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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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“First of all, to have a strong theory that's going to answer all the questions because the idea of logical independence and pervasiveness that we now know exists just wasn't, you know, there was no known, they didn't know anything like that happening ever. And so it's natural to think that it wouldn't happen and also that they would be able to guard against this inconsistency. So it seems like the goals of the Hobbit program are quite natural in that historical context. But when you think a little more about what the nature of it would be like, it shows you this kind of road procedure. And now you're saying, well, that doesn't seem so unlikely maybe. I mean, in the light of the increasing computer power and so on, it's actually maybe turning into our everyday experience where the machines are calculating more and more for us. And in a way that could be alarming.”
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 could answer it by just waiting for either the answer to come out yes from the machine or the answer to come out no. So the nature of mathematical investigation in Hilbert's world is one of just turning the crank of the theorem or enumeration machine devoid of creative thinking or imagination. It's just getting the answer from this by procedure. So Hilbert, in effect, is telling us, I mean, with his program, the fundamental nature of mathematics wrote computation. I mean, the way I think about the Hilbert program seems extremely attractive in the historical context of being worried about the antinomies, the inconsistencies, and so how can we kind of block them. And so it seems natural”
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, exactly right. Let's imagine what it would be like if he had been right. So we would have this finitary theory. And it would prove that the strong theory was free of contradiction. So we could start enumerating proofs from the strong theory. I mean, right now, we can write a computer program that would systematically generate all possible proofs from a given theory. So we could have like this theorem enumeration machine that just spit out theorems all day long in such a manner that every single theorem would eventually be produced by this device. And so, if you had a mathematical question. Of any kind”
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 if we go back to the Hilbert program, so Hilbert has these two goals, produce the strong theory which is going to answer all the questions, and then prove by purely finitary means that that theory will never lead into contradiction. And one can think about, well, the incompleteness theorem should be viewed as a decisive refutation of the Hilbert program. Defeats both of those goals decisively, completely. But before explaining that, maybe one should think about what if Hilbert had been right, what would be the nature of mathematics in the world that Hilbert is telling us to search for?”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“One of the hugely successful theories of the natural numbers and elementary number theory, essentially all of classical number theory, so whatever kind of theorems you want to be proving about the prime numbers or factorization or any kind of finitary reasoning about finite combinatorial objects, all of it can be formalized in piano arithmetic. I mean, that's the basic situation. Of course, one has to qualify those statements in light of the G ⁇ tel incompleteness theorem, but for the most part, the classical number theoretic analysis of the finite number is almost entirely developable inside piano arithmetic.”
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 view it as finitary, but this is a contentious view. I mean, not everyone agrees with that. That's what I stress. I got it. Piano arithmetic 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
“To take a specific example, I mean, I always conceive of the perhaps the most natural finitary theory that one would be called upon to explain. Which is a first order theory of the nature of arithmetic. But okay, so some people say, well, piano arithmetic has these strong first order induction axioms and this much, much weaker versions of arithmetic like Isigma naught or i sigma 1 and so on, which are even more finitary than piano arithmetic. So different philosophical positions take different attitudes about what does it take to be finitary, how finitary do you have to be to be truly finitary.”
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, this is, of course, philosophically contentious, and people have different ideas about what exactly it should mean. And so there's hundreds of papers out exactly that question. But I like to take it just kind of informally. I mean, it means that we're talking about finite sequences of symbols and we're going to have a theory, you know, finite strings of symbols and we affinitary theory would be one whose subject matter is about those kinds of things so that we can conceivably argue about the nature of these finite strings. So a proof is just a finite sequence of statements so that every statement is either one of the axioms or follows by the laws of logic from the earlier statements in some specified manner, like using modus ponens or some other law of logic like that, and such that the last line on the list is, you know, the theorem that you're proving. So that's what a proof is in this kind of way of thinking.”
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 the meaning may be totally absent. I don't think it's necessarily part of the formalist view that there is no meaning behind, but rather it's emphasizing that we can divorce the meaning of the sentences from the process of manipulating those sentences. And then Hilbert wanted to prove in this purely finitary theory that if we follow the rules of that game, we're never going to get a contradiction. So, those were the two aims of the Hilbert program to found the strong infinitary theory, probably set theory, which is going to answer all the questions. And then secondly, prove in the finitary theory that the strong theory is safe, in other words, consistent.”
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 thing about formalism is that you think of the process of doing mathematics, you divorce it from the meaning of the mathematical assertions, right? So the meaning of the mathematical assertions that you make in this infinitary theory has to do with these huge uncountable infinities and so on possibly. And that's a very sort of uncertain realm maybe and the source of the paradoxes and so on in some people's minds. And so, but the reasoning process itself consists of writing down sequences of symbols on your page and undertaking an argument with them, which is following these finitary rules. And so if we divorce the meaning of the symbols from just the process of manipulating the symbols, it's a way of looking at the nature of mathematics as a kind of formal game.”
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 those statements might be referring to infinite, uncountable objects. The statements themselves are not infinite uncountable objects. The statements themselves are just finite sequences of symbols.”
