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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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“I mean, of course, it's a theoretical question about the asymptotic behavior of these problems. I mean, for a problem to be in P means that there is a computable decision procedure that runs in time bounded by some polynomial. But the coefficients on that polynomial could be enormous. And the degree could be incredibly high. And so for small values of inputs, then it doesn't make sense to talk about this polynomial time feasibility with respect to, say, the range of problem inputs that we will ever give it in our lifetime or in the span of human civilization or whatever.”
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 an interesting question. Sometimes people ask about whether it could be independent, which I think is an interesting question for logicians And of course, what one has to say if you're entertaining the idea of independence, you know, over which theory? Because every statement is going to be independent over an extremely weak theory. So that's, you know, it doesn't make sense to say it's independent all by itself. You're only independent relative to a theory, right? So the way I think about PNP 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
“And for most of the NP complete problems, you can prove that there's a polynomial time approximation that solves almost all instances in a feasible amount of time. So like the knapsack problem, packing problems and so on, other kinds of problems, satisfaction problem, depending on how you set up the formalism, you can prove, and I've proven many instances of this But also, I think it's widespread for almost all the NP complete problems, the difficult problems, and these are important problems for industrial application. These are problems that we actually want to solve. We can have feasible algorithms that solve almost every instances of them.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“It's a probabilistic way. I mean, it's probabilistic in the sense that we're solving almost all instances. Computably. This version of this that are maybe more interesting from the point of view of complexity theory and actually useful, I mean there's the whole PNP problem and so on and there's this genre of NP complete problems which are problems that are infeasible they would take exponential time to solve them in the ordinary way and they're not known to be polynomial time solvable although in these cases it's an open question whether there is a polynomial time algorithm a feasible algorithm”
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 you can show using this kind of analysis that the probability one behavior of a random Turing machine is that the head falls off the tape before it repeats a state. And that is the stupid proof that shows how to solve the halting problem. Because when that happens, we can answer the halting problem saying, no, the computation stopped because the machine crashed, not because it halted. So therefore it doesn't count as halting on some accounts. Or, you know, if you want to define that as halting, crashing as halting, but in any case, however it is that you set up your formalism, you're going to be able to answer the question for the behavior of the machine when the head falls off.”
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 a pretty stupid reason. Okay, but that's half of them already, just like that. Okay. And then some of them went right and they changed to a new state. And amongst those, you know, the new state, half of those ones are going left and half are going right from that place. And then most of those are changing to a new state. When there's a lot of states, it's very likely that the next state that you transition to is new. And so you get this random walk behavior if you know what that means, where half go left and half go right at each step. And there's a theorem due to polia, which is called the Polia recurrence theorem, which says, when you have a random walk, a one-dimensional random walk, then it's very likely to come back to where you started. And when that happens for us, then half of them from that place fall off on the next step.”
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 was my goal. I love this. Yeah. So we thought more about it though. We hit the jackpot because we found one gigantic stupid reason that converged to 100%. I mean, in the limit. And so the stupid reason for a program not to halt is that, well, if you think about the behavior, say the head is sitting there, it's on the leftmost cell of the tape at the very beginning. It's in the start state and the head is following an instruction and the instruct state, which it is, and you're reading something on the tape, then you should write something and you should change to a new state and you should either move left and right, left or right. But half of them move left. But if you move left and you are already at the end, then the head falls off. So the conversation stops because the head fell off the tape.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“To show. So that's a kind of trivial reason for non halting, you know, and when I first made that observation, I thought, okay, this is the proof strategy because we wanted, I wanted to say at first the goal was, look, that's a stupid reason for a program not to halt. And I just want to pile up as many stupid reasons as I can think of until it gets more than 50%. And then I can say most.”
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 can say they don't halt because you just look at them and you can understand them. They never change to the whole state so they can't halt.”
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 proportion of programs with N states that don't ever halt because they don't have any instructions saying halt. Those programs obviously never hold because they can't hold, they don't have any instruction that says halt.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“On which the machine writes zeros and ones, and the head moves back and forth according to rigid instructions. And the instructions are all of the form if the machine is in such and such a state and it's reading such and such symbol on the tape, then it should write this symbol on the tape and it should change to this new state specified and it should either move left or right as specified. So a program consists of instructions like that. If you look at a program, one of the states is the halt state, and that's when the program halts. But you can calculate how many programs Don't have any instruction that transitions to the halt state. You can easily calculate the proportion, and in the limit it goes to one over e squared, thirteen and a half percent. If you calculate the limit.”
