From 2afe84ae68654f256f95e31374eb77836204b090 Mon Sep 17 00:00:00 2001
From: Aaron Taylor <ataylor@subgeniuskitty.com>
Date: Mon, 17 May 2021 18:43:10 -0700
Subject: [PATCH] Initial commit of Chapter 4 notes. Just a review, import of
 proof outlines, and update of rhetoric. Proofs tomorrow.

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 notes/notes.tex     |   1 +
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+\newpage
+\section{Chapter 4: Bad Numbers}
+
+\subsection{Review}
+
+This was an interesting chapter, much busier than the previous two. It was nice
+to see that we haven't veered too far into nonsense.
+
+Knuth generates all the elements of generation three and they match our
+explorations from the Chapter 1 notes. He also proves a transitive law holds
+for Axiom \autoref{ax:leq-relation} using the same argument I used, showing
+that younger (simpler) sums reach a contradiction established by the original
+numbers. Then, with transitivity, he notes that all his numbers are similar to
+our Definition \autoref{defi:reduced-form} reduced form numbers. Knuth ends the
+chapter with a wink for meeting him at this point.
+
+
+\subsection{Exploration}
+
+I continue my attempt to define an addition operation which respects the
+implications of the names assigned by Knuth ($-1$, $0$ and $1$) and which
+extends those names in the pattern I think is developing.
+
+From Definition \autoref{defi:addition}, using the symbol $\sgkadd$, we are defining addition as
+$$
+x \sgkadd y = \surreal{X_L}{X_R} \sgkadd \surreal{Y_L}{Y_R}
+\equiv \surreal{\set{X_L \sgkadd y} \cup \set{Y_L \sgkadd x}}{\set{X_R \sgkadd y} \cup \set{Y_R \sgkadd x}}
+.
+$$
+
+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 $0$.
+
+We already know the outputs aren't well behaved in this manner, but we patched
+that up by only considering closure in terms of future generations. Now we
+examine the inputs.
+
+\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}
+
+With the inputs and outputs behaving as needed, we now apply our operation to
+the set, checking if we meet the group axioms.
+
+\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, our operation $(\mathbb{U},\sgkadd)$ is well
+defined, closed, and respects the three group axioms. Our universe $\mathbb{U}$
+is a group under $\sgkadd$, our addition operation. Let's name this group
+$\mathbb{U}_{\sgkadd}$.
+
+Note: This group is also commutative, seen by running transposed symbols
+through Definition \autoref{defi:addition} and noting that sets are inherently
+unordered.
+
+
+\subsection{Conjecture}
+
+Given our definition of addition, it seems natural to consider defining
+multiplication in the same recursively-younger manner. Then see if it's also a
+group and if it relates to $\mathbb{U}_{\sgkadd}$ in the usual manner.
diff --git a/notes/notes.tex b/notes/notes.tex
index 8d7fa5b..7d21862 100644
--- a/notes/notes.tex
+++ b/notes/notes.tex
@@ -12,6 +12,7 @@
 \include{chapter-1}
 \include{chapter-2}
 \include{chapter-3}
+\include{chapter-4}
 
 %===============================================================================
 
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