What are some construction of number system which also works when you use them "within" the surreal numbers?

Question

I've noticed that a few construction we use to construct different number systems also work if we carry them on within the surreal numbers. A few examples of what I mean are bellow. I was wondering if there were other examples than those I have found (also hopefully some which aren't just the natural numbers or real numbers).

Example 1) Peano axiom natural numbers.

If we define 0 := {|} and s(x) := {x|} and do the Peano axiom construction of the natural numbers, the numbers we get have the same form as their surreal number counter part.

E.g Normally when using the Peano axioms, 3 is defined as s(s(s(0))). If we apply s(s(s(0))) using the definition stated above, we get {{{{|}|}|}|}, which is also equal the the surreal number 3. This works for all natural numbers.

Example 2) Ordinal numbers.

For the ordinal numbers 0 := {}, and N is the set off all numbers less than N, e.g 3 = {0,1,2} = { {} , {{}} , {{},{{}}} }. If we instead say that 0 := {|} and that N := {L|} where L is the is the set off all numbers less than N, the surreal numbers we get are the same as the ordinal numbers, e.g {0,1,2|} = {{},{{}|},{{},{{}|}|}|} which is the surreal number 3.

Example 3) Dedekind cut real numbers.

The construction of real numbers by Dedekind cuts also works in the surreal numbers. You need to have the surreal rational numbers already defined, and then partition them in {L|R} the same way you would do in a (A,B) dedekind cut. The surreal number you would get from a cut is the same number you would get from a regular dedekind cut.

E.g $(\{ x \in \Bbb Q st. x\lt 0$ or $x^2\lt 2 \} ,\{ x \in \Bbb Q st. x\ge 0$ or $x^2\ge2 \} )$ is the dedekind form of $\sqrt{2}$ and $\{\{ x \in \Bbb Q st. x\lt 0$ or $x^2\lt 2 \} | \{ x \in \Bbb Q st. x\ge 0$ or $x^2\ge2 \} \}$ (where Q is the surreal rationals and 2 is the surreal 2) is the surreal $\sqrt{2}$.

Counter Example, Cauchy sequence real numbers.

As far as I can tell, this doesn't work with how cauchy sequences construct the real numbers. If you have a cauchy sequence and put all it's elements in L or R in {L|R}, you will get a different real number than the real number the cauchy sequence equals.

E.g 1, 1.4, 1.41, 1.414, is a cauchy sequence for $\sqrt{2}$ but {1, 1.4, 1.41, 1.414, ... |} is a surreal number slightly less than the surreal $\sqrt{2}$.

2, 1.5, 1.42, 1.415, ..., is a cauchy sequence for $\sqrt{2}$ but {2, 1.5, 1.42, 1.415, ...|} is the surreal number 3.

So there seems to be no "direct" method to make it work, but it can work if you change a couple of rules. You need 2 cauchy sequence that in the same equivalence class, put one in L and the other in R. The one in L can't have a biggest element, and the one in R can't have a least element.

E.g {1, 1.4, 1.41, 1.414, ... | 2, 1.5, 1.42, 1.415, ...} is the surreal $\sqrt{2}$.

This is ofcourse a different construction than the regular cauchy sequences real number construction, which is why I didn't consider it an example.

I apologize if part of this post is messy, I've had this question in my head for awhile and this is the best I am able to formulate it. If something is a little ambiguous, feel free to interpret the way that makes most sense. If I've made any mistakes here, or if anything I say seems to be flat out wrong, please point it out.

Answer 1

I'd say the (beautifully phrazed, really) question

What can be done on surreal numbers in a similar way to the way it can be done on natural or real numbers?

is a general open informal question. It must have been, at some point, on the minds of most mathematicians who are interested in surreal numbers. It is too vague, at least for now, to be possibly answered. But it is a good naive idea that comes to mind when discovering surreal numbers.

However, rather than having constructions on $\mathbb{N}$ or $\mathbb{R}$ "immediately work" on surreal numbers, one must often adapt the notions to make them work, albeit sometimes in a straightforward fashion.

For instance, since the field $\mathbf{No}$ of surreal numbers is not isomorphic to the field $\mathbb{R}$ of real numbers (it is not even Archimedean), well almost no interesting result in real-analysis or real topology holds for $\mathbf{No}$ (see James Propp's paper Real Analysis in Reverse). So one must substantialy adapt statements in order to obtain analytically flavored analogies between $\mathbb{R}$ and $\mathbf{No}$.

A clear example is the statuses of "the" exponential function $\exp$ on $\mathbb{R}$ and $\mathbf{No}$. In real numbers, the exponential function is the unique solution of $y'=y$ which takes value $1$ at $0$, and also the unique bijective morphism $f:(\mathbb{R},0,+,<) \rightarrow (\mathbb{R}^{>0},1,\times,<)$ which is differentiable at zero with $f'(0)=1$.

In surreal numbers, there are many such functions, some of which don't really look at all like the real exponential function. Harry Gonshor's exponential function, which the one people use, is but one of them.

Now on the one hand, the exponential function in real numbers is the unique strictly positive function which satisfies a set of inequalities of the form $\exp(x)>\exp(x') \sum \limits_{k=0}^n \frac{(x-x')^k}{k!}$ for all $n \in \mathbb{N}$ and $x'<x$ (one requires three other similar sets of inequalities ot obtain the unicity). On the other hand, the Gonshor's exponential function is the "simplest" function which satisfies the same inequalities. And it does share many properties with the real exponential function.

So exponentiation does work on surreal numbers, yet not all methods of defining the exponential in $\mathbb{R}$ may directly apply or give a good result on $\mathbf{No}$.

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As for your counterexample, I would put it in the category "can be adapted".

Cauchy sequences of real-numbers can be used to generate Dedekind cuts. In fact I find the notion of "binary" expansion with $+1$ and $-1$ rather than $0$ and $1$ to be quite natural if one wants to approximate a real number with dyadic numbers, and this expansion corresponds to the "canonical" representation of surreal numbers using cuts.