1: Are you making fun of me?
0: Yes. This stuff is important to the story. It's silly to toss everything in /opt just because you think someone somewhere might think it's a digression.
1: But you're doing the same thing now.
0: Exactly.
1: Anyways, you were gonna explain why calling us "The Foundational People" isn't just another example of the tribalism thing from /opt/names.
0: Right. Ok, let's start at the beginning.
1: Listening.
0: _(ahem)_
_(Narrator: 0 clears 0's throat.)_
0: Once upon a time, in the late 1800s, a guy named George segfaulted mathematics.
1: Segfaulted mathematics?!
0: Ok well not quite. More like he got root on the universe.
1: _Got ROOT on the UNIVERSE?!_
0: Ok that's not quite right either. His name was Georg Cantor.
---
1: Was he the first Foundational Person?
0: No no, he's not one of ours. Not to exclude him. He could have used a good hug and some proper friends, and I would have loved to know the guy. But in his heart, Cantor was a mathematician. He thought like a mathematician, and he wanted to be accepted by the mathematicians.
1: Why wasn't he accepted?
0: Well, he made the mistake of doing some mathematics.
1: What's wrong with that?
0: Well this mathematics was different. Or more accurately, it was the same. See what Cantor did was to use standard mathematical reasoning, the kind that was generally accepted by the mathematical community, to show that if you accept the standard types of definitions and proofs that are common in mathematics, then you're basically forced to accept that infinitely many different sizes of infinity "exist." There's a whole hierarchy of infinite "numbers" up above the normal numbers like 0, 1, ..., etc. And if you believe in the set of normal positive whole numbers, like if you agree that set is a thing, then it's pretty much impossible within normal mathematical reasoning to avoid being forced to admit that all the rest of these infinite numbers upstairs exist too.
1: Is that bad?
0: Well some people loved it. Like this guy who's quoted incessantly in every book about this stuff.
---
> _"No one shall expel us from the paradise that Cantor has created."_
> -David Hilbert.
---
0: Others people thought Georg was corrupting the youth.
1: C'mon, nobody would actually think infinity is "not suitable for kids."
0: I'm serious. Here's an example.
> Kronecker, who headed mathematics at Berlin until his death in 1891, became increasingly uncomfortable with the prospect of having Cantor as a colleague, perceiving him as a "corrupter of youth" for teaching his ideas to a younger generation of mathematicians.
>
> -The Dynamic Read-Writable Free Encyclopedic Repository of the Modern State of Human Knowledge
1: That's a bit of an over-reaction don't you think?
0: Not at all. Georg was totally corrupting the youth.
1: That's ridiculous.
![[corrupting-the-youth.png]]
1: How are infinite numbers corrupting the youth?
0: Well they're 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.
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:
_Consider the set of natural numbers:_
$0, \; 1, \; \dots$
1: Naturally.
0: 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 one." Cardinals are like "X has 7 things in it."
1: So 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:
_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, Georg 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, the cardinality of $\omega \cdot 2$ is the same as the cardinality of $\omega$. That's a fancy way of saying they have the same size. Because Georg's definition of "same size" is "If you change only the names, do they have the same number?"
1: What do you mean "If you change only the names?"
0: Well no matter what we believe about mathematics, like whether we believe that "infinite stuff exists" or whether we think that's all nonsense, one thing everyone can agree on is names aren't magic.
1: Obviously. I don't know what you mean, but I think I agree.
0: All I meant by magic is that the following situation would be completely unacceptable:
Suppose you have a set of stuff, like:
$0, \; 1, \dots, \; \omega, \; \omega+1, \; \dots$
Then suppose you just change the squiggles we use to write down the numbers without changing anything else about the set. For example, suppose you put hats on everything:
$\hat{0}, \; \hat{1}, \dots, \; \hat{\omega}, \; \hat{\omega}+\hat{1}, \; \dots$
0: Is that the same set?
1: I don't know. Maybe not. I could imagine the hat-numbers might be different objects.
0: Exactly. Now, do the two sets have _the same size_?
1: I don't know. I'm not convinced they do.
0: Explain?
1: Like maybe the hats weigh something. Maybe $\hat{1}$ means "The number $1