#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins
Episode publishedPodcast: Lex Fridman PodcastLex Fridman Podcast
Transcript preview
Preview only
The following is a conversation with Joel David Hamkins, a mathematician and philosopher specializing in set theory, the foundation of mathematics, and the nature of infinity. He is the number one highest rated user on Math Overflow, which I think is a legendary accomplishment. Math Overflow, by the way, is like Stack Overflow, but for research mathematicians. He is also the author of several books, including Proof and the Art of Mathematics and lectures on the philosophy of mathematics. And he has a great blog, infinitely more dot x y z. This is a super technical, super fun conversation about
Well, that's a really great question. I would wanna start talking about infinity and telling the story much earlier than Cantor, actually, because I mean, you can go all the way back to ancient Greek times when Aristotle emphasized the potential aspect of infinity as opposed to the impossibility according to him of achieving an actual infinity. And Archimedes method of exhaustion where he is trying to understand the the area of a region by carving it into more and more triangles, say, and sort of exhausting the area, thereby understanding the total area in terms of the sum of the areas of the
Yeah. Let's talk about the Hilbert's Hotel.
Hilbert's Hotel is a hotel with infinitely many rooms. You know, each room is a full floor suite. So there's Floor 0. I always start with zero because for me, the natural numbers start with zero, although that's maybe a point of contention for some mathematicians. The other mathematicians are wrong.
Like a bunch of Diablo programmers, so starting at zero is a wonderful place to start.
Exactly. So there's Floor 0, Floor 1, Floor 2, or Room 0123, and so on, just like the natural numbers. So Hilbert's hotel has a room for every natural number, and it's completely full. There's a person occupying Room N for every n. But meanwhile, a new guest comes up to the desk and wants a room. Can I have a room, please? And the manager says, hang on a second. Just give me a moment. And you see, when the other guests had checked in, they had to sign an agreement with with the hotel that maybe there would be some changing of the rooms, you know, during the stay. And so the manager sent a mess
So to be just say one more time, the hotel is full The hotel is full. And then you could still squeeze in one more. And that breaks the traditional notion of mathematics and and breaks people's brains about when they try to think about infinity, I suppose. This is a property of infinity.
It's a property of infinity that sometimes when you add up an element to a set, it doesn't get larger. That's what this example shows. But one can go on with Hilbert's hotel, for example. I mean, maybe the next day, you know, 20 people show up all at once. We can easily do the same trick again, just move everybody up 20 rooms. And then we would have 20 empty rooms at the bottom, and those new 20 guests could go in. But on the following weekend, a giant bus pulled up, Hilbert's bus. And Hilbert's bus has, of course, infinitely many seats. There's seat zero, seat one, seat two, seat three, and s
You you separate the hotel into even and odd rooms, and you squeeze in, and you help or bust people into the odd rooms, and the previous occupants go into the even rooms?
Keep exploring with a free account.
A free account preserves the short preview. Upgrade to Pro for the full transcript and episode chat.