In surreal numbers, what exactly is $\omega_1$?

Question

This answer refers to $\omega_1$ in context of surreal numbers, and calls it "first uncountable ordinal".

But what exactly does it mean? How can it be represented in the $\{L|R\}$ form?

How do we know that "first uncountable ordinal" represents one surreal number and not a set of numbers? How is $\omega_1$ distinguished from $2\omega_1$?

I mean, the surreal $\omega$ can be considered equivalent to the germ at infinity of the function $f(x)=x$. But what about $\omega_1$? What is the representation?

According to my reasoning, if we take $\omega$ to be equal to the germ of $x$, the simplest constructed uncountable number is $\alpha=\int_0^1 \omega \, dx$. But it is the numerosity of the interval $[0,\pi)$, and not of the interval $[0,1)$.

On the other hand, we can simply say that the numerosity of the interval $[0,1)$ is the simplest uncountable number, but then it is constructed a bit more complicatedly: $N([0,1))=\frac1\pi\int_0^1 \omega \, dx$. Or, if we want to get rid of the $1/\pi$ coefficient, we should equate $\omega$ with the germ of $x/\pi$, which, I think, is not considered a canonical embedding of Hardy fields into surreals.

So, how is the canonical embedding determined? Could it be that the number $\omega_1$ could represent the numerosity of $[0,\pi)$?

Due to Euclid's principle, numerosities are totally ordered, so should be embedded into surreal numbers. On the other hand, what is the canonical embedding?

Answer 1

There is nothing special about $\omega_1$ or indeed any infinite number in the surreals, and they cannot be defined purely from the field structure of the surreals. What I claim is that all infinite numbers in the surreals are automorphic. That is, all such numbers can be moved from one to the other by automorphisms of the surreal field, and consequently they look exactly the same with respect to the surreal algebraic structure.

For example, every statement in the language of fields that is true of $\omega$ is also true of $\omega+1$ and $\omega_1+\omega-5$ and $\omega_1$, and indeed there is an automorphism of the surreals $\pi:№\to№$ with $\pi(\omega_1)=\omega$.

The reason is that the surreal numbers are a saturated real-closed field. In any real-closed field, by a famous theorem of Tarski's, every assertion in the first-order language of fields is equivalent to a quantifier-free statement. From this it follows that the definable sets of surreals are finite unions of intervals with rational endpoints. Since all infinite numbers will sit the same way in those intervals, it follows that they all have the same $1$-type. By saturation it follows that there is an automorphism $\pi:№\to№$ moving any of them to another.

So there are automorphisms moving $\omega_1$ to any other infinite element, including to $\omega$ or $\sqrt{\omega}$ or $\omega_1+1$ or what have you.

What this argument shows is that the $\{L\mid R\}$ notation is not a field-theoretic concept — it is not respected by automorphisms of the field. The $\{L\mid R\}$ notation is not structural with respect to the surreal field as a field. The $\{L\mid R\}$ notation, in contrast, as well as the underlying tree structure of the surreals, are conceptions arising in a way of constructing the surreals. We are able to name particular numbers arising in that particular construction, and this is often extremely useful. But we cannot recover the name of a surreal number from the number itself in a purely algebraic manner.

Answer 2

Assuming you already know how to define all the at most countable ordinals as surreal numbers, then $\omega_1 = \{ \text{those ordinals} \mid \}$. You can define ordinals in the context of surreal numbers as numbers whose right set is empty and the left set are other ordinal numbers. (This is not a vicious cycle because of regularity.) And a countable ordinal is just an ordinal whose left set is countable.

This number is different from $2 \omega_1$, if what you mean by that is the surreal number $2$ and the surreal number $\omega_1$ under surreal multiplication. In particular it is larger, as one can verify by looking up all the definitions.

I think there are too many questions in this post, not particularly well-connected. So I'll refrain from answering the rest, and recommend you split them off into a new question.