# how to create the set of hyperreal numbers using ultraproduct

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

-

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

-

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.

-
I tried to show in my answer above that the explanation is not beyond the scope of an answer here. – Mikhail Katz Jan 7 at 7:56
@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 7 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 7 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 7 at 12:33
@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 7 at 12:43