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I've been curious about transfinite number systems including infinite ordinals, hyperreals, and surreal numbers. The hyperreals in particular seem particularly appealing for introducing a hierarchy of infinities and infinitesimals that behave algebraically nicely, but I'm not sure why they are defined with something as complicated as ultrafilters.

I've been wondering if we can't get the same basic properties from the following definitions, where ⍵ can be used to write arbitrary expressions which can be rewritten and simplified in the same ways expressions in terms of a finite valued variable can.

To determine order, let f(⍵) and g(⍵) be expressions in terms of ⍵:
f(⍵) > g(⍵) if and only if f(x) > g(x) for all x above some real value.
f(⍵) < g(⍵) if and only if f(x) < g(x) for all x above some real value.
f(⍵) = g(⍵) if and only if f(x) = g(x) for all x above some real value.
You could think of this as taking the limit of the order the two expressions are in.

To find the standard part, let f(⍵) be an expression in terms of ⍵:
st(f(⍵)) = limit of f(x) as x goes to infinity

You could also then introduce ε as a shorthand for 1/⍵ as a basis for infinitesimals. I'm aware that not every expression within this system can be definitively ordered, so indeterminate order will be a fundamental part, but this is a property also shared by surreal numbers with its star numbers. With these definitions, for example, we can say that sin⍵ < 2 but cannot say if sin⍵ < 0.

Is there anything fundamentally wrong with these definitions? It seems too simple for me to be the first person to think about it, so is there a name for this specific basis for a transfinite number system? What can hyperreal numbers do, for example, that these can't?

I'm interested in this because it seems like an interesting system to explore with a prohibitively high barrier of entry, like if someone introduced complex numbers with "I can tell you about complex numbers, but you'll need to be well versed in set theory and formal proofs so you can follow along with my 30 minute presentation" rather than "Well let's just say that the square root of negative one exists, and let's call it i". It seems like this could at least be a more intuitive way of thinking about certain transfinite numbers, and it seems to generate the same order for things that the hyperreal numbers in terms of omega and epsilon do. I'm willing to accept my system is different, but could it also be applied in nonstandard analysis? From what I can tell it works just fine for derivatives, integrals, and can be used to compare divergent limits.

Aidan Simmons
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    You're basically thinking about "germs at infinity" (or "Hardy fields"). The part where nonstandard analysis (for instance) improves on this sort of construction is in generating agreement between the "new" number system and $\mathbb{R}$. For instance, if you allow arbitrary expressions then $<$ fails to be a linear order (consider $\sin(\omega)$ vs. $\cos(\omega)$). In general, getting a high degree of agreement requires technical devices like ultrafilters. – Noah Schweber Apr 30 '24 at 19:48
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    Ultrapowers/Ultraproducts etc. might be very complicated. We use them because they gives us an essy way to produce bigger structures that will have same first order properties. In context of hyperreals really we could make up simmilsr structure without using ultrapowers. Basically we could use compactness theorem, loosely speaking we could extend theory of reals with existing of some infinite element. Then we would eventually get some nonstandard extension of reals (which really is all we need for nonstandard analysis). I think that compactness is genneraly much easier to understand – Antares Apr 30 '24 at 20:17
  • @NoahSchweber Germs at infinity and Hardy fields seem interesting, so I'll look into it. In the system that I described, you could say that $sin(ω)^2 + cos(ω)^2 = 1$ even if you can't tell where either sin(ω) or cos(ω) is within the interval [-1,1]. Is there a reason we don't make the system more broad and then set conditions on when you can make certain kind of statements like those that rely on linear ordering? To me it seems this way generates agreement with a focus on algebraic rules, unless I misunderstand what you mean by that. – Aidan Simmons Apr 30 '24 at 20:28
  • @Antares You mention extending the reals with the existence of an infinite element, which sounds a lot like what I proposed. Do you know if the compactness theorem would be applicable to the rules I described or would the properties of the infinite element need to be different? – Aidan Simmons Apr 30 '24 at 22:10
  • @AidanSimmons I presented some general idea behind but in reality it's much more nuanced than what you are proposing. What we basically do is this. We make some theory $T$ that has same axioms like (first order) theory of reals + we add infinitely many axioms $c>0,c>1,...$ (where $c$ is a constant symbol), and $\pm$ few nuances. By compactness theorem we can show this theory has a model. This model will be nonstandard extension of reals (what we need) and also has infinite element ($c$ is bigger than any real). – Antares Apr 30 '24 at 22:27
  • It is important step to be sure that our new model (structure) has same first order properties. Even existing of such objects isn't that much trivial and relies on some (relatively weak) version of axiom of choice. – Antares Apr 30 '24 at 22:29
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    Hi, welcome to MSE! I think in the analogy with $\Bbb C$, "let's add a square root of $-1$" is like "let's consider a proper field extension of $\Bbb R$ satisfying the transfer principle". "Take an ultrapower of $\Bbb R$" corresponds to "consider the product of the space of Cauchy sequences quotiented by...". It's a detail that can be temporarily avoided, if you want. If all you want is "algebraically nice hierarchy of infinitesimals" you might be happy with this sort of stuff, which is more like "let's add an infinitesimal". – Izaak van Dongen Apr 30 '24 at 22:37
  • @AidanSimmons In an earlier comment, you suggested to "set conditions on when you can make certain kind of statements like those that rely on linear ordering." I suspect that, if you were to work out the details of that suggestion, making sure that your conditions don 't produce things like $f(\omega)<g(\omega)<h(\omega)<f(\omega)$, you'd rediscover ultrapowers. – Andreas Blass May 01 '24 at 01:05
  • @AndreasBlass I don't believe that would be possible from the way I'm defining ordering. For f(ω)<g(ω) to be true would suggest f(x) is strictly less than g(x) for all real values above a certain x, similarly for g(ω)<f(ω). By transitivity this would suggest f(x) is strictly less than itself above some real valued threshold, which isn't possible. Because the order relation is determined by comparing functions over the reals, it shouldn't itself introduce that sort of behavior. I also think some of the complexity is avoided by only assigning definitive orders to things with stable end behavior. – Aidan Simmons May 01 '24 at 01:41
  • @AidanSimmons You won't get cycles of $<$ with yohur first definition of $<$. I was referring to an atttempted modification of that definition, as in one of your comments, setting further conditions to evade the sin-vs.-cos problem in your first definition. – Andreas Blass May 01 '24 at 01:50
  • @AndreasBlass Oh what I meant by that wasn't that the ordering rule would be changed for those cases. I meant that if you wanted to make arguments that rely on all elements being in a linear order that you'd determine the conditions where you could safely make those arguments rather than restrict the number system. Standard form polynomials with ω in place of x would be a safe subset for example. – Aidan Simmons May 01 '24 at 02:00
  • @IzaakvanDongen Thank you! The answer you linked to does appear to discuss a similar idea, so I'm curious if the same thing can be extended to real valued functions as a whole like I'm trying to do rather than just rational functions. I've also been seeing the term 'totally ordered' more and I believe that may apply to my definitions rather than well ordered since it removes the requirement that a least element can always be determined. – Aidan Simmons May 01 '24 at 04:24

