TITLE: Examples of Surreal Numbers that are only Surreal Numbers?
QUESTION [7 upvotes]: 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?

REPLY [3 votes]: 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.