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 the asymptotic density of the programs is one. And the proof was quite fascinating because it's one of these situations where the theorem sounds really surprising, I think, to many people when I first tell it, I mean, to computability experts, then it's sort of intriguing to think that you can solve almost every instance of a halting problem. But then when they hear the proof, it's completely a letdown. Unfortunately, nobody likes the theorem after the proof. And so the proof is so simple, though. If you know what a Turing machine, how a Turing machine operates, there's this infinite paper tape.”
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 question Alexei asked me was Does the halting problem have a black hole? And so if we take, say, the standard model of Turing machines. One way infinite tape with zeros and ones on the tape, and so on the head moving back and forth. And, you know, it stops when it gets into the halt state. Then it turns out we proved that there is a black hole. And what that means is there's a computer procedure that decides correctly almost every instance of the halting problem, even though the halding problem is not decidable. We can decide almost every instance. So more precisely, There's a collection of Turing machine programs such that we can easily decide whether programs in that collection or not. And for the programs in the collection, we can decide the halting problem for those programs easily. And furthermore, almost every program is in the collection in the sense that as the number of states goes to becomes large, the proportion of programs in the collection goes to 100%.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Certain likelihood it will have a certain behavior. And the answer turns out to be extremely interesting. Once years ago, Alexey Myaznikov asked me a question. He had this concept of a decision problem with a black hole. And what that means is it's a decision problem, which is possibly difficult in the worst case. But the difficulty was concentrated in a very tiny region called the black hole, and outside of that black hole it was very easy. And so, for example, this kind of problem is a terrible problem to use if you're basing your encryption scheme, you know, you don't want to use a black hole problem because if someone can rob the bank 95% of the time, then that's not what you want, or even any non-trivial percent of the time is too dangerous. So you don't want to use problems that are almost every case is easy.”
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 main lesson of computability theory, in my view. Is that It's never the case that you can have a thorough understanding of the behavior of a program by looking at the program and that the content of what you learn from a program. I mean, in the most general case is always obtained just by running it and looking at the behavior. And the proof of that is there's a theorem called Rise's theorem, which makes that idea completely robust. But I want to just take a little detour towards another question riffing on something that you just said, namely one can ask the question, what is the behavior of a random program? So you have some formal computing language and you want to look at the collection of all programs of a certain size. Maybe there's only finitely many. And can you say something about the behavior of a randomly chosen one like with this?”
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 the evolution, and you can prove that that question is equivalent to the halting problem. It's computably undecidable. It's semi-decidable in the sense that if it will become alive, then you will know it at a finite stage because you could just run the game of life algorithm and let it run. And if it ever did come alive, you could say, yeah, it was alive. But if you've run it for a thousand years and it hasn't come alive yet, then you don't necessarily seem to have any basis for saying, no, it won't ever come alive if the behavior was very complicated. Maybe if you have a complete understanding of the evolution of the behavior, then you can say no, but you can prove you won't always have that understanding precisely because the problem is equivalent to the halting problem.”
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 game of life is a sort of playground for computably undecidable questions because, in fact, you can prove that the question of whether a given cell will ever become alive is computably undecidable. In other words, given a configuration and you ask, will this particular cell ever be alive?”
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, I don't want to give the impression, though, that the serial numbers are not widely studied because there's thousands of people who are studying it. In fact, Philip Ehrlich, who is one of the world experts on this real number, mentioned to me once that Conway was his own worst enemy with regard to that very issue because in the Conway style everything is a game, and he treated the surreal numbers as a kind of play thing, a toy. And maybe that makes people not take it seriously, although my view is that it is extremely serious and useful and profound. And I've been riding a whole series of essays on the surreal numbers for my substack at infinitely more. And I just find the whole subject so fascinating and beautiful. I mean, it's true. I'm not applying it in engineering, which maybe was part of this Conway ambition.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Is your greatest disappointment in life? I mean, I would never ask a question like that at a conference in a very public setting. But Conway was extremely graceful, and he answered by saying that the surreal numbers, not the numbers themselves, but the reception of the surreal numbers, because he had ambition that the surreal numbers would become a fundamental number system used throughout mathematics and science, because it was able to do nonsense analysis, it was able to do calculus, it unified the ordinals and so on. And it's such a unifying, amazing structure, beautiful structure with elegant proofs and sophisticated ideas all around it. And he was disappointed that it never really achieved that unifying status that he had the ambition for. And this he mentioned as his. Greatest disappointment.”
