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Joel David Hamkins

lines on the record: 286
first: 2025-12-31
most recent: 2025-12-31
sittings or episodes: 1
sources: podcast

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“Arsavin is the nth real number on the list. Basically, our assumption allows us to think of the real numbers as having been placed on a list, R1, R2, and so on. Okay, and now I'm going to define the number Z, and it's going to be the integer part is going to be a zero, and then I'm going to put a decimal place, and then I'm going to start specifying the digits of this number Z. D1, D2, D3. And what I'm going to make sure is that the nth digit after the decimal point of z is different from the nth digit of the nth number on the list. Okay, so to specify the nth digit of z, go to the nth number on the list r sub n, and I look at its nth digit after the decimal point. And whatever that digit is, I make sure that my digit is different from it.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Now, obviously, since the natural numbers are included in the real numbers, we know that the real numbers are at least as large as the natural numbers. And so the claim that we want to prove is that it's strictly larger. So suppose that it wasn't strictly larger. So then they would have the same size. But to have the same size, remember, means by definition that this is a one-to-one correspondence between them. So we suppose that the real numbers can be put into one-to-one correspondence with the natural numbers. So therefore, for every natural number n, we have a real number, let's call it r sub n”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Okay, so Cantor wants to prove that the infinity of the real numbers is different and strictly larger than the infinity of the natural numbers. So the natural numbers are the numbers that start with zero and add one successively. So 0, 1, 2, 3, and so on. And the real numbers, as we said, are the numbers that come from the number line, including all the integers and the rationals and the algebraic numbers and the transcendental numbers and all of those numbers all together.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“How do you even begin? I'm going to prove to you that every natural number is interesting. I mean, Zero is interesting because it's the additive identity, right? That's pretty interesting. And one is the multiplicative identity. So when you multiply it by any other number, you just get that number back, right? And two is the first prime number that's super interesting, right? Okay, so one can go on this way and give specific reasons, but I want to prove as a general principle that every number is interesting. And this is the proof. Suppose toward contradiction that there were some boring numbers.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“In all the different kinds of sets. And if you have a kind of simplicity attitude, then zero and one are looking pretty good too. So, and they're definitely not. Sorry.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“That's right. So some of the famous transcendental numbers would include the number pi, you know, the 3.14159265 and so on. So that's a transcendental number. Also, Euler is constant. The E, like E to the X, the exponential function”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Oh, great. So it was Louisville who first proved that there are transcendental numbers, and he exhibited a very specific number that's now known as the Louisville constant, which is a transcendental number. Candor also famously proved that there are many, many transcendental numbers. In fact, it follows from his argument on the accountability of the real numbers that there are uncountably many transcendental numbers, so most real numbers are transcendental”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“That's right. So with the real numbers, we have the algebraic numbers. We have, of course, all the rational numbers, the integers and the rationals are all part of the real number system, but then also we have the algebraic numbers like the square root of two or the cube root of five and so on, numbers that solve an algebraic equation over the integers. Those are known as algebraic numbers. It was an open question for a long time whether that was all of the real numbers or whether there would exist numbers that are the transcendental numbers. The transcendental numbers are real numbers that are not.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But it's not true, and that's the profound achievement that Cantor made is proving that the set of real numbers is not accountable infinity. It's a strictly larger infinity, and therefore there are more than one concept of infinity, more than one size of infinity.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And yet, the rational numbers are also still only accountable 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 numbers, then you know what fraction I'm talking about. I mean, plus the sine issue, I mean, if it's positive or negative. But if you just think about the positive fractions, then you have the numbers of the form P over Q, where Q is not zero. So you can still do 3 to the P times 5 to the Q if 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 a...”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“So Maybe there's one more step I want to insert before doing that, which is. The rational numbers, so we did pairs of natural numbers That's the train car, basically. But maybe it's a little bit informative to think about the rational, the fractions, the set of fractions or rational numbers. Because a lot of people maybe have an expectation that maybe this is a bigger infinity because the rational numbers are densely ordered between any two fractions you can find another fraction, right? The average of two fractions is another fraction. And so sometimes people, it seems to be a different character than the integers which are discreetly 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.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Overly arithmetic way to think about it, but there's a kind of direct way to understand that it's still accountable 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 you're going to accommodate everyone in any of the sets in one list.