+\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.
+
projects / surreal-numbers / commitdiff