More updates to Chapter 1 notes exploring order and uniqueness.

diff --git a/notes/chapter-1.tex b/notes/chapter-1.tex
index 1d7a06f..78834af 100644 (file)
--- a/notes/chapter-1.tex
+++ b/notes/chapter-1.tex
@@ -10,17 +10,15 @@ of those symbols, and a binary relation on that selected subset.
 \begin{axiom} \label{ax:symbol-generation}
 Every symbol is comprised of a left and right set of other symbols. The initial
 symbol is named $0$ and is comprised of the empty set for both its left and
 \begin{axiom} \label{ax:symbol-generation}
 Every symbol is comprised of a left and right set of other symbols. The initial
 symbol is named $0$ and is comprised of the empty set for both its left and

-right sets.
-
-All further symbols are generated by repeatedly selecting (potentially empty)
-subsets of the set of all existing symbols, using these selections as left and
-right sets to define a new symbol, and then adding that symbol the the set of
-all existing symbols.
+right sets. All further symbols are generated by repeatedly selecting
+(potentially empty) subsets of the set of all existing symbols, using these
+selections as left and right sets to define a new symbol, and then adding that
+symbol to the set of all existing symbols.

 \end{axiom}
 
 From this set of all symbols, growing with each iteration, we select the subset
 of symbols which satisfy the next axiom, calling them \emph{numbers}. This
 \end{axiom}
 
 From this set of all symbols, growing with each iteration, we select the subset
 of symbols which satisfy the next axiom, calling them \emph{numbers}. This

-subset forms our \emph{universe}.
+subset forms our \emph{universe}, also denoted $\mathbb{U}$.

 
 \begin{axiom} \label{ax:number-definition}
 Every number corresponds to two sets of previously created numbers, such that
 
 \begin{axiom} \label{ax:number-definition}
 Every number corresponds to two sets of previously created numbers, such that


@@ -83,8 +81,9 @@ number $0$, generation-1 consists of the numbers $-1$ and $1$, etc.
 \end{defi}
 
 \begin{defi} \label{defi:generation-function}
 \end{defi}
 
 \begin{defi} \label{defi:generation-function}

-For a surreal number $x$, the \emph{generation function} $g(x)$ returns the
-generation in which $x$ was created. For example, $g(0) = 0$ and $g(-1) = 1$.
+For a surreal number $x$, the \emph{generation function} $g\colon \mathbb{U}
+\rightarrow\mathbb{N}$ returns the generation in which $x$ was created. For
+example, $g(0) = 0$ and $g(-1) = 1$.

 \end{defi}
 
 \begin{defi} \label{defi:younger}
 \end{defi}
 
 \begin{defi} \label{defi:younger}


@@ -169,7 +168,7 @@ Two numbers $x$ and $y$ are \emph{similar} iff, per Axiom
 \end{defi}
 
 Applying this new definition to the twenty numbers from generations 0-2, we
 \end{defi}
 
 Applying this new definition to the twenty numbers from generations 0-2, we

-find that they start to collapse into equivalency classes, like the
+find that they start to collapse into equivalency classes like the

 aforementioned $\surreal{}{} \similar \surreal{-1}{1}$. More on this later.
 
 Returning to our investigation of order, now that we have `proven'
 aforementioned $\surreal{}{} \similar \surreal{-1}{1}$. More on this later.
 
