1: What's going on down here? 0: Corruption. Just jump over it. --- ## Or: Gen ## Or: 根 ## Or: 木艮 ## Or: MuGen ## Or: 無限 ## Or: ∞ ### Or: Questionable Derivations --- ## Corruption 1: Corruption? 0: 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 the files with only numbers and no names. 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 they were at least changing the norms. And 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. I mean damn, 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 "stable mathematics" that more or less allowed the execution of arbitrary code. At least 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 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, ... nth. 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: For finite numbers it's not. For Cantor's infinite numbers, they're slightly different. He called them Ordinals and Cardinals. 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: Weird. 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, they allowed him to do this: $0, \; 1, \dots, \; \omega, \; \omega+1, \; \dots \;$ 1: Then what? 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.