# Talk:Hyperreal number

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## 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)

## 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)

## 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)

## 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)

## 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."

## 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)

## 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)

## 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)

## 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)

## 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
${\displaystyle {\frac {1}{h}}(f(x+h)-f(x))\cong L}$
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 xa = − 2 a is infinitesimal.
${\displaystyle {\frac {f(x)-f(a)}{x-a}}={\frac {|-a|-|a|}{-a-a}}={\frac {a-a}{-2a}}=0}$
Let x= 0, xa is infinitesimal.
${\displaystyle {\frac {f(x)-f(a)}{x-a}}={\frac {0-|a|}{0-a}}={\frac {-a}{-a}}=1}$
Thus there is no hyperreal L such that
${\displaystyle {\frac {f(x)-f(a)}{x-a}}\cong L}$
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
${\displaystyle {\bigg |}{\frac {f(x)-f(a)}{x-a}}-f'(a){\bigg |}\leq \epsilon }$
if x is such that
${\displaystyle |x-a|<\delta }$
--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)

## 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)

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)

## Isomorphic as a field to R

I didn't get this:

in which case F is isomorphic as a field to R, but is not order isomorphic to R.

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)

Yes, the claim in the article is incorrect, as has been discussed before. Someone tried to remove the claim from the article, but missed a second occurrence of it. I've now removed it completely (I hope). --Zundark 12:35, 6 July 2007 (UTC)

## Another construction

Is this a valid construction of the hyperreals?

Define, in the ring of polynomials F[x], that a non-zero polynomial is positive if the highest-order term is positive
This makes F[x] into an ordered ring. Also, F[x] is an integral domain
So, the field of quotients of F[x], F(x) is an ordered field
The order in F(x) induces a topology, the order topology
Now this is the tricky part. I don't know if there is a generalization of completion (topology) that works for something that is not a metric. If there is this analogue, then the completion of this topological field would be a set of hyperreals, since x would play the role of an infinitely big hyperreal and 1/x would be an infinitesimal.

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)

I doubt the accuracy of that as a construction; we'll see what it actually says, once it's published. — Arthur Rubin | (talk) 16:33, 13 May 2007 (UTC)
On the other hand, R(((ε))) may very well work for most purposes.... — Arthur Rubin | (talk) 16:35, 13 May 2007 (UTC)
What is the square root of x? Tlepp 08:19, 8 November 2007 (UTC)
• Why not compute it the same way we compute the square root of 2? Start with the approximation a1 = 1. This is too little, so take the average of a1 and x / a1, and we get ${\displaystyle a_{2}={\frac {x+1}{2}}\,}$. This is too much, so take ${\displaystyle a_{3}={\frac {1}{2}}(a_{2}+{\frac {x}{a_{2}}})\,}$. So we get a sequence ${\displaystyle a_{n+1}={\frac {1}{2}}(a_{n}+{\frac {x}{a_{n}}})\,}$. The tricky part (which I don't know if it's ok or not) is to prove that it converges. If it converges, then its limit would be ${\displaystyle a={\frac {1}{2}}(a+{\frac {x}{a}})\,}$, or a2 = x. Albmont (talk) 12:28, 16 April 2008 (UTC)
I'll ignore the question of whether this series actually converges (I have suspicions that it doesn't, but don't care enough to try); it is, however, relatively easy to show, on this very hint, that Albmont's particular construction does not actually represent the hyperreals. (Assuming I'm interpreting it correctly anyway.)
Specifically, suppose there is a Laurent series (or rational function, or inverse Laurent series) expansion for the square root of x. Then, by a simple substitution, we could change the formula to represent the square root of -x. Similarly, if one of these roots does not exist then neither does the other.
Meanwhile, there is a formula on R that states that for every nonzero a, exactly one of a and -a has a square root in R; it is easily written in a logical form, so must thus hold in the actual hyperreals by the transfer principle. And we just saw that it is not true for a=x in this construction, so this construction is not the hyperreals, Q.E.D. Is that proof enough? --85.141.137.223 (talk) 00:32, 29 June 2014 (UTC)
(No, it doesn't converge). However, repeating my comment of 16:35, 13 May 2007, the Levi-Civita field (or some variant such as a superset such as the field of Hahn series, or a subset such as the field of Puiseux series), at least form non-Archimedean real-closed fields. — Arthur Rubin (talk) 09:30, 29 June 2014 (UTC)