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 you can still do calculus with them because you have infinitesimals if you use these non standard methods, the infinitesimal based methods to calculate. And people do that. I once organized a conference in New York and we had John Conway as a speaker at the conference and there was a question session and someone asked him, I mean, it's a bit rude question, I think, but they asked it. And the question was,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Right, so the surreal numbers have a property that they form a non-standard model of the real field, which means that they provide a notion of infinitesimality that one can use to develop calculus on the grounds of Robinson's nonstandard theory that I had mentioned earlier. But they don't have the least upper bound properties. No non trivial set of serial numbers has at least Eborah round. And there are no convergent sequences in the serial numbers. And so for the sort of ordinary use in calculus based on limits and convergence, that method does not work in the surreal numbers at all. So that's what I mean when I say the surreal numbers are fundamentally discontinuous. They have a fundamental discontinuity going on.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Which is true in the real numbers because if you think about, say, a cubic or a fifth degree polynomial, then you know it's going to cross the axis because it has opposite behaviors on the two infinities because it's an odd degree polynomial. So on the positive side, it's going to the positive infinity, on the negative side it would be going to minus infinity. So it has to cross. So we know in the real numbers every odd degree polynomial has a root. And that's also true in the surreal numbers. makes it what's called a real close field which is a very nice mathematical theory so it's really quite interesting how we can find copies of all these other number systems inside the serial 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
“They make the surreal numbers into what's called an ordered field. So they satisfy the field axioms, which means that you have distributivity and commutativity of addition and multiplication, and also you can have reciprocals for every non-zero number you can divide by the number. So you can add and multiply and divide and subtract. And furthermore, you can take square roots, and furthermore every odd degree polynomial has a root.”
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 ordinal omega itself is the firstborn number that's bigger than all those finite numbers, and minus omega is the firstborn number that's less than all those finite numbers. But also we have the number epsilon, which is the firstborn number that's strictly bigger than zero and strictly less than all the positive rational numbers. So that's going to be an infinitesimal number in that gap. And so on. On day omega plus one, we get more numbers and then omega plus two and so on. And the numbers just keep coming forever. So this is how you build the surreal number system. And then it turns out you can define the arithmetic operations of addition and multiplication in a natural way that is engaging with this recursive definition. So we have sort of recursive definitions of plus and times for the surreal numbers. And it turns out you can prove”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Going to create a lot of new surreal numbers. So every real number will be born at that stage because every real number fills a gap in the previously born rational numbers that we had just talked about. It's not all the rationals because actually the rational numbers that are born at the finite stages are just the rationals whose denominator is a power of two, it turns out. Those are called the dyadic rationals. So the real numbers are all born on day omega, but also some other numbers are born on day omega, namely”
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 now we have three numbers minus one, zero, and one. And they have four gaps because there could be a number below minus one or between minus one and zero or between 0 and 1 or above 1. And so we create those four new numbers. The first number above one is called 2. The first number between 0 and 1 is called 1 half. And then on the negative side, we have minus a half and minus two and so on. So now we have, what is that seven numbers? So there's eight gaps between them. So at the next birthday, they call them the next stage will be born all the numbers between those gaps. And then between those and between those and so on. And as the days progress, we get more and more numbers. But those are just the finite birthdays because as I said, it's a transfinite process. So at day omega, that's the first infinite day.”
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 now we have this number zero, and so therefore we now can define new gaps because if we put zero into the left set and have an empty right set, then we should create a new number that's bigger than zero and less than everything in the empty set. And that number is called the number one. And similarly, at that same stage, we could have put zero into the right set. And so that would be the firstborn number that's less than zero, which is called minus one”
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 for example, we could start. Well, at the beginning, we don't have any numbers. We haven't created anything yet. And so, well, we could take nothing and we could divide it into two sets, the empty lower set and the empty upper set. I mean, the two empty sets. And everything in the empty set is less than everything in the empty set because that's a vacuous statement. We satisfy the conditions and we apply the number generation rule, which says we should create a new number. And this is what I call the Big Bang of numbers, the surreal genesis, when the number zero is born. Zero is the firstborn number that is bigger than everything in the empty set and less than everything in the empty set.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Finite sequence of stages. And at every stage, we take the numbers that we have so far in all possible ways, we divide them into two sets, a lower set and an upper set, or a left set and a right set. So we divide them into these two sets so that everything in the left set is less than everything in the right set. And then at that moment, we create a new number that fits in the gap between L and R. That's it. That's all we do. So let me say it again. The rule is we proceed in stages and at any stage then in all possible ways we divide the numbers we have into two collections the left set and the right set so that everything in the left set is less than everything in the right set and we create a new number, a new surreal number that will fit in that gap.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Yes, absolutely. I really admire his style of mathematical thinking and working in mathematics. And the surreal number sister is a good instance of this. So the way I think about the surreal numbers is what it's doing is providing us a number system that unifies all the other number systems so it extends the real numbers, well, not only it extends the integers, the natural numbers and the integers and the rational numbers and the real numbers, but also the ordinals and the infinitesimals. So they're all sitting there inside the surreal numbers. And it's this colossal system of numbers. It's not a set even. It's a proper class. It turns out because it contains all the ordinal numbers. But it's generated from nothing by a single rule. And the rule is, so we're going to generate the numbers in stages in transitions.”
