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1 | \newpage |
2 | \section{Chapter 4: Bad Numbers} | |
3 | ||
4 | \subsection{Review} | |
5 | ||
6 | This was an interesting chapter, much busier than the previous two. It was nice | |
7 | to see that we haven't veered too far into nonsense. | |
8 | ||
9 | Knuth generates all the elements of generation three and they match our | |
10 | explorations from the Chapter 1 notes. He also proves a transitive law holds | |
11 | for Axiom \autoref{ax:leq-relation} using the same argument I used, showing | |
12 | that younger (simpler) sums reach a contradiction established by the original | |
13 | numbers. Then, with transitivity, he notes that all his numbers are similar to | |
14 | our Definition \autoref{defi:reduced-form} reduced form numbers. Knuth ends the | |
15 | chapter with a wink for meeting him at this point. | |
16 | ||
17 | ||
18 | \subsection{Exploration} | |
19 | ||
20 | I continue my attempt to define an addition operation which respects the | |
21 | implications of the names assigned by Knuth ($-1$, $0$ and $1$) and which | |
22 | extends those names in the pattern I think is developing. | |
23 | ||
24 | From Definition \autoref{defi:addition}, using the symbol $\sgkadd$, we are defining addition as | |
25 | $$ | |
26 | x \sgkadd y = \surreal{X_L}{X_R} \sgkadd \surreal{Y_L}{Y_R} | |
27 | \equiv \surreal{\set{X_L \sgkadd y} \cup \set{Y_L \sgkadd x}}{\set{X_R \sgkadd y} \cup \set{Y_R \sgkadd x}} | |
28 | . | |
29 | $$ | |
30 | ||
31 | Since we have defined this operation in terms of specific forms, we must ensure | |
32 | the operation behaves identically with respect to all similar forms. It would | |
33 | be a shame if, for example, $0+0=0$ only held for certain values of $0$. | |
34 | ||
35 | We already know the outputs aren't well behaved in this manner, but we patched | |
36 | that up by only considering closure in terms of future generations. Now we | |
37 | examine the inputs. | |
38 | ||
39 | \begin{theorem} \label{thm:sgkadd-welldefined} | |
40 | The binary operation $\sgkadd$ on $\mathbb{U}$ is well defined. That is, for | |
41 | numbers $x, x', y, z \in \mathbb{U}$ such that $x \sgkadd y = z$ and $x | |
42 | \similar x'$, $\exists z' \in \mathbb{U}$ such that $x' \sgkadd y = z' | |
43 | \similar z$. | |
44 | \end{theorem} | |
45 | ||
46 | \begin{proof} | |
47 | TODO | |
48 | \end{proof} | |
49 | ||
50 | With the inputs and outputs behaving as needed, we now apply our operation to | |
51 | the set, checking if we meet the group axioms. | |
52 | ||
53 | \begin{theorem} \label{thm:sgkadd-identity} | |
54 | The number $0 = \surreal{}{}$ is the identity element for the binary | |
55 | operation $\sgkadd$ on $\mathbb{U}$. That is, for any number $x \in | |
56 | \mathbb{U}$, $x \sgkadd 0 \similar 0 \sgkadd x \similar x$. | |
57 | In this behavior, the number $0$ is unique up to similarity. | |
58 | \end{theorem} | |
59 | ||
60 | \begin{proof} | |
61 | TODO | |
62 | \end{proof} | |
63 | ||
64 | \begin{theorem} \label{thm:sgkadd-associative} | |
65 | For all $x, y, z \in \mathbb{U}$, it holds that | |
66 | $$(x \sgkadd y) \sgkadd z \similar x \sgkadd (y \sgkadd z).$$ | |
67 | \end{theorem} | |
68 | ||
69 | \begin{proof} | |
70 | TODO | |
71 | \end{proof} | |
72 | ||
73 | \begin{defi} \label{defi:inverse} | |
74 | For a number $x$, let \emph{negation} be defined as | |
75 | $$ | |
76 | -x = -\surreal{X_L}{X_R} \equiv \surreal{-X_R}{-X_L} | |
77 | $$ | |
78 | where $-A$ means the set of numbers $-a$ for all $a \in A$. | |
79 | \end{defi} | |
80 | ||
81 | \begin{theorem} \label{thm:sgkadd-inverse} | |
82 | For every number $x \in \mathbb{U}$, there exists a number $-x \in | |
83 | \mathbb{U}$ such that $x \sgkadd -x = 0$. | |
84 | In this behavior, the number $-x$ is unique up to similarity. | |
85 | \end{theorem} | |
86 | ||
87 | \begin{proof} | |
88 | TODO | |
89 | \end{proof} | |
90 | ||
91 | Putting that all together, our operation $(\mathbb{U},\sgkadd)$ is well | |
92 | defined, closed, and respects the three group axioms. Our universe $\mathbb{U}$ | |
93 | is a group under $\sgkadd$, our addition operation. Let's name this group | |
94 | $\mathbb{U}_{\sgkadd}$. | |
95 | ||
96 | Note: This group is also commutative, seen by running transposed symbols | |
97 | through Definition \autoref{defi:addition} and noting that sets are inherently | |
98 | unordered. | |
99 | ||
100 | ||
101 | \subsection{Conjecture} | |
102 | ||
103 | Given our definition of addition, it seems natural to consider defining | |
104 | multiplication in the same recursively-younger manner. Then see if it's also a | |
105 | group and if it relates to $\mathbb{U}_{\sgkadd}$ in the usual manner. |