Curiosity with surreal numbers

Question

I'm a high school student who is interested in surreal numbers and studying it by myself. Suddenly a weird idea popped in my mind. For example, can we define an event $A$ that has $P(A)=\frac1{\omega}$? Or can we construct a geometric system with surreal numbers?

Answer 1

For probability, yes $0<1/\omega<1$, but the problem is that there isn't a good theory of surreal integration, so you can't handle things like integrating a probability density function. That just leaves you with things like "this super-unfair coin has probability $1/\omega$ of landing on heads and probability $1-1/\omega$ of landing on tails", which isn't particularly interesting, but I suppose causes no logical problems.

What do you mean by "geometric system"? How about the following?: I define the "surreal plane" to be ordered pairs of surreals. Then I can talk about the triangle with vertices $(1/\omega,3)$, $(\sqrt{\omega},-\omega^2)$, and $(0,0)$. It has "area" $(\omega+3\sqrt{\omega})/2$ by the shoelace theorem. Again, the lack of surreal integration limits what one can do with this idea.