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As title says, can anyone explain how to create the set of hyperreal numbers using ultraproduct from the set of real umbers?

Mikhail Katz
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2 Answers2

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The explanation may be a bit simpler than seems to be implied by earlier answer. Namely, the hyperreals are a certain quotient of the space $\mathbb{R}^{\mathbb N}$ of sequences of real numbers. More specifically, consider sequences $(x_n)$ that vanish for almost all $n$, and let $I\subset \mathbb{R}^{\mathbb N}$ be a maximal ideal containing all such sequences (the existence of a maximal ideal is proved in undergraduate algebra courses). Then the hyperreals are a quotient field $\mathbb{R}^{\mathbb N}/I$. This is equivalent to a certain construction using an ultrafilter on $\mathbb N$. A more detailed summary of the construction can be found on page 911 in https://www.math.wisc.edu/~keisler/calc.html

Mikhail Katz
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I’d say that a full explanation is beyond the scope of an answer here. Section $3$ of this PDF is a fairly gentle but very incomplete start; this PDF goes into much more detail and seems to be about as accessible as a thorough treatment is likely to be, though I’ve only skimmed it.

Brian M. Scott
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  • I tried to show in my answer above that the explanation is not beyond the scope of an answer here. – Mikhail Katz Jan 07 '16 at 07:56
  • @user72694: And I disagree: in all likelihood your answer is useless to anyone who doesn't already have some idea of how to construct them. Pointing the reader to a real exposition is far more useful than an answer that, although correct, is almost completely uninformative in terms of useful detail. – Brian M. Scott Jan 07 '16 at 12:24
  • I am interested in your point of view. My point was that extending an ideal to a maximal one is within the scope of a serious undergraduate algebra class. – Mikhail Katz Jan 07 '16 at 12:27
  • @user72694: Bluntly, so what? It's an abstraction that gives very little insight into the structure of the hyperreals, at least at the level of an undergraduate algebra class. (Which may in any case do rather less than you think, depending on the institution; that explanation would make no sense to the overwhelming majority of students who'd had the one offered where I taught, I'm sorry to say.) – Brian M. Scott Jan 07 '16 at 12:33
  • Brian, will all due respect we are currently teaching a freshman calculus class using infinitesimals and I can tell you that they find the approach appealing. Note that we don't construct the real number system in a freshman calculus class, either, whether in Thomas-Finney or in Keisler. I think you will agree with me that the construction of the real numbers via equivalence classes of Cauchy sequences will make little sense to a majority of freshmen sitting in a typical calculus course. If you like I can send you a recent study of the issue. – Mikhail Katz Jan 07 '16 at 12:36
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    @user72694: Jerry Keisler was one of my instructors in grad school, and I'm quite familiar with his book; I don't doubt that many of them, at least, find the approach appealing. So do I, speaking as a teacher. Indeed, I have no quarrel with any assertion in your comment. However, it does not appear to be at all relevant. – Brian M. Scott Jan 07 '16 at 12:43
  • Brian, the OP has not been very active but at any rate the tone of his question as formulated 3 years ago did not indicate that he is a freshman. More advanced students in analysis and algebra may be be quite ready for this type of quotient construction. Notice also that Skolem's construction of a nonstandard model of $\mathbb{N}$ is of the same type, without using any choice. The perception that this is only accessible to those who already have a PhD in model theory is I think exaggerated :-) – Mikhail Katz Jan 07 '16 at 12:47
  • @user72694: I do not have any such perception, and nothing that I've written here can reasonably be so interpreted. In any case your answer does not really address the OP's question, which was specifically about the ultrafilter construction. For that reason alone I wonder what the upvoters were thinking. I don't dowvote, ever, but I'd certainly understand it if someone did. – Brian M. Scott Jan 07 '16 at 12:55
  • Brian, a construction is generally understood in a series of successive approximations where the first step is to get an intuitive idea of what is going on. Most students are more familiar with maximal ideals than with ultrafilters. If you can formulate the construction in a way they can understand, it will be easier for them to understand a different take on this. The construction is described is actually a complete definition of a hyperreal line. The definition using an ultrafilter can come at a next stage. But what is wrong with starting with a more intuitive stage? – Mikhail Katz Jan 07 '16 at 12:58
  • @user72694: Nothing, if you're teaching a course. Here, however, you're supposed to be answering a specific question, and you neither did so nor pointed the OP at a source that does. – Brian M. Scott Jan 07 '16 at 13:00
  • I think many people come to SE seeking to understand. Besides, I provided a source already. – Mikhail Katz Jan 07 '16 at 13:09