Talk:Hyperreal number

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)

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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)

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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

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Differential forms living in cotangent spaces are not the same thing as

infinitesimals even though the notation may or may not be identical.

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Sorry, my mistake, I thought you meant a description of the formal construction of hyperreal numbers. --Jan Hidders

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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)

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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)

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I would like to change the displayed formulas in this article to LaTex, unless there is a compelling reason not to. User:CSTAR

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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)