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“But if we think about it in this grid manner, then I can imagine a kind of winding path winding through these grid points like up and down the diagonals winding back and forth. So I start at the corner point and then I go down up into the left and then down into the right, up into the left, down into the right, and so on, in such a way that I'm going to hit every grid point on this path. So this gives me a way of assigning room numbers to the points because Every grid point is going to be the nth point on that path for some end and that gives a correspondence between the grid points and the natural numbers themselves. So it's a kind of different picture. I mean, before we use this three to this C5 times 5 to the S, which is a kind of”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yeah, that's exactly right. I mean, I guess. Because when you work with these notions that the argument of Hilbert Sortel becomes kind of clear, there's many, many other ways to talk about it too. For example, let's think about, say, the integer lattice, the grid of points that you get by taking pairs of natural numbers, say, so the upper right quadrant of the integer lattice. So there's the 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? Those dots, if you think about them as dots, are really the same as the train cars. If you think about each column in that integer lattice, it's accountable infinity. It's like one train car and then the next train car next to it. And then the next column next to that, the next train car.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Once again, right? So the new set that we built has many more elements than the old set in the sense that there's additional elements, but it doesn't have many more elements in terms of its size because it's still just accountable infinity and it fits into Hilbert's hotel.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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, 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 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 infinitely many infinities adding up still to the very same infinity, which is a strong instance, a strong violation of Euclid's principle.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Three to the C times 5 to the S. To the sea, three to the car number, so three times three times three, you know, the number of the car. You multiply three by itself the number of the train car, and then you multiply five by itself the seat number times, and then you multiply those two numbers together. So 3 to the seat times five to the S. That's always an odd number because the prime factorization has only threes and fives in it. There's no two there. So therefore it's definitely an odd number. And it's always different because of the uniqueness of prime factorization. So every number can be factored uniquely into prime. So if you have a number of that form, then you can just factor it and that tells you the exponent on three and the exponent on five. And so you know exactly which prime.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And so we have an infinity of infinities of the train passengers together with the current occupants of the hotel and everybody on the train wants to check in to Hilbert's hotel So the manager can again, of course, send a message up to all the rooms, telling every person to double their room number again. And so that will occupy all the even numbered rooms again, but free up again the odd numbered rooms. So somehow we want to put the train passengers into the odd numbered rooms. And so, well, every train passenger is on some car, let's say car C and seat S. Somehow we have to take these two coordinates c, s, the car number and the seat number, and produce from it an odd number in a one-to-one way. And that's actually not very difficult. In fact, one can just use, say, an easy way to do it is to just use the number.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“You have two countably infinite sets, then their union is also countably infinite. If you put them together and form a new set with all of the elements of either of them, then that union set is still only countably infinite. It didn't get bigger. And that's a remarkable property for a notion of infinity to have, I suppose. But if you thought that there was only one kind of infinity, then it wouldn't be surprising at all because if you take two infinite sets and put them together, then it's still infinite. And so if there were only one kind of infinity, then it shouldn't be surprising that the union of two countable sets is countable. So, that's another way to push this a bit harder, and that is when. When Hilbert's train arrives, and Hilbert's train has infinitely many train cars, and each train car has infinitely many seats”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“That's right. So, what it really shows, I mean, another way of thinking about it is that what 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 that a set is countable if it fits into Hilbert's hotel because Hilbert's hotel basically is the set of natural numbers in terms of the room numbers. So to be equanumerous 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”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“That's exactly right. So, I mean, that's a very easy way to do it. If you just tell all the current guests to double their room number, so in room N, you move to room two times n. So they're all going to get their own private room, the new room, and it will always be an even number because two times n is always an even number. And so all the odd rooms become empty that way. And now we can put the bus occupants into the odd numbered rooms.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“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 so on. And so one wants to, you know, all the people on the bus want to check into the hotel, but the hotel is completely full. And so what is the manager going to do? And when I talk about Hilbert's hotel, when I teach Hilbert's Hotel in class, I always demand that the students provide the explanation of how to do it. Maybe I'll ask you can you tell me what is your idea about how to fit them all in the hotel? Everyone on the bus and also the current occupants.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“The 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 Satill, for example. I mean, maybe the next day, 20 people show up all at once. We can easily do the same trick again, just move everybody up.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Of course, and so he can put the new guest in that room. So even when you have infinitely many things, then the new guest can be accommodated. And that's a way of showing how the particular infinity of the occupants of Hilbert's hotel violates Euclid's principle. I mean, it exactly illustrates this idea because adding one more element to a set didn't make it larger because we can still have a one-to-one correspondence between the total new guests and the old guests by the room number, right?”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Exactly. So there's floor zero, floor one, floor two, or room zero, one, two, three, 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 the hotel that maybe there would be some changing of the rooms during the stay. And so the manager sent a message up to all the current occupants and told every person, hey, can you move up one room, please? So the person in room five would move to room six and the person in room six would move to room seven and so on and everyone moved at the same time. And of course, we never want to be placing two different guests in the same room and we won't want everyone to have their own private room. But when you move everyone up one room, then the bottom room zero becomes available.