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 it's a real number system is an amazing and amazingly beautiful mathematical system that was introduced by John Conway.”
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 I view this kind of forcing argument that I was just describing in a similar way. You start in set theory and you go to this land of nonsense in the forcing extension, this imaginary world, and you argue and you come back. I mean, you make a consequence in the ground model. And it's such a beautiful way of arguing.”
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 the middle of their construction, they were led to the square root of minus five or something, you know, in the construction. And they didn't have any meaning for that, but they would just do it symbolically, and eventually it would turn in, you know, because of the methods that they had, they would combine and they would cancel and so on. And all the complex parts would cancel out and they'd end up with this actual answer, you know, three plus square to 17 or whatever. And they could check it and it worked. It was a solution of the original equation. And so it must have been bewildering to them because they would start with this question purely in the real numbers, an algebraic question, and they would march on their method and proceed through the land of nonsense, you know, with these square roots of negative numbers and then end up with an answer that was real again that they could verify was correct.”
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 they would have these algebraic equations that they're trying to solve, you know, and they would have the tools and methods of doing it. But then in the course of, you know, so they would have to do things to the polynomial and change the factors and so on and produce other polynomials and solve them and so on. And sometimes they could produce solutions.”
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, absolutely. And that's a really powerful argument method, actually. People often want to do that. Suppose you're in some set theoretic context, you know, you could think about it as living in a set theoretic universe, and you want to prove something in that universe only. But maybe one way to do it is to first construct this forcing extension and then use the features about this forcing extension to realize that certain things must have already been true in the ground model. And then you throw the forcing extensions away and you. Yeah. So this can happen to pick a more elementary example. If you think about the early days of people reasoning with the complex numbers before they really understood them.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Nevertheless, the central ideas of geology have now been picked up by the people with the research program in the universe, because it turns out that set theoretic geology is helping them or us to discover the nature of the one true universe relates to its mantle. There's this concept of the Cetheetic mantle that I had introduced in a way that is extremely interesting. And so it's historically quite funny, I think, because this research program that grew entirely out of the pluralist point of view ended up being picked up by the universe point of view research program in a way that is quite important.”
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, something like that. Forcing is a way of producing a new universe, and so you could start somewhere and go to that new universe, or you could look where you are and say, well, look, I got here by doing that already in the past. So we define models of the bedrock model and ground, you know, sort of undoing the forcing. And really it was quite fruitful. And I view this as part of the sort of pluralist perspective, except the difference is that set theoretic geology is amenable to the universe view. So even though the work was inspired by this philosophical view on the multiverse view,”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“He said, I want to undo forcing. I want to go backwards. And at first said, but Joan, is it doesn't work that way? You start in the model, in the ground model, and you go out, you go to the bigger one. You know, that's how forcing works. And he said, no, no, I want to go backwards. And so he was quite persistent, actually. And so finally, I said, okay, let's do it. Let's take it seriously. And so we sat down and started thinking more precisely and carefully and deeply about the nature of Taking a Cetherdic universe and seeing where did it come from by forcing, which was a new way of thinking about forcing at the time”
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 their potentialist in the sense that we could have more sets the universe could be wider and taller and so on, you know, by forcing or by extending upward. And so we want to understand the nature of this realm of set theoretic universes. And that's quite some exciting work. And so with Benedict Loeva and I, we proved some theorems on the modal logic of forcing and set theoretic potentialism under end extension. I've done a bunch of work on this topic. And also, I mounted together with Gunther Fuchs and Jonas Reitz, who was one of my own PhD students, the topic of set theoretic geology, which is studying, it's taking the metaphor of forcing, I mean, enforcing you have the ground model and the forcing extension. And when I was first working with Jonas.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“This fundamental dispute on this question. But he has a very strong and successful research program sort of trying to give legs to finding the nature of the one true set theoretic universe. And it's driving the question he's asking and the mathematical programs that he's pursuing. Whereas if you have a pluralist view as I do, then you're going to be led and attracted to questions that have to do with the interaction of different set theoretic universes, or maybe you want to understand the nature of how are the models of set theory related to their forcing extensions and so on. So this led to things that I call, say, set theoretic potentialism where you think about a set theoretic universe in a potentialist way, not in the sense of potential infinity directly because all of these universes have infinite sets inside them already.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“My view is that the choice of the philosophical perspective doesn't actually have to do with the mathematical developments directly at all. Rather, it tells us where should set theory go, what kind of set theory should we be looking at, what kind of questions should we be asking? So if you have a universe mentality, the universe view, then you're going to be pushed to try to find and articulate the nature of the one true set theoretic universe. And I think that remark is really well borne out by the developments with Hugh Wooden, who's one of the most prominent mathematicians and philosophers with the universe view and his theory of ultimate L and so on. And he's really striving.”