 Returning to our investigation of order, now that we have `proven'


@@ -237,20 +236,18 @@ The twenty numbers from generations 0-2 break down into seven similarity-based
 equivalence classes, as shown below.
 
 $$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$
 equivalence classes, as shown below.
 
 $$\surreal{}{-1} \similar \surreal{}{-1,0} \similar \surreal{}{-1,1} \similar \surreal{}{-1,0,1}$$

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

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

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

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

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

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

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

 $$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$
 
 Thus, whereas Knuth's universe has 20 numbers by generation-2, our universe
 $$\surreal{1}{} \similar \surreal{0,1}{} \similar \surreal{-1,1}{} \similar \surreal{-1,0,1}{}$$
 
 Thus, whereas Knuth's universe has 20 numbers by generation-2, our universe

-only has seven numbers. For convenience, we introduce the following two
-definitions (and theorems).
-
-% ==================================================================================================
-% ==================================================================================================
-% ==================================================================================================
+only has seven numbers. For convenience when writing our program which creates
+a (finite set based) surreal numberline, we desire a unique representation for
+each equivalence class. Toward that end, we introduce the following 
+definition and theorem.

 
 \begin{theorem} \label{thm:reduced-form}
 Given an arbitrary number $x$, there exists a number $x' =
 
 \begin{theorem} \label{thm:reduced-form}
 Given an arbitrary number $x$, there exists a number $x' =


@@ -259,45 +256,28 @@ cardinality of both $X'_L$ and $X'_R$ are one or zero.
 \end{theorem}
 
 \begin{proof}
 \end{theorem}
 
 \begin{proof}

-TODO
-% Since a finite, total ordered set contains a maximum and minimum element, it
-% follows that we can define $X'_L$ as containing only the maximum element of
-% $X_L$ and define $X'_R$ as containing only the minimum element of $X_R$. If
-% either $X_L$ or $X_R$ are the empty set, the corresponding $X'_L$ or $X'_R$ is
-% also the empty set.
-% 
-% Then, to show that $x \similar x'$, we apply Axiom \autoref{ax:leq-relation} in
-% both directions.
-% 
-% In the forward direction, $x \leq x'$ means that, $\forall x_L \in X_L$, it
-% must hold that $x_L \leq x'$. Applying \autoref{thm:strong-self-relation}, we
-% conclude that $x_L \leq x'_L < x'$. (this isn't quite right. reconsider my
-% approach)
-% 
-% ...
-% 
-% By Axiom \autoref{ax:leq-relation} and \autoref{thm:weak-self-relation}, for
-% $x'_l \in X'_L$, 
-%
-% ==================
-%
-% The current ten equivalence classes were created by noting that every number
-% with a left or right set containing more than one element is similar to another
-% number whose sets do NOT contain more than one element. Since similarity is
-% built atop the LEQ relation from Axiom \autoref{ax:leq-relation} which itself
-% is based upon element-wise comparisons, we see that the only elements of
-% importance to these relations are the largest element of the left set and the
-% smallest element of the right set. Of course, for such a statement to be
-% sensical, we must first prove that Axiom \autoref{ax:leq-relation} imposes a
-% total ordering, otherwise there might not be a largest or smallest element in a
-% set.
-% 
-% From this we see that, since Axiom \autoref{ax:leq-relation} makes its
-% comparison element-wise, every surreal number generated by our current, finite
-% methods must be similar to a surreal number containing one or zero elements in
-% its left 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.
+Let $x = \surreal{X_L}{X_R}$. Since a finite, total ordered set contains a
+maximum and minimum element, it follows that we can define $X'_L$ as containing
+only the maximum element of $X_L$ and define $X'_R$ as containing only the
+minimum element of $X_R$. If either $X_L$ or $X_R$ are the empty set, the
+corresponding $X'_L$ or $X'_R$ are also the empty set.
+
+Use these two newly defined sets to construct $x' = \surreal{X'_L}{X'_R}$. This
+clearly satisfies Axiom \autoref{ax:number-definition} by inheritance.  Then,
+to show that $x \similar x'$, we apply Axiom \autoref{ax:leq-relation} in both
+directions.
+
+In the forward direction, $x \leq x'$ requires that, $\forall x_L \in X_L$, it
+must hold that $x_L \ngeq x'$. Indeed, from \autoref{thm:strong-self-relation},
+\autoref{thm:transitive}, and our explicit construction of $X'_L$ as the
+maximum element of $X_L$, the desired relation holds since $X_L \leq X'_L <
+x'$.
+
+Also in the forward direction, $x \leq x'$ requires that, $\forall x'_R \in
+X'_R$, it holds that $x'_R \nleq x$. By the same theorems/construction as
+before, the desired relation holds since $x < X_R \leq X'_R$.
+
+The reverse direction follows via a similar pair of arguments.