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Hubert's hotel is a hotel with infinitely many rooms. You know, each room is a full floor suite. So there's floor zero. 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”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And it really wasn't fully resolved, I think, until Cantor. He's the one who really explained so clearly about these different sizes of infinity and so on in a way that was so compelling. And so he exhibited to different infinite sets and proved that they're not equinumerous. They can't be put into one-to-one correspondence. And it's traditional to talk about the uncountability of the real numbers. So Canto's big result was that the set of all real numbers is an uncountable set. So maybe if we're going to talk about countable sets, then I would suggest that we talk about Hilbert's hotel, which really is...”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“A principle that Euclid appealed to in the elements, I mean, many times when he's calculating area and so on, it's a kind of basic idea that if something is just a part of another thing, then the whole is greater than the part. And so what Galileo was troubled by was this tension between what we call the Cantor-Hume principle and Euclid's principle.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Attitude about this situation is that those two infinities are exactly the same and that Galileo was right in those observations about the equanumerosity. And the way we would talk about it now is appeal to what I call the Cantor-Hume principle or some people just call it Hume's principle, which is the idea that if you have two collections, whether they're finite or infinite, then we want to say that those two collections have the same size. They're equinumerous if and only if there's a one-to-one correspondence between those collections. And so Galileo was observing that line segments of different lengths are equinumerous and the perfect squares are equinumerous with the whole, all of the natural numbers and to any two circles are equinumerous and so on. The tension between the candor Hume principle and what could be called Euclid's principle, which is that the whole is always greater than the part, which”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“And the midpoints are matched, and so on, so spreading out the lines as you go. And so every point on the shorter line would be associated with a unique, distinct point on the longer line in a one-to-one way. And so it seems like the two line segments have the same number of points on them because of that, even though the longer one is longer. And so it makes, again, a kind of confusion of our ideas about infinity. And also with two circles, if you just place them concentrically and draw the rays from the center, then every point on the smaller circle is associated with a corresponding point on the larger circle, you know, in a one to one way. And again, that seems to show that the smaller circle has the same number of points on it as the larger one precisely because they can be put into this one-to-one correspondence. Now, of course, the contemporary”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Seems like on the basis of this one to one correspondence that there should be exactly the same number of squares, perfect squares as there are numbers. And yet there's all the gaps in between the perfect squares, right? And this suggests that there should be fewer perfect squares, more numbers than squares because the numbers include all the squares plus a lot more in between them, right? And Galileo was quite troubled by this observation because he took it to cause a kind of incoherence in the comparison of infinite quantities, right? Another example is if you take two lines segments of different lengths and you can imagine drawing a kind of foliation, a fan of lines that connect them so the endpoints are matched from the shorter to the longer segment.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“In many ways, Galileo was anticipating Cantor's developments, except he couldn't quite push it all the way through and ended up throwing up his hands in confusion, in a sense. I mean, the Galileo paradox is the idea or the observation that if you think about the natural numbers, I would start with zero, but I think maybe he would start with one, the numbers 1, 2, 3, 4, and so on. And you think about which of those numbers are perfect squares. So zero squared is zero and one squared is 1 and 2 squared is 4, 3 squared is 9, 16, 25, and so on. And Galileo observed that the perfect squares can be put into a one-to-one correspondence with all of the numbers. I mean, we just did it. I associated every number with its square. And so it's...”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Infinity at all. Galileo is an extremely prominent exception to this, though he argued against this sort of potentialist orthodoxy in the dialogue of two new science is really lovely account there that he gave.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“I would want to 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 area of a region by carving it into more and more triangles, say, and sort of exhausting the area and thereby understanding the total area in terms of the sum of the areas of the pieces that he put into it. It proceeded on this kind of potential under this potentialist understanding of infinity for hundreds of years, thousands of years, almost all mathematicians were potentialists only and thought that it was incoherent to speak of an actual”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Personal side, Cantor's own breakdown. He literally went mad, spending his final years in and out of Sanatorium's obsessed with proving the continuum hypothesis. So laying that all out on the table, can you explain the idea of infinity that some infinities are larger than others? And why was this so transformative to mathematics? Well, that's a really great question.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Some infinities are bigger than others. This idea from Canter at the end of the nineteenth century, I think it's fair to say, broke mathematics before rebuilding it. And I also read that this was a devastating and transformative discovery for several reasons. So one, it created a theological crisis because infinity is associated with God. How could there be multiple infinities? And also cantor was deeply religious himself. Second, there was a kind of mathematical civil war, the leading German mathematician chronicler called Cantor, a corruptor of youth and tried to block his career. Third, many fascinating paradoxes emerged from this, like Russell's paradox about the set of all sets that don't contain themselves and those threatened to make all of mathematics inconsistent. And finally, on the psychological side,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source