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I'm teaching myself about hyperreal numbers. My main motivation for doing so is that they include infinite numbers, whose existence I hear disputed & doubted often as "quantifying the unquantifiable".

In this YouTube video, around twelve minutes in, it states that

$$\Bbb R(x)=\left\{ \frac{f(x)}{g(x)}\,\middle|\, f(x), g(x)\in \Bbb R[x], g(x)\neq 0\right\},$$

where $\Bbb R[x]$ is the set of polynomials in $x$ with real coefficients, is an instantiation of the hyperreal numbers ${}^*\Bbb R$.

What is meant by "instantiation" here?

Please note that hyperreal numbers are not rigorously defined in the video.

I have only a rudimentary understanding of hyperreal numbers; I'm not sure about what they are exactly. Perhaps that is where my confusion lies.

The Wikipedia article on hyperreals doesn't contain a definition either.

For questions on what hyperreals are, see the following:

Their answers make sense but I wouldn't say I have a firm grasp of them.

I'm given to understand that hyperreals are of use to model theory. That might explain where the terminology in question comes from. However, I don't remember where I heard this claim. (I think it was in a Numberphile video.)

A suspicion: I think $\Bbb R(x)$ is to ${}^*\Bbb R$ what the quotient $\Bbb R[x]/(x^2+1)$ is to $\Bbb C$.

Please help :)

Shaun
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    it doesn't mean much; in the last part of the video he picked an infinite $x\in{}^\Bbb R$ which then gives you $\Bbb R(x)\subset{}^\Bbb R$, and he briefly mentioned how to compare (the sizes of) two elements of $\Bbb R(x)$ (by expanding them to a Laurent series in $x^{-1}$). – user8268 May 19 '22 at 13:23
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    The creator of the video seems to not understand what hyperreal numbers actually are. I would not recommend it as anything more than a very loose layman's introduction to some of the ideas involved. – Eric Wofsey May 19 '22 at 13:36
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    $\mathbb R(x)$ with the order defined in video is a model for the axiom $A,B$ and $C$ in part 1A of Keisler 2007, but not the axioms from part 1C. It is not a field of hyperreal numbers. – SolubleFish May 19 '22 at 13:42
  • "Instantiation" (or "An instance") is computer science jargon for what in mathematical logic jargon is a "model". See more on mathematical logic in Ebbinghaus, Flum, Thomas, for elementary logic. Particularly Chap VI.4. – p.co May 19 '22 at 15:24
  • @p.co Unfortunately, this does not actually model the hyperreals. – Akiva Weinberger May 19 '22 at 17:38
  • Numberphile is full of junk. Please don't refer to them if you want to learn proper mathematics! – user21820 May 22 '22 at 13:07
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    "The Hyperreal number system" has a reasonably good gloss of what "hyperreals" mean, which is consistent with the wikipedia article, and of course contrary to your quoted video. – user21820 May 22 '22 at 13:29
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    No, @user21820, as it doesn't discuss $\Bbb R(x)$. – Shaun May 22 '22 at 13:30
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    @Shaun: It doesn't have to discuss it; it mathematically forbids it; as it is not an elementarily equivalent structure! – user21820 May 22 '22 at 13:30
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    Come on, now, @user21820; there must be some sense in which the concepts are related - that's what I'm interested in. I invite you to answer this question. – Shaun May 22 '22 at 13:34
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    I do not understand why you believe that the video you cited is correct. Just ask any other model theorist if you don't believe what I and Alex Kruckman said already here and at the linked thread... – user21820 May 22 '22 at 13:59
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    A useful discussion of some of the ideas involved in hyperreal numbers can be found here (in a chat room). – Shaun May 22 '22 at 14:54
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    "there must be some sense in which the concepts are related - that's what I'm interested in." Why must there be, exactly? – Noah Schweber Jun 26 '22 at 03:33
  • You may find helpful this answer (and its links to expositions on this and related topics). – Bill Dubuque Jun 26 '22 at 04:15
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    I have no idea how related this actually is, but Terry Tao proposed a "cheap non-standard analysis" that doesn't use ultrafilters in this blog post. – Calvin Khor Jun 26 '22 at 10:11
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    @CalvinKhor The method Tao describes (using a Frechet filter) is very old - see the citations I give on Schmieden and Laugwitz's approach here (link in my prior comment) for its many disadvantages vs. a free ultrafilter (update: this is also mentioned in the comments on Tao's blog post - so see there too). – Bill Dubuque Jun 29 '22 at 14:08

1 Answers1

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The claim in the video you link to is incorrect; to be honest, no video making such a claim should be used as a reference for technical topics. There is no substantive sense in which $\mathbb{R}(x)$ is related to hyperreals.

The most that can be said is that $\mathbb{R}(x)$ is an ordered (well, fine, orderable) field with infinitesimal elements - that is, a non-Archimedean field. Hyperreal fields, however, are much more than merely non-Archimedean fields. A hyperreal field must have a "high degree of similarity" to $\mathbb{R}$ itself, in a particular technical sense. $\mathbb{R}(x)$ demonstrably lacks this, most obviously since it does not satisfy the property "$\forall u\ge 0\exists v(v^2=u)$." So we don't even have $\mathbb{R}(x)\equiv\mathbb{R}$, let alone that $\mathbb{R}(x)$ is a hyperreal field (see below)!

To someone new to the subject of nonstandard analysis, this may feel like uncharitable hair-splitting. I want to emphasize as strongly as possible that the difference between mere non-Archimedeanness and hyperrealness is huge. Remember that the whole point of hyperrealness is that a hyperreal field should let you "transfer" results to the standard real numbers $\mathbb{R}$, so that infinitesimal methods can still provide real results. The fact that this is possible at all is really neat and awesome, and ignoring precisely what makes hyperrealness special in the interest of a superficial gain in accessibility is a pedagogical disservice.

It may help at this point to see the (most common in my experience) precise definition of hyperreal-ness, in order to contrast it with $\mathbb{R}(x)$. Let $\Sigma$ be the expansion of the language of fields by a symbol for every finite-arity relation on the reals, and let $\mathcal{R}$ be the reals construed as a $\Sigma$-structure in the obvious way. A hyperreal field is then just a proper elementary extension $\mathcal{F}$ of $\mathcal{R}$.

Note that strictly speaking a "hyperreal field" isn't a field but rather an expansion of a field (compare $\mathcal{R}$ with the much-less-complicated $(\mathbb{R};+,\times,0,1)$. This is a bit messy, so we describe $\mathcal{F}$ as a field equipped with a third-order operation ${}^*$ which assigns to each finite-arity relation $A$ on $\mathbb{R}$ an extension $^*A$ on the underlying set of $F$ which satisfies some simple properties. This amounts to the same thing as the above, but is arguably more convenient in two significant ways: pdagogically this approach is more accessible to non-logicians, and technically it lets us pay more careful attention to exactly how much transfer is needed for a given application.

Noah Schweber
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    For further elaboration on the points above see the papers I cite here, which includes highly accessible Monthly expositions, as well as pertinent historical remarks from Dauben's biography of Abby Robinson. – Bill Dubuque Jun 26 '22 at 07:26