 \end{proof}
 
 \begin{defi} \label{defi:reduced-form}
 \end{proof}
 
 \begin{defi} \label{defi:reduced-form}


@@ -305,26 +285,49 @@ Given an arbitrary number $x$, the $x'$ guaranteed to exist by
 \autoref{thm:reduced-form} is termed the \emph{reduced form} of $x$.
 \end{defi}
 
 \autoref{thm:reduced-form} is termed the \emph{reduced form} of $x$.
 \end{defi}
 

-\begin{theorem} \label{thm:named-form}
-TODO: Without defining a named form, prove that it is unique.
-\end{theorem}
-
-\begin{proof}
-TODO
-\end{proof}
-
-\begin{defi} \label{defi:named-form}
-TODO: Define a named form as the oldest reduced form of an equivalence class determined by similarity.
-\end{defi}
-
-% ==================================================================================================
-% ==================================================================================================
-% ==================================================================================================
-
-Between \autoref{thm:strong-self-relation} and Definition
-\autoref{defi:named-form} which follows from \autoref{thm:named-form}, we
-finally possesss the tools necessary to finish our Go program and use it to
-generate and examine a surreal number line.
+Note that reduced forms are not unique since $\surreal{}{} \similar
+\surreal{-1}{1}$ but $\surreal{}{} \neq \surreal{-1}{1}$.
+
+Note also that uniqueness is not trivially achieved by applying the generation
+function to an equivalence class. Although all elements are comparable, the set
+is finite, and the natural relation inherits the reflexive and transitive
+properties from $\mathbb{N}$, the relation is not antisymmetric since
+$\surreal{-1}{0} \neq \surreal{-1}{}$ and yet both are from generation-2. Thus,
+stealing well ordering from $\mathbb{N}$ and using it to define a unique
+element to represent the equivalence class turns out to be harder than I had
+hoped.
+
+I suspect the first time a reduced form number from a given equivalence class
+is generated, it will be alone. I base that on viewing the growing set as being
+comprised of two types of numbers, first a primary set forming a sparse line
+bookended by the empty set, making the set longer with every generation.
+
+$$\set{}$$
+$$\set{} < 0 < \set{}$$
+$$\set{} < -1 < 0 < 1 < \set{}$$
+
+Within this set a secondary set is forming, sitting directly in the middle
+between every existing number, making the set denser with every generation.
+Although harder to whip up in \LaTeX, I visualize this like a house of cards
+viewed end-on, a honeycomb lattice linking two parents to each child, adding a
+full spatial dimension to the honeycomb with each generation.
+
+If this visualization is true, then uniqueness of the reduced form representing
+an equivalence class in the first generation said class is encountered, follows
+naturally, even though this result doesn't hold for any following generations.
+
+Regardless, for the purpose of writing a program that generates numbers and
+orders them, the truth of this conjecture is irrelevant. We can simply identify
+the first number encountered in an equivalence class as the canonical
+representation of that class. The naturally serial nature of the program will
+serialize creation of numbers, adding a finer grained age relation and making
+it trivial to choose a `first' element to represent each class.
+
+All this started because I wanted a program to generate some numbers and
+investigate the relation. We finally have the requisite tools to return to that
+task, namely the strong self relation in \autoref{thm:strong-self-relation}
+which solves our comparison problem and the uniqueness of reduced forms in the
+subset of numbers(/equivalence classes) our program will actually generate.

 
 
 \subsection{Conjecture}
 
 
 \subsection{Conjecture}