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\section{Notes: Chapter 1}

In the first chapter, Knuth provides two axioms, reproduced here.

\begin{axiom}
Every number corresponds to two sets of previously created numbers, such that
no member of the left set is greater than or equal to any member of the right
set.
\end{axiom}

For example, if $x = \surreal{X_L}{X_R}$, then $\forall x_L \in X_L$, it must
hold $\forall x_R \in X_R$ that $x_L \ngeq x_R$.

\begin{axiom}
One number is less than or equal to another number if and only if no member of
the first number's left set is greater than or equal to the second number, and
no member of the second number's right set is less than or equal to the first
number.
\end{axiom}

For example, the relation $\surreal{X_L}{X_R} = x \leq y = \surreal{Y_L}{Y_R}$
holds true if and only if both $X_L \ngeq y$ and $Y_R \nleq x$.

With no surreal numbers yet in our possession, we construct the first surreal
number using the null set (or void set, as Knuth calls it) as both the left and
right set. Although we have not yet examined its properties, Knuth names this
number ``zero''. Thus, $\surreal{}{} = 0$.

As his final trick, Knuth defines a second generation of surreal numbers using
$0$ in the left and right set, naming them $1$ and $-1$ and claiming the
following relation.

$$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$

Before we try generating some more surreal numbers, we should look for useful
equivalence classes.
