1: What's going on down here? 0: Corruption. Just jump over it. goto: [[#Corruption]] --- ## (1,2): Numbers ### Or: Gen[^1] [^1]: gen. (root) The origin, source, or agent of creation of something. From the French -gène, which derives from the Ancient Greek -γενής (-genḗs), as in the Greek word génesis (γένεσις), meaning "origin," "creation," or "generation", and deriving from the Proto-Indo-European root \*ǵenh₁- referring to birth, procreation, and lineage. ### Or: 根[^2] [^2]: 根. (char) A CJKᵃ character meaning "root" and transliterated as "Gen" in Mandarin pinyin.ᵇ ᵃ CJK is a collective term for graphemes used in the Chinese, Japanese, and Korean writing systems, which each include Chinese characters. ᵇ Hereafter zh, following [[ISO 639-1]], in any genre of the surrounding book for which a distinction between language and writing system would come across as somewhere between pointless and insufferably pedantic. ### Or: 木艮[^3] [^3]: 木艮. (chars) The radicalsᵃ of the character 根. ᵃ radical. (noun). Late 14c., "originating in the root or ground;" of body parts or fluids, "vital to life," from Latin radicalis "of or having roots," From Latin radix (genitive radicis) "root" (from PIE root \*wrād- "branch, root"). Found as a root (pun inten.) of various technical terms throughout biology, philology, politics, and mathematics. The basic sense of the word in all meanings is "pertaining or relating to roots, origins, or fundamentals." -Dictionary of the Standard Modern Dialect of Distributed Colloquial Vernacular ### Or: MuGen[^4][^5] [^4]: 木. (char) A character meaning "tree" and written "Mu" in zh. [^5]: 艮. (char) A character representing the sound "Gen", and written "Gen" @ zh. One of the Eight Trigrams, symbolizing mountains. (☶) ### Or: 無限[^6][^7][^8] [^6]: 無. (char) A character meaning "none" or "nothingness", and transliterated as "Mu" in Japanese romaji (hereafter ja, following Ibid). [^7]: 限. (char) A character meaning "limit", and transliterated as "Gen" @ ja. [^8]: 無限. (word) Infinity. ### Or: ∞ #### Or: Questionable Derivations --- ## Corruption 1: Corruption? 0: Corruption. The lost+found directory is where things that have been corrupted are put. 1: Why keep corrupted things? 0: In the hope that they might be salvageable, with some work. 1: So what was all that up there? 0: One often finds incomprehensible markings in here. Streams of raw bytes, in places they don't belong. It comes with the territory. Same as how the files here only have numbers instead of names. Typical lost+found thing. That's corruption. 1: Speaking of numbers and corruption, you never explained. 0: Explained what? 1: Why did people think infinite numbers were "corrupting the youth"? 0: Well, corruption is tricky, it's got more than one meaning. > corrupt. /kəˈrəpt/ > (adjective, verb) > 1. having or showing a willingness to act dishonestly in return for money or personal gain. > 2. made unreliable by errors or alterations. > 3. in a state of decay; rotten or putrid. > 4. cause to act dishonestly in return for money or personal gain. > 5. change or debase by making errors or unintentional alterations. > 6. infect; contaminate. 0: And as far as George and his infinite numbers were concerned, the definition sort of fits. He was at least changing the norms. There are some downsides of infinite numbers for some purposes. But when I said "corrupting the youth," I meant it as a compliment. The youth always make sure to find ways to get corrupted each generation. It's part of what it means to be "the youth." And of all the ways to get corrupted, getting corrupted with taboo mathematics isn't exactly the worst way to get corrupted. Life is about tradeoffs, and mathematics isn't exactly the worst thi--- 1: Fair. 0: But he definitely changed the norms. Georg really stretched the boundaries of the mathematical universe about as far as they could be stretched without bursting. But he wasn't making imprecise or poorly thought out arguments. His arguments were very much "normal mathematics." That's part of why they were so controversial. 