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1 | \newpage |
2 | \section{Notes: Chapter 1} | |
3 | ||
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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. |