- That's right. So what it really shows... I mean, another way of thinking about it is that, well, we can define that a set is countable if it is equinumerous with a set of natural numbers. And a kind of easy way to understand what that's saying in terms of Hilbert's Hotel is thata set is countable if it fits into Hilbert's Hotel, 'cause Hilbert's Hotel basically is the set of natural numbers in terms of the room numbers. So to be
occupants of the hotel could, you know, have the same number as any of the train cars. So putting countably many countable sets together to make one big union set is still countable . It's quite remarkable, I think. I mean when I first learned this many, many years ago, I was completely shocked by it and transfixed by it. It was quite amazing to me that this notion of countable infinity could be closed under this process of
- So let's talk about the real numbers. What are the real numbers? Why do they break infinity? The countable infinity. Looking it up on Perplexity, real numbers include all the numbers that can be represented on the number line, encompassing both rational and irrational numbers. We've spoken about the rational numbers, and the rational numbers, by the way, are by definition, the numbers that can be represented
You can literally count them, one, two, three and so on. So Cantor called these countable infinities, but then there are bigger infinities, Cantor called them uncountable. These infinities like the set of all real numbers, the complex numbers, they can't be matched one-to-one with the natural numbers.
You can't even count. It's just non-countable , the ways of expressing the feeling. And also, traveling around and finding new waves, exploring and going on surf trip with friends, for me,
correspondence. It's traditional to talk about the uncountability of the real numbers. Cantor's big result was that the set of all real numbers is an uncountable set. Maybe if we're going to talk about countable sets, then I would suggest that we talk about Hilbert's Hotel, which really makes that idea perfectly clear. - Yeah, let's talk about Hilbert's Hotel.
countable if it is equinumerous with a set of natural numbers. And a kind of easy way to understand what that's saying in terms of Hilbert's Hotel is thata set is countable if it fits into Hilbert's Hotel, 'cause Hilbert's Hotel basically is the set of natural numbers in terms of the room numbers. So to be equinumerous with a set of natural numbers is just the same thing as to fit into Hilbert's Hotel. And so what we've shown is that
- Exactly right. We've proved that if you have countably many countable sets, then the union of those sets, putting all those sets together into one giant set, is still countable . You know, because the train cars are each countable , plus the current hotel. It's sort of like another train car, if you want to think about it that way. The current occupants of the hotel could, you know, have the same number as any of the train cars. So putting countably many countable sets together to make one big union set is
But all those things are good. there were a countable number of palm trees.
infinity, then it shouldn't be surprising... that the union of two countable sets is countable . So there's another way to push this a bit harder, and that is when Hilbert's train arrives, and Hilbert's train has infinitely many train cars...
So there's the, you know, row zero, row one, row two and so on, column zero, column one, column two and so on, and each row and column has a countable infinity of points on it, right? So those dots, if you think about them as dots, are really the same as the train cars if you think about each column
perspective maybe that you're adopting, but it's not true, and that's the profound achievement that Cantor made is proving that the set of real numbers is not a countable infinity. It's a strictly larger infinity, and therefore there's more than one concept of infinity, more than one size of infinity. - So let's talk about the real numbers. What are the real numbers? Why do they break infinity?
set that we built is... has many more elements than the old set in the sense that there are additional elements, but it doesn't have many more elements in terms of its size because it's still just a countable infinity and it fits into Hilbert's Hotel. - Have you been able to sort of internalize a good intuition about countable infinity? Because that is a pretty weird thing. You can have a countably
more elements in terms of its size because it's still just a countable infinity and it fits into Hilbert's Hotel. - Have you been able to sort of internalize a good intuition about countable infinity? Because that is a pretty weird thing. You can have a countably infinite set of countably infinite sets, and you can shove it all in and it still is a countable infinite set.
train, infinite number of cars, with each car having infinite number of seats. - Exactly right. We've proved that if you have countably many countable sets, then the union of those sets, putting all those sets together into one giant set, is still countable . You know, because the train cars are each countable , plus the current hotel. It's sort of like another train car, if you want to think about it that way. The current
still countable . It's quite remarkable, I think. I mean when I first learned this many, many years ago, I was completely shocked by it and transfixed by it. It was quite amazing to me that this notion of countable infinity could be closed under this process of infinitely many infinities adding up still to the very same infinity, which is a strong instance, a strong violation of Euclid's principle once again, right? So, the new
So those dots, if you think about them as dots, are really the same as the train cars if you think about each column of... in that integer lattice, it's a countable infinity. It's like one train car and then there's the next train car next to it, and then the next column next to that, the next train car.
3 to the P times 5 to the Q. The same idea works. with the rational numbers. So this is still a countable set. And you might think, "Well, every set is going to be countable because there's only one infinity." I mean, if that's a kind of perspective maybe that you're adopting, but it's not true, and that's the profound achievement that Cantor made is proving that the set of
His proof also used an uncountable number of steps, was that even allowed? Some mathematicians argued proofs should be finite, others accepted infinity, but only the countable kind and then things got worse. When mathematicians played around with the axiom of choice, it created disturbing results.
And I just remember feeling like I'd been punched in the gut. The idea that the amount of time you have left with the people that you care about most in the world is that finite and countable that you can place it onto a few hands just shook me to the core.
is still countable . You know, because the train cars are each countable , plus the current hotel. It's sort of like another train car, if you want to think about it that way. The current occupants of the hotel could, you know, have the same number as any of the train cars. So putting countably many countable sets together to make one big union set is still countable . It's quite remarkable, I think. I mean when I first learned this many, many years ago, I was completely shocked by it and
- Have you been able to sort of internalize a good intuition about countable infinity? Because that is a pretty weird thing. You can have a countably infinite set of countably infinite sets, and you can shove it all in and it still is a countable infinite set. - Yeah, that's exactly right. I mean, I guess, of course, when you work with these notions, the argument of
themselves. So it's a kind of different picture. Before, we used this 3 to the C, 5 times 5 to the S, which is a kind of, you know, overly arithmetic way to think about it. But there's a kind of direct way to understand that it's still a countable infinity when you have countably many countable sets, because you can just start putting them on this list. And as long as you give each of the infinite collections a chance to add one more person to the list, then
integers, which are discretely ordered, right? From any integer, there's a next one and a previous one, and so on. But that's not true in the rational numbers. And yet, the rational numbers are also still only a countable infinity. And the way to see that is actually it's just exactly the same as Hilbert's train again, because every fraction consists of two integers: the numerator and the denominator. And so if I tell you two natural
'fewer audience injuries than last year'. And I guess the marketing people had originally written it as 'less audience injuries than last year' and he made them change it to be grammatically correct. And so we took our picture together with the thumbs up on 'fewer audience injuries' because the audience injuries are countable . Another thing I see people having trouble with, especially in business, you see this all the time in business, is improper capitalization. So capitalizing the name of your award or,