Adding some more thoughts to my chapter-1 notes/speculations.

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

\newpage
\section{Notes: Chapter 1}

\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}?