Are all nimbers included in the surreals?

Question

I guess the question says it all. The **nimber* (https://en.wikipedia.org/wiki/Nimber) concept, sometimes called "Sprague-Grundy numbers" embodies the "values" of positions in impartial games which can be added together to form larger games. The quintessential example is that in the game Nim, a pile with $n$ markers has a nim-value of $n$, sometimes denoted (in the notation used by Cnway, Berlekamp, and Guy) as $\star n$.

Unimpartial ("partizan") games can also have values of other nimbers which are formed by adding ordinary numbers to nimbers; also, values such as $\uparrow$ occur when a game presents a left-player move to $0$ and a right-player move to $\star$. So for example, ${\uparrow} > 0$ and ${\uparrow} > \star$but for any positive real number $p$, we have $p > {\uparrow}$.

But partizan games can also be used to define the surreal numbers (https://en.wikipedia.org/wiki/Surreal_number. These include equivalents of all the reals, as well as infinities (and all the ordinals) and infinitessimals.

But in the pages about the surreals, there is a conspicuous lack of mention of concepts like $\star$ and $\uparrow$. So are these nimbers also included in the surreals? And if so, how does $\uparrow$ compare with $+\epsilon$? (If they are comparable I think one must say $+\epsilon > {\uparrow}$ but I'm not really sure -- and it is hard for me to construct a game model that would answer the question.)

Answer 1

Surreal numbers (or just "numbers", for short) are very special kinds of games, namely those which can be written in the form $x=\{S\mid T\}$, where every element of $S$ and $T$ is a surreal number and $s<t$ for all $s\in S$ and $t\in T$ (this is an inductive definition, since you have to already know that the elements of $S$ and $T$ are numbers to learn that $x$ is a number). You can prove by induction that the usual partial order on games restricts to a total order on numbers; i.e. any two numbers are comparable. So since $0$ is a number and $\star$ is incomparable with $0$, $\star$ cannot be a number. The same holds for all other nimbers: the only nimber which is a surreal number is $0$.

The game ${\uparrow}=\{0\mid\star\}$ is also not a number, but it is comparable with all numbers. In fact, it is strictly smaller than all positive numbers (this implies it cannot be a number, since it is positive). To prove this, let $x=\{S\mid T\}$ be any positive number; we will show that the Left player can always win $x-{\uparrow}=\{S\mid T\}+\{\star\mid 0\}$. If Left goes first, they can move to $x+\star=\{S\mid T\}+\{0\mid 0\}$. Right can then move to either $t+\star$ for some $t\in T$ or $x+0$. In the second case Left wins because $x+0=x>0$. In the first case, Left can then move to $t+0=t$, and then wins since $t>x>0$ (since $x$ is a number, it satisfies $s<x<t$ for all $s\in S$ and $t\in T$).

Now suppose Right goes first. They can move to either $t-{\uparrow}$ for some $t\in T$ or $x+0=x$. In the first case, Left wins by induction since $t$ is a positive number. In the second case, Left wins since $x>0$.