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

diff --git a/notes/chapter-1.tex b/notes/chapter-1.tex
index 9a1e41c..e340d75 100644 (file)
--- a/notes/chapter-1.tex
+++ b/notes/chapter-1.tex
@@ -30,19 +30,19 @@ 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
 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 claiming the
+$0$ in the left and right set, naming them $1$ and $-1$ and establishing the

 following relation.
 
 following relation.
 

-$$-1 = \surreal{}{0} \leq 0 = \surreal{}{} \leq 1 = \surreal{0}{}$$
+$$-1 \equiv \surreal{}{0} \leq 0 \equiv \surreal{}{} \leq 1 \equiv \surreal{0}{}$$

 
 
 \subsection{Exploration}
 
 \begin{defi} \label{defi:generation}
 A \emph{generation} shall refer to the numbers generated by applying Axiom
 
 
 \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 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.
+\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
 \end{defi}
 
 Working by hand with Axiom \autoref{ax:number-definition}, generation-2


@@ -60,11 +60,8 @@ Two numbers $X$ and $Y$ are \emph{similar} iff, per Axiom
 \similar Y$.
 \end{defi}
 
 \similar Y$.
 \end{defi}
 

-From this point forward, we will refer to similar surreal numbers
-interchangeably.
-

 Using this definition, the twenty numbers from generations 0-2 break down into
 Using this definition, the twenty numbers from generations 0-2 break down into

-ten equivalence classes based on similarity, as shown below.
+no more than ten equivalence classes based on similarity, as shown below.

 
 $$\surreal{0}{} \similar \surreal{-1,0}{}$$
 $$\surreal{}{}$$
 
 $$\surreal{0}{} \similar \surreal{-1,0}{}$$
 $$\surreal{}{}$$


@@ -80,7 +77,8 @@ $$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \s
 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
 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. This motivates the following definition.
+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
 
 \begin{defi} \label{defi:reduced-form}
 The \emph{reduced form} for a surreal number $X = \surreal{X_L}{X_R}$ is


@@ -93,20 +91,42 @@ 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.
 
 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
 
 \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 $n-1$.
+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
 
 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.
-
+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}?