Initial commit of some notes related to chapter 1.

[surreal-numbers] / notes / chapter-1.tex

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 \newpage
 \section{Notes: Chapter 1}
 
 \newpage
 \section{Notes: Chapter 1}
 

-Hello from 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.