## Defining order (section "The ultrapower construction")

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'm afraid not. Although this may not be the appropriate venue for discussion, it may help clarify the text, so I'll comment further.
• ${\displaystyle A\leq B\leftrightarrow \{i|A_{i}\leq B_{i}\}\in U}$
Where U is a free ultrafilter. Leting U be the set of all infinte subsets leads to failure of basic properties of the hyperreals, as the anon notes above. — Arthur Rubin (talk) 19:19, 5 April 2008 (UTC)
Does "I'm afraid not" refer to my question "Can anyone explain in more colloquial terms the comparison of two Hyperreals?" Answering that question is absolutely relevant here, since such a description could be integrated into the article for those (like me) less familiar with such concepts as "Ultrafilters". --69.91.95.139 (talk) 19:51, 5 April 2008 (UTC)
"I'm afraid not" means I can't answer it, except to say that you're not close to what the article says or the intent. Another way of looking at it is that XU can be looked at as an abstraction of "almost all natural numbers are in X", which satisfies:
1. It is not the case that almost all numbers are in the empty set.
2. If almost all numbers are in X, and X is contained in Y, then almost all numbers are in Y.
3. If almost all numbers are in X, and almost all numbers are in Y, then almost all numbers are in the intersection of X and Y.
4. If it is not the case that almost all numbers are not in X, then almost all numbers are in X.
Properties 1-3 correspond to U being a filter, and 1-4 correspond to U being an ultrafilter. See ultraproduct as to why this is important. — Arthur Rubin (talk) 22:41, 5 April 2008 (UTC)
Ok, I'm starting to get a vague understanding of the whole Ultrafilter thing. My next question: why doesn't it matter which Ultrafilter we use? Either A above is B or it's C (or another number between -2 and 2), and the Ultrafilter decides which. Obviously, it has a significant impact on such comparisons. If, as I think is the case if I understand correctly, the Ultrafilter chosen does matter for the above comparison, why does the page say it doesn't?
I'm sorry if I seem a little insistent with these questions; I am here to try and improve this page so noobs who read it in the future can have less of a struggle figuring out the tricky bits than I did. I appreciate your efforts to help my understanding; I really do. --69.91.95.139 (talk) 23:14, 5 April 2008 (UTC)
(Yes, A is either 2 or -2, depending on the ultrafilter.)
I find your questions challenging. I learned this material in model theory many years ago, but I think this is the first time I've attempted to explain it. If we (Wikipedians) can simplify the explanation, it would improve the article.
Which page says the ultrafilter doesn't matter? I'd like to check that. The general properties of the ultraproduct don't depend on which ultrafilter is used, but the specifics might, unless they are isomorphic (in the appropriate sense). — Arthur Rubin (talk) 01:50, 6 April 2008 (UTC)
In parentheses, under the section "The ultrapower construction", it says "The good news is that the axiom of choice guarantees the existence of many such U, and it turns out that it doesn't matter which one we take; the bad news is that they cannot be explicitly constructed." I know that the value of certain Hyperreals like the one above depends on the Ultrafilter chosen, so it does matter which one we take.
Let me make sure I understand the concept of a 'free' Ultrafilter correctly. If all sets found in an Ultrafilter contain a particular element then it is a principal Ultrafilter. I can see how an Ultrafilter on a finite set would necessarily be principal by that definition. However, I know there must be more than just that to it, because otherwise, the definable Ultrafilter on the Natural numbers characterized by preferring infinite subsets over finite subsets, and then preferring subsets containing some number n, would be a free Ultrafilter. Can you possibly explain why such a definition would not be a free Ultrafilter?
Thanks! --69.91.95.139 (talk) 13:08, 6 April 2008 (UTC)
I'll have to look into "it doesn't matter". Thanks.
The construction you describe turns out not to be a filter, I believe, although I'm not sure exactly what you're saying. — Arthur Rubin (talk) 16:43, 6 April 2008 (UTC)
I'll explain my construction in more detail.
For every subset S of the natural numbers, and its relative complement T, building the Ultrafilter U, with n being the preferred natural number:
If S is finite in size, then T∈U
If T is finite in size, then S∈U
If both are infinite in size and n∈S, then S∈U
If both are infinite in size and n∈T, then T∈U
I hope that makes more sense. --69.91.95.139 (talk) 18:04, 6 April 2008 (UTC)
No, that's not an ultrafilter. Let A be the union of the set of even numbers with {n} (that is, start with the even numbers, and throw n in if it's not already there). Let B be the union of the set of odd numbers with {n}. Now by your rules A and B must both be in the ultrafilter. But then the intersection of A and B must be in the ultrafilter. But the intersection is just {n}, which is a finite set, so by your rules it can't be in the ultrafilter. Contradiction. --Trovatore (talk) 20:27, 6 April 2008 (UTC)
Hey, same person here, just got an account. Since my construction is not even an ultrafilter, can you (anyone), by any chance, provide an example of an explicit construction of a principle ultrafilter on the Natural Numbers? Or is such a construction impossible?
As I've said before, I'm still struggling with the concepts. I appreciate your patience. --Onto09 12:33, 7 April 2008 (UTC)
The article ultrafilter#Types and existence of ultrafilters suggests the following:
• ${\displaystyle U_{n}=\{X\subseteq \mathbb {N} |n\in X\}}$
which turns out to work. — Arthur Rubin (talk) 12:58, 7 April 2008 (UTC)
Just by the way, it's spelled principal ultrafilter. --Trovatore (talk) 16:32, 7 April 2008 (UTC)

