Examples of Surreal Numbers that are only Surreal Numbers?

Question

I was just reading through the construction of the surreal numbers on wikipedia, and I read through some of the examples. I noticed that all of the examples were how certain types of already existing numbers (such as reals or hyperreals) could be constructed. However, I'm curious about examples of numbers that are unique to the set of surreal numbers. I've heard that they are the largest possible ordered field, so I would imagine that many if not most of surreal numbers are of this type. So what are some examples?

Answer 1

A few remarks:

$\DeclareMathOperator{\Noo}{\mathbf{No}}$ -$\Noo$ is not "the largest possible ordered field", but rather "a universal ordered field" with nice properties. Universality is to be understood in the weak sense: every ordered field embeds in it, not in a unique way in general.

-The differents fields (or frameworks whithin which one can talk) of hyperreals do not compare in a canonical way with surreals. That $\Noo$ is universal requires some choice, and for instance, no explicit embedding of $^*\mathbb{R}$ into $\Noo$ is known. Thus it makes only little sense to compare which number lies in which field, and it is more fruitful to understand global properties of the different fields.

-Although in hyperreal fields constructed by ultrafilters on $\mathbb{N}$ one can embed ordinals below $\varepsilon_0$ in a somewhat natural way preserving natural aritmetic (send $\omega$ to the class of $id_{\mathbb{N}}$ modulo the ultrafilter and embeds other ordinals using operations on ordinals and on hyperreals), it is not usual to consider that hyperreals contain distinguished infinite elements such as "$\omega$".

-However, for any infinite element $x$ in $^*\mathbb{R}$, $x-1 \in ^*\mathbb{R}$ is strictly below $x$ but infinite.

-While an hyperreal field does not canonically contain all ordinals, it is closed under extensions of real functions, most of which aren't explicitly known on $\Noo$. IN this sense many expressions such as $\sin(H)$ make sense in $^*\mathbb{R}$ but not in $\Noo$.

-To finish on a positive note, a somewhat similar question is asked here.

Answer 2

For a simple answer to the original question, per Gerry Myerson's comment, $$ \sqrt{\omega} \equiv \left\{1,2,3,\ldots \;\vert\; \omega, \omega/2, \omega/3, \ldots \right\} $$ is a surreal number that isn't part of the usual hyperreals. (Its name is justified because you can verify that $\sqrt{\omega}\cdot\sqrt{\omega}=\omega$.)

My understanding is that surreal numbers like $\omega^\omega\equiv\left\{\omega,\omega^2,\omega^3,\ldots\;\vert\;\right\}$ are also outside the usual hyperreals.

Answer 3

Another example I've just learned. Let $\Omega$ be the set of all hyperreals (whatever your definition for that is). Then $$\omega_1=\{\ \Omega\ |\ \}$$ defines a new number $\omega_1$ larger than every hyperreal, which will be reached on the $\omega_1$-th day.