#cites-dyn-dict #footnotes-builtin #headings-verbose #one-cooperative #math-latex #genres-doc-hyp #genres-heavenly-script #genres-legal-latin #genres-jump-over 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. Same reason 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. > 2. made unreliable by errors or alterations. > 3. in a state of decay; rotten or putrid. > 4. cause to act dishonestly. > 5. change or debase. > 6. infect; contaminate. 0: And as far as George and his infinite numbers were concerned, the definition sort of fits. 1: Which definition? 0: At least 5, or even 2, 3, and 6, according to some people at the time. 1: I'm confused. 0: I just mean whether you liked his stuff or not, 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 into taboo mathematics isn't exactly the worst way to get corr--- 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". One 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'm only half convinced that's different. 0: Well our minds seem to distinguish them at some level. We use totally different words for the two ideas. \ --- After all, there's: - "first, second, third" vs "one two three" in english."[^9] - "premier, deuxième, troisième" contre "un deux trois" en anglais."[^10] - "primero, segundo, tercero" contra "uno dos tres" en inglés."[^11] - "первый, второй, третий" против "один два три" на английском."[^12] - "पहला, दूसरा, तीसरा" बनाम "एक दो तीन" अंग्रेज़ी में।."[^13] - "اول، دوم، سوم" در مقابل "یک دو سه" به انگلیسی."[^14] - "перший, другий, третій" проти "один два три" англійською."[^15] - "første, andre, tredje" versus "en to tre" på engelsk."[^16] - "erste, zweite, dritte" gegen "eins zwei drei" auf englisch."[^17] - "primeiro, segundo, terceiro" contra "um dois três" em inglês."[^18] [^9]: This passage has troubled scholars since the discovery of the documents that have come to be known as Sudocode. Here the Zero character shows a marked departure from 0's usual erudite if rather informal manner of speech, and in this section appears to have simply copy-pasted the sentence into some translation software, with zero (pun unint.) regard for the fact that the translation of "English" into French et al. is not in fact the French word for French. A minority of scholars in the more fundamentalist interpretive tradition known as Paradocs (\[sic\] not to be confused with Paradox, but rather meaning those whose interpretive tradition defines itself in opposition to the Orthodocs (\[sic\] not to be confused with Orthodox (though in this case they happen to mean roughly the same thing))) have argued that the original Author(s) of Sudocode made this mistake intentionally, in order to make a point about the insufficiently strong type systems of natural languages in general, (Given that the book (or whatever it is) tracks the development of computing from its earliest history through at least the 20XY era in part through a series of increasingly strong type systems, (eventually climaxing with (the rather hard to grasp or take in in their entirety) CoC style systems such as Coq (now called Rocq) and other hard to grok systems at the top of the so-called lambda cube) this position is not without merit. Whatever the origin of this passage, it is very clear that the documents that have come to be called Sudocode are not the work of a single Author with a consistent thesis. Though there is evidence that certain core documents within the volume may have been written by one or a small number of Authors, the document we have at present appears to have been corrupted (or at least contributed to) by latter day Authors and Redactors about which little can be known.) by making a point that Nat. Lang. Soft. Trans. systems lack sufficient precision in the ability to specify the types of words that they (the nat. lang.s) are all but incapable of expressing even concepts as simple as `"The word for $word in $lang is $(trans $word $lang)"` without the use of considerable circumlocutions on the part of the speaker, or else, of course, pseudo-code. [^10]: Ibid. [^11]: Op. cit. [^12]: Cf. supra, F.N. 1. [^13]: See discussion in n.1 above. [^14]: Vide supra n.1. [^15]: Loc. cit. [^16]: As previously noted (see F.N. 1). [^17]: Contra, but ultimately identical in import; see F.N. 1. [^18]: Scholars generally treat this as repeating the pattern established in F.N. 1. --- 1: How so? 0: Like "first second third" vs "one two three". 1: Oh ok, that makes sense. 0: One's 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]]