## deletion

I deleted some material from the transfer principle section, as per discussion at talk:transfer principle. Katzmik (talk) 10:03, 15 September 2008 (UTC)

## no model theory??

I found the following comment in the article and I find it remarkable:

Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic.

Is this true? Can it be documented? Is this referring to IST? Katzmik (talk) 12:05, 15 September 2008 (UTC)

I'm not sure if this answers your question, but ... The ultrafilter approach is more constructive than Robinson's original treatment, but it does depend on the existence of an ultrafilter, which can't be explicitly constructed. I don't understand the remark that the transfer principle can be proved "as a consequence of the definitions," if "the definitions" refers to the usual axioms of the real number system such as commutativity and so forth. The axioms make statements within a certain system such as the reals or hyperreals; the transfer principle relates one such system to another. I don't see how the transfer principle can be stated without referring to first order logic, since it fails for some statements that are outside of first order logic. It's true that you don't need to know a lot of model theory to understand nonstandard analysis, but the transfer principle is a statement about two different models. At the very least, I think this statement should be clarified, and a reference should be cited.--76.167.77.165 (talk) 18:53, 31 October 2008 (UTC)

## Simple summary of properties near the start of the article

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)

## merge with non-standard analysis

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)

## revert by Alansohn

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)

## Homeomorphic to real numbers?

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)

Covering a closed interval [a,b] by open intervals of infinitesimal length produces a cover which does not contain a finite subcover. Tkuvho (talk) 07:07, 9 December 2009 (UTC)

## Newton and Leibniz knew about Ultraproducts?

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)

Well, the language wasn't as clear as it could have been, but I read it as "infinite and infinitesimal numbers had been widely used ... by Newton and Leibniz ...", and the hyperreals represent a way of treating these quantities precisely, not that N and L used hyperreals specifically. I note that your new language has the same problem, if it is a problem, as it claims that the quantities "had been in widespread use", which also is not true of the hyperreals before they were described. --Trovatore (talk) 07:44, 12 December 2009 (UTC)

## Ultrapower

${\displaystyle ^{*}\mathbb {R} }$ is not a field extension of ${\displaystyle \mathbb {R} }$ 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)

