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1 | \newpage |
2 | \section{Notes: Chapter 3} | |
3 | ||
4 | \subsection{Review} | |
5 | ||
6 | In the third chapter, Knuth begins to embrace similarity exactly as we did in | |
7 | Definition \autoref{defi:similar}, noting that different forms exist which are | |
8 | similar to each other. He also more carefully tracks through the relations in | |
9 | $\surreal{}{0} \leq \surreal{}{} \leq \surreal{0}{}$. | |
10 | ||
11 | Finally, at the end of the chapter, Knuth begins speculating about the form of | |
12 | numbers to come and their relation to each other, pointing out, via an | |
13 | erroneous categorization of positive number forms, that categorization isn't as | |
14 | simple as it first appears | |
15 | ||
16 | ||
17 | \subsection{Exploration} | |
18 | ||
19 | My goal is that of assigning meaningful names to newly generated numbers under | |
20 | the assumption that Knuth's three names ($-1$, $0$ and $1$) were naturally | |
21 | meaningful. Since that implies some notion of `distance from zero' (after all, | |
22 | I'm attempting to build a numberline), defining some form of addition and | |
23 | subtraction seems like a natural starting point. | |
24 | ||
25 | Remember that we're still only considering finite generations for our universe. | |
26 | That means our addition operation will never be closed since our finite | |
27 | universe always has a largest element and nobody can stop us from adding $1$ to | |
28 | it. However, as such closure-violating elements are members of our universe in | |
29 | a later generation, only when considering closure, we will mentally think of | |
30 | our universe as something like $\mathbb{U} \equiv \bigcup_{n \in \mathbb{N}} | |
31 | \mathbb{U}_n$ and otherwise ignore the problem. | |
32 | ||
33 | Thus, updating my caveat from my Chapter 1 notes: | |
34 | ||
35 | \begin{framed} | |
36 | \noindent All definitions and theorems only consider the case of finite left | |
37 | and right sets. In these finite cases, we're pretending that addition is | |
38 | closed, but it's not. | |
39 | \end{framed} | |
40 | ||
41 | Based on my Chapter 2 notes, when considering my numbers in the line defined by | |
42 | my total order, I want every reduced form number that has both left and right | |
43 | ancestors to be the geometric mean of those ancestors. And any reduced form | |
44 | number missing one or both ancestors is similar to another, more suitable | |
45 | reduced form number. | |
46 | ||
47 | Also in my Chapter 2 notes, I hand-wavingly defined such a geometric mean | |
48 | recursively, just to build a mental picture. However, since I don't see a way | |
49 | to define the same concept using only the left and right sets of the operands, | |
50 | a recursive definition might be the right approach as long as it reduces the | |
51 | overall age of the numbers involved, allowing me to use the same | |
52 | sum-of-generations type argument used when proving transitivity. | |
53 | ||
54 | \begin{defi} \label{defi:addition} | |
55 | For two numbers $x$ and $y$, define \emph{addition} as | |
56 | $$ | |
57 | x \sgkadd y = \surreal{X_L}{X_R} \sgkadd \surreal{Y_L}{Y_R} | |
58 | \equiv \surreal{\set{X_L \sgkadd y} \cup \set{Y_L \sgkadd x}}{\set{X_R \sgkadd y} \cup \set{Y_R \sgkadd x}} | |
59 | $$ | |
60 | where $\set{A \sgkadd b}$ means the set of numbers $a \sgkadd b$ for all $a \in A$. | |
61 | ||
62 | We are using the symbol $\sgkadd$ in order to keep our addition distinct | |
63 | from whatever Knuth eventually defines. | |
64 | \end{defi} | |
65 | ||
66 | Given the obvious symbolic symmetry, we won't bother explicitly proving this | |
67 | operation is commutative. | |
68 | ||
69 | Keep in mind that this is only guaranteed to produce a valid number (per Axiom | |
70 | \autoref{ax:number-definition}) subject to our caveat regarding closure. | |
71 | ||
72 | Since we have defined this operation in terms of specific forms, we must ensure | |
73 | the operation behaves identically with respect to all similar forms. It would | |
74 | be a shame if, for example, $0+0=0$ only held for certain values of $0$. | |
75 | ||
76 | \begin{theorem} \label{thm:sgkadd-welldefined} | |
77 | The binary operation $\sgkadd$ on $\mathbb{U}$ is well defined. That is, for | |
78 | numbers $x, x', y, z \in \mathbb{U}$ such that $x \sgkadd y = z$ and $x | |
79 | \similar x'$, $\exists z' \in \mathbb{U}$ such that $x' \sgkadd y = z' | |
80 | \similar z$. | |
81 | \end{theorem} | |
82 | ||
83 | \begin{proof} | |
84 | TODO | |
85 | \end{proof} | |
86 | ||
87 | \begin{theorem} \label{thm:sgkadd-identity} | |
88 | The number $0 = \surreal{}{}$ is the identity element for the binary | |
89 | operation $\sgkadd$ on $\mathbb{U}$. That is, for any number $x \in | |
90 | \mathbb{U}$, $x \sgkadd 0 \similar 0 \sgkadd x \similar x$. | |
91 | In this behavior, the number $0$ is unique up to similarity. | |
92 | \end{theorem} | |
93 | ||
94 | \begin{proof} | |
95 | TODO | |
96 | \end{proof} | |
97 | ||
98 | \begin{theorem} \label{thm:sgkadd-associative} | |
99 | For all $x, y, z \in \mathbb{U}$, it holds that | |
100 | $$(x \sgkadd y) \sgkadd z \similar x \sgkadd (y \sgkadd z).$$ | |
101 | \end{theorem} | |
102 | ||
103 | \begin{proof} | |
104 | TODO | |
105 | \end{proof} | |
106 | ||
107 | \begin{defi} \label{defi:inverse} | |
108 | For a number $x$, let \emph{negation} be defined as | |
109 | $$ | |
110 | -x = -\surreal{X_L}{X_R} \equiv \surreal{-X_R}{-X_L} | |
111 | $$ | |
112 | where $-A$ means the set of numbers $-a$ for all $a \in A$. | |
113 | \end{defi} | |
114 | ||
115 | \begin{theorem} \label{thm:sgkadd-inverse} | |
116 | For every number $x \in \mathbb{U}$, there exists a number $-x \in | |
117 | \mathbb{U}$ such that $x \sgkadd -x = 0$. | |
118 | In this behavior, the number $-x$ is unique up to similarity. | |
119 | \end{theorem} | |
120 | ||
121 | \begin{proof} | |
122 | TODO | |
123 | \end{proof} | |
124 | ||
125 | Putting that all together, $(\mathbb{U},\sgkadd)$ is well defined, closed, and | |
126 | respects the three group axioms. It's a group. Let's name it | |
127 | $\mathbb{U}_{\sgkadd}$. It's also commutative. | |
128 | ||
129 | ||
130 | \subsection{Conjecture} | |
131 | ||
132 | $\mathbb{U}_{\sgkadd}$ really is a group. | |
133 |