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37beaed0 | 1 | \newpage |
4751e940 | 2 | \section{Chapter 2: Symbols} |
37beaed0 AT |
3 | |
4 | \subsection{Review} | |
5 | ||
6 | In the second chapter, Knuth doesn't add much new information. The only | |
7 | interesting bits I see are the following two quotes: | |
8 | ||
9 | \begin{framed} | |
10 | ``The Stone's version is a little different, but $x \leq y$ must mean the same | |
11 | thing as $y \leq x$.'' | |
12 | \end{framed} | |
13 | ||
14 | \begin{framed} | |
15 | ``On the first day of creation, Conway ``proves'' that $0 \leq 0$. Why should | |
16 | he bother to prove that something is less than or equal to itself, since it's | |
17 | obviously equal to itself.'' | |
18 | \end{framed} | |
19 | ||
20 | Why, indeed? It seems our prior efforts are headed in a useful direction. | |
21 | ||
22 | ||
23 | \subsection{Exploration} | |
24 | ||
25 | This week I'm interested in defining an addition operation on the surreal | |
26 | numbers generated during my Chapter 1 explorations. | |
27 | ||
28 | When crafting generation-2, we noted that $\surreal{}{} \similar | |
29 | \surreal{-1}{1}$, leading us to conjecture that each number takes on a value | |
30 | directly between the value of its left and right sets, ignoring the null set | |
31 | for now, and assuming the number is in reduced form. | |
32 | ||
33 | Let's begin by attempting to create an addition rule satisfying | |
34 | ||
35 | $$-1 + 1 = 0.$$ | |
36 | ||
37 | Since $-1 = \surreal{}{0}$ and $1 = \surreal{0}{}$, and since $\surreal{0}{0}$ | |
38 | is not a valid number per Axiom \autoref{ax:number-definition}, we're pushed | |
39 | toward a definition for | |
40 | ||
41 | $$x + y = z$$ | |
42 | ||
43 | which discards $X_R$ and $Y_L$. This rules out simple definitions like $Z_L = | |
44 | X_L \cup Y_L$ and $Z_R = X_R \cup Y_R$. | |
45 | ||
46 | We could try something like $Z_L = X_L \cup y$ and $Z_R = Y_R \cup x$, but then | |
47 | addition is non-commutative and non-closed, creating non-numbers if $y > x$. | |
48 | But, if we could rework that definition to function regardless of the relative | |
49 | magnitude of $x$ and $y$, then we might be on to something useful. | |
50 | ||
51 | ||
52 | \subsection{Conjecture} | |
53 | ||
54 | Geometrically, and allowing myself a recursive definition, I want something | |
55 | like $Z_L = x$ and $Z_R = x + y + y$, but expressed in terms of set operations. | |
56 |