+\subsection{Review}
+
+In the first chapter, Knuth provides two axioms.
+
+\begin{axiom} \label{ax:number-definition}
+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} \label{ax:leq-comparison}
+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 establishing the
+following relation.
+
+$$-1 \equiv \surreal{}{0} \leq 0 \equiv \surreal{}{} \leq 1 \equiv \surreal{0}{}$$
+
+
+\subsection{Exploration}
+
+\begin{defi} \label{defi:generation}
+A \emph{generation} shall refer to the numbers generated by applying Axiom
+\autoref{ax:number-definition} to all possible combinations of extant numbers.
+Generations are numbered sequentially such that generation-0 consists of the
+number $0$, generation-1 consists of the numbers $-1$ and $1$, etc.
+\end{defi}
+
+Working by hand with Axiom \autoref{ax:number-definition}, generation-2
+contains the numbers shown below.
+
+$$\surreal{-1}{}, \surreal{1}{}, \surreal{-1,0}{}, \surreal{0,1}{},$$
+$$\surreal{-1,1}{}, \surreal{-1,0,1}{}, \surreal{-1}{1}, \surreal{0}{1},$$
+$$\surreal{-1,0}{1}, \surreal{}{1}, \surreal{-1}{0}, \surreal{}{-1},$$
+$$\surreal{-1}{0,1}, \surreal{}{0,1}, \surreal{}{-1,0}, \surreal{}{-1,1},$$
+$$\surreal{}{-1,0,1}$$
+
+\begin{defi} \label{defi:similar}
+Two numbers $X$ and $Y$ are \emph{similar} iff, per Axiom
+\autoref{ax:leq-comparison}, $X \leq Y$ and $Y \leq X$. This is denoted by $X
+\similar Y$.
+\end{defi}
+
+Using this definition, the twenty numbers from generations 0-2 break down into
+no more than ten equivalence classes based on similarity, as shown below.
+
+$$\surreal{0}{} \similar \surreal{-1,0}{}$$
+$$\surreal{}{}$$
+$$\surreal{}{0} \similar \surreal{}{0,1}$$
+$$\surreal{-1}{}$$
+$$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$
+$$\surreal{-1}{1}$$
+$$\surreal{0}{1} \similar \surreal{-1,0}{1}$$
+$$\surreal{}{1}$$
+$$\surreal{-1}{0} \similar \surreal{-1}{0,1}$$
+$$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$
+
+From this we see that, since Axiom \autoref{ax:leq-comparison} makes its
+comparison element-wise, every surreal number generated by our current methods
+must be similar to a surreal number containing one or zero elements in its left
+and right sets since only the largest member of the left set and smallest
+member of the right set are important. This motivates the following definition.
+
+\begin{defi} \label{defi:reduced-form}
+The \emph{reduced form} for a surreal number $X = \surreal{X_L}{X_R}$ is
+defined as $\surreal{x_l}{x_r}$ where $x_l$ is the largest element of $X_L$ and
+$x_r$ is the smallest element of $X_R$. If either $X_R$ or $X_L$ are the empty
+set, then the corresponding $x_r$ or $x_l$ also become the empty set.
+\end{defi}
+
+Note that we are guaranteed largest and smallest elements of the corresponding
+non-empty sets from Axiom \autoref{ax:leq-comparison} and our definition of
+similarity. We are only building a one-dimensional number line.
+
+Note that the reduced form is not unique since $\surreal{-1}{1} \similar
+\surreal{}{}$.
+
+
+\subsection{Conjecture}
+
+If we can build an addition operation which holds for $1 + (-1) = 0$, then we
+could start trying to assign meaningful names to some of the elements from
+generation-2. It appears that numbers of the form \surreal{n}{} behave like the
+number $n+1$ and similarly, \surreal{}{n} behaves like $n-1$. That seems to
+build the integers.
+
+If we write a program to generate a bunch of new surreal numbers and graph them
+as ``generation vs magnitude'', perhaps we can assign some meaning to numbers
+which don't fit the pattern mentioned in the previous paragraph. Maybe these
+behave like $1/n$? It sort of feels like surreal numbers constructed via finite
+repetitions of our current process will end up building something vaguely like
+the dyadic rationals.
+
+I think some of our equivalence classes are similar. I suspect that the number
+line maintains symmetry with each generation. Since we have ten equivalence
+classes, an even number, symmetry is broken if none of the equivalence classes
+are similar. In fact, since it appears that $\surreal{-1}{1} \similar
+\surreal{}{}$, I'm sure that our equivalence classes can be collapsed further.
+
+I think I can use Definition \autoref{defi:generation} to start proving some
+surreal number properties inductively.
+
+I'm having a lot of problems inserting generation-n into the numberline
+containing generation-0 to generation-(n-1). Since Axiom
+\autoref{ax:leq-comparison} requires comparing against the number itself rather
+than just comparing the sets which define the number, it's hard to slot a new
+generation's number in. For example, how do I test \surreal{}{-1} and
+\surreal{}{1} without working the existing numberline from both ends? I'm
+tempted to note that no number can contain itself, and that the two sets must
+be `less than on the left' and `greater than on the right' to allow just
+comparing sets and finding the right spot on the numberline by working from
+both directions inward, rather than just left to right. Can I make that both
+rigorous and equivalent to Axiom \autoref{ax:leq-comparison}?
projects / surreal-numbers / blobdiff