The link to ultrafilter may not be the most appropriate one. Internally speaking, R* is the "same" field as R, in the sense that the first order properties are the same. However, externally, R* is an extension of R. This is clearly not a side remark but an essential part of the story. The standard part function is not internal, but is an essential tool in establishing the foundations of analysis. Tkuvho (talk) 12:48, 3 March 2010 (UTC)
The introduction should contain an elementary introduction to R*, avoiding technical terms. Therefore I would prefer to change back to a field extension, which is a more familiar term than an ultrapower, to a non-specialist reader. Tkuvho (talk) 14:21, 3 March 2010 (UTC)
I don't agree.The topics must be accurate and then we have to explain it to non-familiar reader.I think ultrapower is better than field extension because field extension referrs to the galois theory and may induce the reader in error--Titi2 (talk) 10:13, 9 April 2010 (UTC)
There is no sense whatsoever in which R* is the same field as R. Titi2, you have a very profound misunderstanding on this point. The first-order properties of a structure do not determine the structure. --Trovatore (talk) 20:59, 3 March 2010 (UTC)
I also prefer field extension; Tkuvho's point is well-taken, and in addition I don't think it's actually necessary that R* be an ultrapower of R. I'd have to review the literature to be sure whether the term the hyperreals is specific to the ultrapower construction. A priori, though, it seems that it should be sufficient that R* be an elementary extension of R realizing a certain type. --Trovatore (talk) 21:22, 3 March 2010 (UTC)
If you're going to take an axiomatic approach to the hyperreals, I think what you need is that R* is a nontrivial elementary extension of R, where the language of the structure R has a name for every n-ary real function (nontriviality ensures that the infinite and infinitesimal types are realized, since R is order-complete). This is the approach taken by Keisler in his textbook, though he also notes that the ultraproduct construction gives a structure satisfying these axioms. He needs the full language so he can use the natural extensions to R* of arbitrary real functions to talk about continuity and differentiability of those functions. Algebraist 21:53, 3 March 2010 (UTC)
Cool, yes, I forgot the stipulation about the language. Did Keisler call the elements of such an elementary extension the hyperreals? --Trovatore (talk) 22:24, 3 March 2010 (UTC)
All I'm looking at is his undergraduate textbook, which is rather vague on the matter. He presents it as being like the situation with the reals, in that we have the axiomatic approach (which he seems to take to be the definition of the reals, and possibly of the hyperreals) backed up by the constructive approach (which he seems to treat mostly as a source of reassurance that the axiomatic approach has some content). As far as I can see, he ignores the fundamental problem that (unlike with the reals) his axioms for the hyperreals fail to determine them up to isomorphism. Algebraist 23:01, 3 March 2010 (UTC)
As you correctly pointed out, the definite article may be inappropriate as the model is not unique (except special circumstances, such as CH and cardinal=${\displaystyle \aleph _{1}}$). On the other hand, certain aspects of it are unique, such as the first order theory. The expression "the hyperreals" may be common in the literature even though the definite article is somewhat problematic. Incidentally, Keisler's companion volume on the foundations (also available at his homepage) gives considerably more information than the textbook. Tkuvho (talk) 09:54, 4 March 2010 (UTC)
Right. In that volume, Keisler uses "a hyperreal number system" for anything satisfying the axioms I gave above, and reserves "the hyperreal number system" for the unique-up-to-isomorphism structure that satisfies those axioms, is saturated, and has size the first inaccessible cardinal. Algebraist 15:20, 4 March 2010 (UTC)

## ultrapower or field extension?

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)

As far as I can tell, Titi2 is just wrong. As you say, the lead can change, but I don't think there is any reason to change it at all on the basis of Titi2's complaint.
Here's what Lang says about the term extension field:
Let F be a field. If F is a subfield of a field E, then we also say that E is an extension field of F.Algebra, 2nd Edition, p. 265.
Lang doesn't mention field extension per se. I think extension field is what I have heard more often, and I would support replacing field extension by extension field. But that's as far as I'd go. --Trovatore (talk) 21:52, 11 April 2010 (UTC)
I agree with you. As I think you mentioned earlier, Titi is influenced by the viewpoint of Nelson's internal set theory, where the usual set-theoretic axioms are modified in such a way that the usual construction of R does indeed produce what is recognizably the hyperreal line. Still, the ultrapower construction is one of the most accessible ones (and was historically the first: Hewitt 1948). It might be worth mentioning it in the lead somehow (the French wiki does, see Titi's comments there). Tkuvho (talk) 07:55, 12 April 2010 (UTC)
Hmm — no, actually I hadn't really considered that he might have been influenced by IST. I don't really know much about IST, but if that is the case, then that might reflect somewhat better on Titi2 than I had thought. I thought it was just some sort of naive version of formalism, making him think that elementarily equivalent structures are really the same thing. --Trovatore (talk) 09:18, 12 April 2010 (UTC)
One of the editors at the French wiki pointed out the IST connection. At any rate, it would go too far afield to elaborate on IST in the lead here, but perhaps something could be mentioned about u.p. Tkuvho (talk) 09:22, 12 April 2010 (UTC)

