X-Git-Url: http://git.subgeniuskitty.com/surreal-numbers/.git/blobdiff_plain/213fa8b5b4ce7a2531f07197927b3b4c06a2714b..4751e94034485f2d53bbb171fe9762b3a279cbaf:/notes/chapter-3.tex?ds=sidebyside

diff --git a/notes/chapter-3.tex b/notes/chapter-3.tex
index 22ad5b0..5e923ce 100644
--- a/notes/chapter-3.tex
+++ b/notes/chapter-3.tex
@@ -1,5 +1,5 @@
 \newpage
-\section{Notes: Chapter 3}
+\section{Chapter 3: Proofs}
 
 \subsection{Review}
 
@@ -63,71 +63,7 @@ sum-of-generations type argument used when proving transitivity.
     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.
-
+I'll explore this definition more in the next chapter.
