\newpage
\section{Notes: Chapter 3}

\subsection{Review}

In the third chapter, Knuth begins to embrace similarity exactly as we did in
Definition \autoref{defi:similar}, noting that different forms exist which are
similar to each other. He also more carefully tracks through the relations in
$\surreal{}{0} \leq \surreal{}{} \leq \surreal{0}{}$.

Finally, at the end of the chapter, Knuth begins speculating about the form of
numbers to come and their relation to each other, pointing out, via an
erroneous categorization of positive number forms, that categorization isn't as
simple as it first appears 


\subsection{Exploration}

My goal is that of assigning meaningful names to newly generated numbers under
the assumption that Knuth's three names ($-1$, $0$ and $1$) were naturally
meaningful.  Since that implies some notion of `distance from zero' (after all,
I'm attempting to build a numberline), defining some form of addition and
subtraction seems like a natural starting point.

Remember that we're still only considering finite generations for our universe.
That means our addition operation will never be closed since our finite
universe always has a largest element and nobody can stop us from adding $1$ to
it. However, as such closure-violating elements are members of our universe in
a later generation, only when considering closure, we will mentally think of
our universe as something like $\mathbb{U} \equiv \bigcup_{n \in \mathbb{N}}
\mathbb{U}_n$ and otherwise ignore the problem.

Thus, updating my caveat from my Chapter 1 notes:

\begin{framed}
    \noindent All definitions and theorems only consider the case of finite left
    and right sets.  In these finite cases, we're pretending that addition is
    closed, but it's not.
\end{framed}

Based on my Chapter 2 notes, when considering my numbers in the line defined by
my total order, I want every reduced form number that has both left and right
ancestors to be the geometric mean of those ancestors. And any reduced form
number missing one or both ancestors is similar to another, more suitable
reduced form number.

Also in my Chapter 2 notes, I hand-wavingly defined such a geometric mean
recursively, just to build a mental picture. However, since I don't see a way
to define the same concept using only the left and right sets of the operands,
a recursive definition might be the right approach as long as it reduces the
overall age of the numbers involved, allowing me to use the same
sum-of-generations type argument used when proving transitivity.

\begin{defi} \label{defi:addition}
    For two numbers $x$ and $y$, define \emph{addition} as
    $$
    x \sgkadd y = \surreal{X_L}{X_R} \sgkadd \surreal{Y_L}{Y_R}
                \equiv \surreal{\set{X_L \sgkadd y} \cup \set{Y_L \sgkadd x}}{\set{X_R \sgkadd y} \cup \set{Y_R \sgkadd x}}
    $$
    where $\set{A \sgkadd b}$ means the set of numbers $a \sgkadd b$ for all $a \in A$.
    
    We are using the symbol $\sgkadd$ in order to keep our addition distinct
    from whatever Knuth eventually defines.
\end{defi}

Given the obvious symbolic symmetry, we won't bother explicitly proving this
operation is commutative.

Keep in mind that this is only guaranteed to produce a valid number (per Axiom
\autoref{ax:number-definition}) subject to our caveat regarding closure.

Since we have defined this operation in terms of specific forms, we must ensure
the operation behaves identically with respect to all similar forms. It would
be a shame if, for example, $0+0=0$ only held for certain values of $0$.

\begin{theorem} \label{thm:sgkadd-welldefined}
    The binary operation $\sgkadd$ on $\mathbb{U}$ is well defined. That is, for
    numbers $x, x', y, z \in \mathbb{U}$ such that $x \sgkadd y = z$ and $x
    \similar x'$, $\exists z' \in \mathbb{U}$ such that $x' \sgkadd y = z'
    \similar z$.
\end{theorem}

\begin{proof}
    TODO
\end{proof}

\begin{theorem} \label{thm:sgkadd-identity}
    The number $0 = \surreal{}{}$ is the identity element for the binary
    operation $\sgkadd$ on $\mathbb{U}$. That is, for any number $x \in
    \mathbb{U}$, $x \sgkadd 0 \similar 0 \sgkadd x \similar x$.
    In this behavior, the number $0$ is unique up to similarity.
\end{theorem}

\begin{proof}
    TODO
\end{proof}

\begin{theorem} \label{thm:sgkadd-associative}
    For all $x, y, z \in \mathbb{U}$, it holds that
    $$(x \sgkadd y) \sgkadd z \similar x \sgkadd (y \sgkadd z).$$
\end{theorem}

\begin{proof}
    TODO
\end{proof}

\begin{defi} \label{defi:inverse}
    For a number $x$, let \emph{negation} be defined as
    $$
    -x = -\surreal{X_L}{X_R} \equiv \surreal{-X_R}{-X_L}
    $$
    where $-A$ means the set of numbers $-a$ for all $a \in A$.
\end{defi}

\begin{theorem} \label{thm:sgkadd-inverse}
    For every number $x \in \mathbb{U}$, there exists a number $-x \in
    \mathbb{U}$ such that $x \sgkadd -x = 0$.
    In this behavior, the number $-x$ is unique up to similarity.
\end{theorem}

\begin{proof}
    TODO
\end{proof}

Putting that all together, $(\mathbb{U},\sgkadd)$ is well defined, closed, and
respects the three group axioms. It's a group. Let's name it
$\mathbb{U}_{\sgkadd}$. It's also commutative.


\subsection{Conjecture}

$\mathbb{U}_{\sgkadd}$ really is a group.