1: How can something be "controversial" in normal mathematics? I thought mathematics had some pretty clear-cut rules about what is and isn't rigorous. 0: Not even close. But that's a story for another file. For now, just know that Cantor sort of found a security vulnerability in the "stable" branch of "standard mathematics" that more or less allowed the execution of arbitrary code. 1: Arbitrary code? 0: Well not anything. He didn't break mathematics to the point of causing contradictions. That would have to wait for the next part of the story. But in George's case, he at least found enough of a vulnerability that he could keep calling `+=1` across dot-dot-dots, and that's sort of arbitrary code execution as far as numbers are concerned. 1: What do you mean "across dot-dot-dots." 0: Well everybody agrees it's ok to write this: _(Narrator: Zero's voice changes to a hyperbolically pretentious tone for the next six words. It's not exactly clear why.)_ 0: _"Consider the set of natural numbers"_ $0, \; 1, \; \dots$ 1: Naturally. _(Narrator: 0's voice goes back to normal. By the way, is there a canonical way to spell your names or---)_ 0: Ok so what Cantor did was come up with definitions of "th" and "size" for sets such that--- 1: What's "th"? 0: Like 1th, 2th, 3th, 4th, 5th, 6th, 7th, 8th, 9th, ... nth. 1: 1th? 0: Exactly. Order. There's two kinds of numbers. For example, there's two kinds of seven. 1: How are there two kinds of seven? 0: Well you can be the 7th person in line. That's a different idea from there being 7 people in line. 1: I don't see how that's different. 0: One is about counting, the other's about order. Like in programming, you can have a data structure called (say) `container` that has a `container.size()` method, even if you don't have a `container.get(n)` method that gives you the $n^{th}$ thing in the container. 1: Ah ok of course. 0: Same idea for Cantor's infinite numbers. Counting numbers are the sort of thing `container.size()` returns. Indexing numbers are the sort of thing you pass into the `container.get(n)` method. He called them Cardinals and Ordinals. Ordinals are like "X is the 7th guy in line." Cardinals are like "X has 7 things in it." 1: Makes sense, I think. But wait you forgot to explain what you meant by "across dot-dot-dots." 0: Ok right. So everybody agrees it's ok to write this: _(Narrator: 0 does the voice thing again. Don't ask, idk.)_ 0: _"Consider the set of natural numbers"_ $0, \; 1, \; \dots$ 1: Naturally. We did this already. 0: What Cantor did was come up with definitions of order and size for sets that allowed him to give a precise meaning to this: $0, \; 1, \dots, \; \omega$ 1: What's $\omega$? 0: The $\omega^{th}$ number. 1: That's not helpful. 0: It's the infinity-th number. 1: The _what?_ 0: That's what I mean by saying he could keep calling `+=1` across dot-dot-dots. It's called a "limit ordinal." Cantor came up with some basic definitions in standard mathematical style that let him write that. Then after that, those same definitions allowed him to do this: $0, \; 1, \dots, \; \omega, \; \omega+1, \; \dots \;$ 1: What's after the second dot-dot-dots? 0: Naturally: $0, \; 1, \dots, \; \omega, \; \omega+1, \; \dots, \; \omega \cdot 2, \; \omega \cdot 2 + 1$ 1: Is that two times infinity? 0: Infinity times two. Slightly different once you get into the implementation details, but that's the idea. 1: How is "infinity times two" not just "infinity"? 0: It is!... In terms of size. Remember, George has two definitions of number. They're the same for finite numbers, but slightly different for infinite ones. So $\omega \cdot 2$ has the same _size_ as $\omega$, but it's different in terms of the order it shows up in the big infinite list. 1: This math stuff seems pretty imprecise. 0: It is. At least sometimes. But not in this case. In this case, it makes sense. It's all about names. 1: Names? 0: Names. Here follow me. goto: [[lost+found/1/3|./3]]