Correctness. Please, write an article about it. michaelliv 05:19, 20 September 2005 (UTC)
I apologize if my comment seemed out of place and unwarranted. Thanks.--CSTAR 15:58, 20 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)

Please, don't worry, I know quite well what tzar means, in both languages and in both countries, for I lived in both of them longer than you. If you had got offended, I would have considered it your problem, not mine. In any case, it was CSTAR who tipped me off with his rather uncivilized remarks that he made without even bothering to read carefully what I wrote. As for you, commenting on half a sentence is not a good practice. Messing up someone's writing is not a good practice. Playing know-it-all and patronizing the people who contribute is not a good practice. And last, but not least, bickering with you, guys, is not my favorite pastime, I'd rather contribute some more to this encyclopedia. michaelliv 01:27, 21 September 2005 (UTC)

[edit] redundant sections

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)

[edit] Other sets containing infintesimals and infinities

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)

[edit] incorrect statement?

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

Seconded. As pointed out above, the order is definable in the field language - so an isomorphism of real closed fields as fields is necessarily an isomorphism of ordered fields. A correct statement would be that the ultraproduct is elementarily equivalent to the reals, but not isomorphic. But perhaps it would be better to remove the sentence entirely. Mbays 11:43, 10 December 2006 (UTC)

[edit] Continuity of derivatives

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)

What is in the article is obvious nonsense. The correct statement is that if f is S-differentiable near x (that is for values of y infinitely close to x) then the S-derivative of f is S continuous at x. The value of that entire section is dubious.--CSTAR 15:52, 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)

No. It means something more like "uniformly differentiable". Formally, f is S-differentiable at x is a different concept: it means there is an L such that for all non-zero infinitesimal h
\frac{1}{h}(f(x+h)-f(x)) \cong L
Your example is obviously not S differentiable at near 0.--CSTAR 14:13, 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)

The function in your example is not differentiable near 0 (Yes, you are correct, I erroneously said not differentiable at 0 and I've corrected myself). Please note the following facts:
  • S-differentiability for standard functions at a standard value is equivalent to differentaibility.
  • S-differentiability of a standard function on a standard interval is equivalent to uniform differentiability.
  • S-differentaibility near a value has no standard equivalent.
The proposition I stated earlier viz
If f is S-differentiable at values of y infinitely close to x then the S-derivative of f is S continuous at x
is entirely trivial.
--CSTAR 18:01, 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)

Re: Is g S-differentiable at values y infinitely close to 0?.
No.
Fact. For any positive infinitesimal value a, f is not S-differentiable at a.
Proof: Consider two values of x infinietly close to a:
Let x= −a, thus x − a = − 2 a is infinitesimal.
\frac{f(x) -f(a)}{x-a} = \frac{|-a| -|a| }{-a-a} = \frac{a -a}{-2 a} = 0
Let x= 0, x − a is infinitesimal.
\frac{f(x) -f(a)}{x-a} = \frac{0 -|a| }{0-a} = \frac{ -a}{-a} = 1
Thus there is no hyperreal L such that
\frac{f(x) -f(a)}{x-a} \cong L
for all x infinitely close to a as claimed.
A similar argument proves:
Fact. f is not S-differentiable at 0.
Definition: f is uniformly differentiable iff for a and for all ε >0 there is a δ >0 such that
\bigg|\frac{f(x) -f(a)}{x-a} -f'(a)\bigg| \leq \epsilon
if x is such that
| x − a | < δ
--CSTAR 21:32, 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)

[edit] Complete?

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)

[edit] Something to think about

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)

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