Could a coordinate space be defined over the surreal numbers?

Question

Self-explanatory. I am aware of coordinate spaces being defined over the field of real and complex numbers, but I've wondered whether you could do something similar with the field of surreals: As in, constructing a space consisting of all $n$-tuples of surreal numbers as its coordinates.

This is largely out of curiosity, as whether or not the surreals' nature as a proper class (collections which are not defined in a given universe of sets, and are treated informally for the most part, as far as ZFC is concerned) would allow for such a construction is a question that's been in the back of my mind for a while.

If the answer is "Yes", then I also extend the question: What would the properties of such a space be? Would it differ in any way from the more ordinary manifolds and coordinate spaces which we normally address using the reals/complex numbers?

Answer 1

When working in ZFC or similar, a "class" basically amounts to a condition.

For example, there are various methods, but one nice bunch of ways to set up the surreals is to build them up inductively. Then they are the class of sets such that for some ordinal $\alpha$, they're in the set "surreals born by day $\alpha$".

So we can set up "a (finite dimensional) coordinate space" simply enough: the class of sets that, for some natural $n$ and ordinal $\alpha$, they are functions from $n$ to the set of surreals born by day $\alpha$.

For the properties, the big issue is that $\mathbb R^n$ and $\mathbb C^n$ are complete metric spaces because $\mathbb R$ is one. But the surreals with the order topology is very far from a metric space.

That said, in several works of Norman L. Alling on the Surreals, including the paper Conway's Field of Surreal Numbers, Alling has defined a sense of "limit" different than the usual one, for which certain sequences (and similar - the indexing set could be any limit ordinal) have unique surreal limits. Very briefly, if you have a sequence that is "very Cauchy" (my phrasing) in the sense that the difference between $|a_k-a_j|$ is infinitesimal compared to $|a_j-a_i|$ whenever $i<j<k$, then there are many "pseudo-limits", but there is a unique pseudo-limit that is the simplest (in the sense of simplicity/birthday of the surreals), which Alling calls "the limit".

Answer 2

As a complement to the answer of Mark S., I claim that properties of those structures would differ widely from those of the corresponding $\mathbb{R}$-valued structures.

For any ordered field $F$, one can adapt the definition of continuity, differentiability, norm and distance, so as to make sense of say $F$-normed vector spaces over $F$, $F$-manifolds or $F$-differentiable manifolds, and so on... The sequential formulation of properties (for instance "sequential continuity") can also be generalized by considering certain ordinals instead of $\mathbb{N}$.

If one wants those structures to have some properties that can be stated in the first order language of ordered fields, then one need only impose that $F$ be a real-closed field. Indeed those are the fields with the same first order properties as $\mathbb{R}$. There are many real-closed fields besides $\mathbb{R}$, and $\mathbf{No}$ (as well as certain natural set-sized subsystems thereof) is one of them.

For instance, Philip Ehrlich described the properties of the surreal plane $\mathbb{No} \times \mathbb{No}$ seing it as a model of Tarski's theory for elementary euclidean geometry (whose models are exactly cartesian squares of real-closed fields). The order saturation property of $\mathbf{No}$ translates into a somewhat nicer form of Tarski's continuity axiom schemata, and as can be expected $\mathbf{No}^2$ is universal for models of Tarski's theory. Unfortunately, I cannot find the article I am alluding to. I think P. Ehrlich argues that this surreal plane is a sort of absolute two dimensional continuum.

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Now most of what we care about when working with manifolds or normed vector spaces actually relies on analytic properties of $\mathbb{R}$ which cannot be stated in the first order language of ordered fields. In fact, any ordered field which has one of the following properties (the implicit topology here is the order topology) is uniquely isomorphic to $\mathbb{R}$:

1. $F$ has the least upper bound property

2. $F$ is connected

3. every closed bounded subset of $F$ is compact

4. every injective continuous function is strictly monotonous

5. $F$ satisfies the intermediate value theorem

6. $F$ satisfies the mean value theorem or Rolle theorem

7. every differentiable function with zero derivative is constant

8. $F$ satisfies the limit of the derivative theorem

9. $F$ satisfies L'Hospital's rule.

   and so on... See James Propp's Real analysis in reverse for most of the proofs. Basically every non-elementary theorem in analysis, such as the Banach-Picard's fixed-point theorem, the implicit function theorem or the Cauchy-Lipschitz theorem, fails dramatically in any other ordered field than $\mathbb{R}$. So you may have nice $F$-manifolds, but you won't be able to say anything interesting about regular maps between them.

One can construe this problem as having too many open sets, continuous functions or differentiable functions. A possible solution can thus be to restrict the objects we are looking at so as to select some which behave very nicely. For instance, one gets good analytic properties in a real-closed field if one replaces the quantification in every second order statement by "for every definable set/function..." or "there exists a definable set/function...". This is basically rephrasing the fact that real-closed fields have the same first order properties as $\mathbb{R}$. But one can do the same in richer languages for which the theory of $\mathbb{R}$ is nice (o-minimal for instance). See here.

The problem is that whereas the set of $F$-valued continuous or differentiable functions is stable under various operations, this is not the case for the sets of functions which satisfy this and that theorem (for instance functions which satisfy the mean value theorem). So one cannot directly select the theorems one would want to retain. One is seemingly stuck having to chose one side of the dichotomy [very few regular functions/sets and good properties thereof] / [many regular functions but very bad properties in general]. The only thing lying in the border is the field of real numbers.