| 1 | \newpage |
| 2 | \section{Notes: Chapter 1} |
| 3 | |
| 4 | In the first chapter, Knuth provides two axioms, reproduced here. |
| 5 | |
| 6 | \begin{axiom} |
| 7 | Every number corresponds to two sets of previously created numbers, such that |
| 8 | no member of the left set is greater than or equal to any member of the right |
| 9 | set. |
| 10 | \end{axiom} |
| 11 | |
| 12 | For example, if $x = \surreal{X_L}{X_R}$, then $\forall x_L \in X_L$, it must |
| 13 | hold $\forall x_R \in X_R$ that $x_L \ngeq x_R$. |
| 14 | |
| 15 | \begin{axiom} |
| 16 | One number is less than or equal to another number if and only if no member of |
| 17 | the first number's left set is greater than or equal to the second number, and |
| 18 | no member of the second number's right set is less than or equal to the first |
| 19 | number. |
| 20 | \end{axiom} |
| 21 | |
| 22 | For example, the relation $\surreal{X_L}{X_R} = x \leq y = \surreal{Y_L}{Y_R}$ |
| 23 | holds true if and only if both $X_L \ngeq y$ and $Y_R \nleq x$. |
| 24 | |
| 25 | With no surreal numbers yet in our possession, we construct the first surreal |
| 26 | number using the null set (or void set, as Knuth calls it) as both the left and |
| 27 | right set. Although we have not yet examined its properties, Knuth names this |
| 28 | number ``zero''. Thus, $\surreal{}{} = 0$. |
| 29 | |
| 30 | As his final trick, Knuth defines a second generation of surreal numbers using |
| 31 | $0$ in the left and right set, naming them $1$ and $-1$ and claiming the |
| 32 | following relation. |
| 33 | |
| 34 | $$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$ |
| 35 | |
| 36 | Before we try generating some more surreal numbers, we should look for useful |
| 37 | equivalence classes. |