| 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. |