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 answer regrettably is apparently not because in calculus, even with that lousy, creaky foundation of infinitesimals, not even well understood that Newton and Leibniz had, they proved all the fundamental theorems of calculus. And they had all the main insights in those early days with that extremely bad foundation. And so that shows you something about the relevance of the kind of foundational views on mathematics and how important they are for mathematical developments and progress and insight. Because I view those early mathematical developments in calculus as genuinely mathematical and extremely important and insightful, even though the foundations weren't any good by contemporary perspectives. Okay, so rather, when it comes to the philosophy of set theory and the dispute between the universe view and the pluralism,”
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 the foundations really were kind of completely suspect, I think, at the time. And that foundations of infinitesimal calculus really only became rigorous in the 1950s or so with the development of non-centered analysis and Robinson's work. Okay, so the point I'm trying to make is that do you need A robust, rigorous foundation of mathematics to make enduring insights in mathematics.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“First of all, I guess one should say that these different philosophical positions that you might take in the philosophy of set theory like the multiverse view or the universe view, we don't ever disagree about the mathematics. We're all agreeing on what the theorems are. It's a question of philosophical perspective on the underlying meaning or the context or really what is a philosophy of mathematics for, right? And I mean, if you look back in history, for example, like to the time of calculus with Newton and Leibniz, right? They famously developed the ideas of calculus using their concepts of infinitesimals and those foundations were roundly mocked by Bishop Barclay and so on who talked about what are these same evanescent increments and shall we not call them the ghosts of departed quantities.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Pluralist truth, namely the fundamental nature of set theoretic truth as this plural character in that there isn't a singular meaning to the fundamental terms, but rather there's this choice of alternative set theoretic universes that have different truths.”
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 fundamental truths are differing from one to the other, and that is the answer to the continuum hypothesis question, the fact that given any model of Seth theory, there's a forcing extension where the continued hypothesis is true and another one where it's false. You can sort of turn it on and off like a light switch. And that's the fundamental nature of the continuum hypothesis is that you can have it or you can have the negation as you like within a very closely related set theoretic world. Wherever you happen to be living, there's a closely related one where ch is true, where the continuum hypothesis is true, and one where it's false. And that itself is a kind of answer. It's not a singularist answer, a universe view answer. It's a pluralist answer. And this led me to my views on the multiverse view of set theory and”
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, well, it's part of my multiverse view, I guess, which we started by, I was describing the universe view, which is the view that, look, there are facts of the matter about all of these questions and that it will turn out if you're a universe view person, which I'm not, but if you are, then you will hold that there is a right answer to the continuum hypothesis question, and there's a right answer to the large cardinal questions and so on, and that what we should be aiming to do is figure out this one true set theory. In contrast, I take the developments of set theory over the past half century or more as evidence that there isn't such a unique set theoretic reality, rather what we've been doing for decades now is producing more and more alternative set theoretic universes in 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
“More powerful than CFC, and then more powerful than that, more powerful than that, and so on. It keeps going forever, and it will never be finished.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“large cardinal axioms. So none of the large cardinal axioms we can prove none of them can settle the continu hypothesis. So the independence phenomenon is still there for things like the continuum hypothesis and the cardinal combinatorics that are building”
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, how lucky we are to find the large cardinal axioms that instantiate exactly this feature of increasing consistency strength, this unending, an extremely tall hierarchy of consistency, strength of axioms, and it exactly fulfills the prediction that Girdle's theorem makes about that kind of thing. Except the axioms in the large conarchy aren't metallical self-referential statements of the form that sometimes arise in the girdle analysis, but rather they're professing existence of big infinities, these large cardinal axioms. And so it's such a welcome development. And yet, it's also known that the continuum hypothesis is independent of all of the known.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source