## The Informal Infinite 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. > _I don't know what predominates in Cantor's theory, philosophy or theology, but I am sure there is no mathematics there._ > -Leopold Kronecker 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. --- ## Not suitable for kids 0: Other 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: I'll explain, follow me.