# Is this a valid basis for a transfinite number system?

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.

• 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. Commented Apr 30 at 19:48
• 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 Commented Apr 30 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. Commented Apr 30 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? Commented Apr 30 at 22:10
• 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". Commented Apr 30 at 22:37

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.
• 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. Commented May 1 at 11:05
• 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 $\omega$. Commented May 1 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. Commented May 1 at 13:12