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065c48ee | 1 | \newpage |
4751e940 | 2 | \section{Chapter 3: Proofs} |
065c48ee AT |
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 | ||
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66 | Keep in mind that this is only guaranteed to produce a valid number (per Axiom |
67 | \autoref{ax:number-definition}) subject to our caveat regarding closure. | |
68 | ||
4751e940 | 69 | I'll explore this definition more in the next chapter. |