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Using the set-theoretic ordinal $\omega$ to ultimately create a number system including infinite numbers is basically the idea behind the surreal numbers. The weakness of the surreals as compared to the hyperreals is that they don't possess a strong enough version of the transfer principle. All this has been discussed a number of times both at MSE and MO. If the complexity of the model-theoretic approach to nonstandard analysis is what bothers you, you may want to consider the axiomatic/syntactic approach, where instead of extending $\mathbb R$ to $\mathbb R^\ast$, one finds infinitesimals (and unlimited numbers) within $\mathbb R$ itself. The key is to work in a st-$\in$-language in place of the usual $\in$-language of ZFC; see this introduction for details.

Mikhail Katz
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  • I've been reading into the transfer principle more, and while I'm not certain that I'm understanding it correctly I do think my system would satisfy the transfer principle, at least for the results from operations and statements of equality and inequality. I considered a more broad definition of "a statement about ω is defined to be true if and only if it's true for all real numbers above some fixed real number," which gives the general idea but could also lead to statements like "ω is not ω" or "ω is a real number" if you're not careful. I'll read that introduction you sent. – Aidan Simmons May 01 '24 at 10:57
  • For starters, you need to be able to extend the functions of interest in, say, analysis to the new field. One might be able to do this using formal power series with $\omega$ etc., for analytic functions, but if one wants to be able to use arbitrary functions, extending them to your new field seems completely hopeless. It's the same problem with the surreals and other possible surrogates for nonstandard analysis. In a field like differential geometry, one needs arbitrary smooth functions to be able to use such a basic technique as partition of unity argument. – Mikhail Katz May 01 '24 at 11:05
  • To be a little more specific, I'm only interested in assigning meaningful values to functions that are defined with real outputs for arbitrarily large real inputs. Ordering then would be determined by proving one function has a stable order relationship with another above a certain real input. – Aidan Simmons May 01 '24 at 12:16
  • I think the objection still stands. If you are interested in behavior at infinity of smooth functions (rather than, say, analytic ones), you are going to face a hopeless task of trying to evaluate them at $\omega$. – Mikhail Katz May 01 '24 at 12:48
  • Doesn't that say more about our ability to prove stable end behavior relative to other functions than the validity of the concept in general? With what I'm proposing, there's no requirement to convert $f(ω)$ into an alternate form, even though we can in cases there is a formal power series like you mentioned. – Aidan Simmons May 01 '24 at 13:12
  • If I understand you correctly, you want to leave $f(\omega)$ as is, rather than "evaluating" it. The difficulty remains that it becomes a hopeless task to compare it to $g(\omega)$ if functions $f$ and $g$ are sufficiently general. – Mikhail Katz May 01 '24 at 13:14
  • I understand it may be difficult to evaluate some functions at $ω$, so it's a worthwhile consideration, but does that itself lead to any contradictions? Could you give me an example of a function that's easy to evaluate at hyperreal $ω$ that would be impractical to evaluate at this $ω$? – Aidan Simmons May 01 '24 at 13:30
  • As I already mentioned, any function can be evaluated at any hyperreal by the transfer principle. As far as evaluating at $\omega$, let $f(x)=x$ and let $g(x)=x+h(x)$ where $h$ is an arbitrary bounded smooth function, say. How are you going to compare $f(\omega)$ and $g(\omega)$? – Mikhail Katz May 01 '24 at 13:49
  • For that example you'd just need to determine if $h(x)$ is eventually strictly positive, strictly negative, or strictly 0. Similarly you could rephrase the order condition to be based on whether $f(x)-g(x)$ is eventually strictly positive, negative, or 0. I also know you said that, but I was asking for an example. I'm trying to see a specific case where a function produces a result from hyperreal $ω$ that can't be reached from my $ω$. – Aidan Simmons May 01 '24 at 14:30