## oversimplification

in the subsection on the "intuitive approach", the following claim is made: "the true infinitesimals are the classes of sequences that contain a sequence converging to zero." This is an oversimplification that will not always be true. Depending on foundational assumptions, certain infinitesimal classes may not be representable by sequences tending to zero. Tkuvho (talk) 13:01, 23 June 2010 (UTC)

The sentence "The infinitesimals can be represented by the non-vanishing sequences converging to zero in the usual sense, that is with respect to the Fréchet filter" is just plain wrong. Tkuvho (talk) 13:08, 23 June 2010 (UTC)

## Connection to P-point filter

The phrase "However, there may be infinitesimals not represented by null sequences; see P-point" was deleted in a recent edit. Why was it deleted? Tkuvho (talk) 11:51, 7 May 2013 (UTC)

What does P-point have to do with it. An infinitesimal in an ultra-power of R or Q (or any separable space) must, considered as a sequence, have zero as a limit point. It doesn't necessarily have to be "null" (converge to 0). On the other hand, a sequence converging to 0 does have to be infinitesimal (or 0). — Arthur Rubin (talk) 00:11, 8 May 2013 (UTC)
I see you said something else above in 2010. Could you explain? If U is a non-principal ultrafilter over N, then an infinitesimal in RU must have (in RN) a subsequence (with coordinate set in U) converging to 0; conversely if a sequence in RN converges to 0, and has no zero components, then it is an infintesimal in RU. — Arthur Rubin (talk) 00:17, 8 May 2013 (UTC)
This is assuming classical logic; I don't know how ultrafilters work in intuitionistic logic. — Arthur Rubin (talk) 00:19, 8 May 2013 (UTC)
The background logic is classical. The question is whether each infinitesimal is representable by a null sequence. In other words, whether a subsequence can be chosen which is supported on a member of the ultrafilter. For this to be true requires a special type of ultrafilter namely P-point (whose existence cannot be proved in ZFC). Tkuvho (talk) 11:19, 8 May 2013 (UTC)
Never mind. You're right. The sequence (an) corresponds to an infinitesimal iff
${\displaystyle (\forall \epsilon >0)(\left\{n\mid ||a_{n}|<\epsilon \right\}\in U).}$
It does follow that (an) has a null subsequence, but it doesn't follow that the subsequence is in U. I don't see that the topological definition of P-point corresponds to the necessary property of an ultrafilter so that all infinitesimals correspond to a null sequence; It appears to be that:
${\displaystyle (\forall n)(U_{n}\in U)\rightarrow (\exists V\in U)(\forall n)((V\smallsetminus U_{n}){\text{ is finite}}).}$
Arthur Rubin (talk) 20:16, 8 May 2013 (UTC)
This would mean that U, in ${\displaystyle 2^{\mathbf {N} }/{\text{finite}}}$, is closed under countable intersections, which might correspond to a P-point filter, although not exactly a P-point. — Arthur Rubin (talk) 20:21, 8 May 2013 (UTC)
Found it. If we use [[Ultrafilter#Ultrafilters on ω|P-point]], rather than the current redirect at [[Glossary of topology#P|P-point]], then the statement as you wrote it makes sense, although could use a a more detailed argument and a source. — Arthur Rubin (talk) 21:58, 8 May 2013 (UTC)
The source is Cutland, Nigel; Kessler, Christoph; Kopp, Ekkehard; Ross, David, On Cauchy's notion of infinitesimal. British J. Philos. Sci. 39 (1988), no. 3, 375–378. Tkuvho (talk) 12:01, 9 May 2013 (UTC)

## Properties of infinite numbers missing.

The section headed "Properties of infinitesimals and infinite numbers" does not mention any properties of infinite numbers. Shame, because that's what I wanted to know about. Tesspub (talk) 10:28, 29 August 2014 (UTC)

## "The derivative of a function y(x) is defined not as dy/dx but as the standard part of dy/dx"

This is incorrect; using Keisler's treatment ${\displaystyle \mathrm {d} x}$ and ${\displaystyle \mathrm {d} y}$ are infinitesimal increments along the tangent line while ${\displaystyle \Delta x}$ and ${\displaystyle \Delta y}$ are infinitesimal increments along the curve. So ${\displaystyle y'(x)={\frac {\mathrm {d} y}{\mathrm {d} x}}=\mathrm {st} {\frac {\Delta y}{\Delta x}}}$. 58.169.240.244 (talk) 15:17, 4 May 2015 (UTC)