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 a very weak arithmetic, purely finitistic theory, we want to prove that the reasoning process of the strong theory is safe. So in order to make sense of that point of view, you basically have to invent the philosophy of formalism where we can look at what is the proof, what is the nature of mathematical reasoning. And on Hilbert's way of thinking about this, a proof is basically itself a finitistic kind of object. It's a sequence of if you think about the nature of what a proof is, it's a sequence of assertions which can be viewed as sort of sequences of symbols that conform with certain rules of logical reasoning. And this is a formalist way of understanding the nature of proof. So we think about a proof in a kind of syntactic, formal way, even though the content”
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 have this strong theory, this set theory that we want to be proving our theorems in. But I mean, on the one hand, we want it to be as strong as possible. We would like it to answer all the questions. There's another famous quote of Hilbert in his retirement address where he proclaims Vermusen Wissen, Ver Verden Wissen. So we must know, we will know, in which he's very optimistic about the ability of mathematics to answer all of the questions of mathematics that we have posed. We have all these problems we want to solve, and he is saying we're going to do it, we're going to solve all these problems. So we want to propose this strong theory, and one has the sense that he had in mind set theory, in which all the questions are going to be answered. But secondly, we want to combine 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
“A minefield that's a really good way of describing the situation. And so Hilbert said, well, look, we have to fix this problem. We want to use the set theory foundations, but we want to do it in a way that is trustworthy and reliable. We can't allow that the foundations of mathematics are in question. This is a kind of attitude, I think, that underlies Hilbert and the Hilbert program. And so he proposed, look, look.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Absolutely. It's one of the most profound developments in mathematical logic. I mean, the incompleteness theorems is when mathematical logic, in my view, first became sophisticated. A kind of birth of the subject of mathematical logic. But to understand the theorems, you really have to start a little bit earlier with Hilbert's program because at the time, you know, with the Russell paradox and so on, there were these various contradictions popping up in various parts of set theory and the Barrelli-40 paradox and so on. And Hilbert was famously supportive of set theory. I mean, this quote of him saying, no one shall cast us from the paradise that Canter has created for us. And what I take him to mean by that is he was so captured by the idea of using set theory as a foundation of mathematics and it was so powerful and convenient and unifying in a way that was extremely important. And he didn't want to give that up. Despite the danger of these paradoxes, these contradictions”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Logical and so on. But I think if you adopt the view that the principles of ZFC have to do with the principles of abstract set formation, which is a fundamentally logical in character, then it's complete success for logicism. So the fact that set theory is able to serve as a foundation means that mathematics can be founded on logic”
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, the project of logicism did not die with Rege, and it was continued. And there's a whole movement, the neologists and so on in contemporary times even. But my view of the matter is that really we should view the main goals of logicism are basically completely fulfilled in the rise of said theoretic foundationalism. I mean, when you view ZFC as a foundation of mathematics and in my view, the principles of ZFC are fundamentally logical in character, including the axiom of choice, as I mentioned, as the principle of logic. This is a highly disputed point of view, though, because a lot of people...”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Frigg ahead put in the appendix of his work a response to Russell's letter in which he explained what happened and he wrote very gracefully hardly anything more unwelcome can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This is the position into which I was put by a letter from Mr Burgund Russell as the printing of this volume was nearing completion. And then he goes on to explain the matter concerns his basic law five and so on and so 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
“And so Russell wrote this letter to Freget, and it was just at the moment when Frege was finishing his work it was already at the publishers and in press basically but it's completely devastating. I mean, it must have been such a horrible situation for Frege to be placed in because he's finished this monumental work years of his life dedicated to this and Russell finds this basically one line proof of a contradiction in the fundamental principles of the thesis that completely destroys the whole system.”
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 he was appealing to the principles that support that axiom throughout his work. I mean, it was really, it wasn't just an incidental thing. He was really using this principle. And Russell rode him a letter when he observed the work in progress, that there was this problem. Because if you accept the principle that for any property whatsoever, you can make the set of objects with that property, then you could form the set of all sets that are not members of themselves. That's just an instance of the general comprehension principle. The set of all sets that aren't elements of themselves can't be a set because if it were then it would be an element of itself if and only if it's not a member of itself, and that's a contradiction.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Before that time, Frege was working on his monumental work undertaking, implementing the philosophy of logicism, which is the attempt to reduce all of mathematics to logic. So Frager wanted to give an account of all of mathematics in terms of logical notions. And he was writing this monumental work and had formulated his basic principles. And those principles happened to imply that for any property whatsoever, you could form the set of objects with that property. This is known as the general comprehension 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
“Because we can form the diagonal class, the class of all sets that are not elements of themselves. If that were a set, Then it would be an element of itself if and only if it was not an element of itself. It's exactly the same logic in all four of those arguments. So there can't be a class of all sets because if there were, then there would have to be a class of all sets that aren't elements of themselves. But that set would be an element of itself if and only if it's not an element of itself, which is a contradiction. So this is the essence of the Russell paradox. I don't call it the Russell paradox actually when I teach it. I call it Russell's theorem. There's no universal set. And it's not really confusing anymore. At the time, it was very confusing, but now we've absorbed this nature of set theory into our fundamental understanding of how sets are and it's not confusing anymore. I mean, the history is fascinating, though, how the Russell paradox because”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Might have just some in not all. So that diagonal salad would have to be named after some fruit. So let's suppose it's named after durian, meaning that it was associated with durian in the one-to-one correspondence. And then we ask, well, is durian In the salad that it's named after. And if it is, then it shouldn't be, and if it isn't, then it should be. And so it's again the same contradiction. So all of those arguments are just the same as Cantor's proof that the power set of any set is bigger than the set. And this is exactly the same logic that comes up in Russell's paradox because Russell is arguing that the class of all sets can't be a set. Because if it were, then we could form the set of all sets that are not elements of themselves. So basically he's what Russell is proving is that there are more collections of sets than elements.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“For any collection of fruits, there are more possible fruit salads than there are fruits So if not, then you can put a one to one correspondence between the fruits and the fruit salads, so you could name every fruit salad after a fruit might not be, that fruit might not be in that salad. It doesn't matter. It's a naming, a one-to-one correspondence. And then, of course, we form. The diagonal salad, which consists of all the fruits that are not in the salad that's named after them. And that's a perfectly good salad. It might be the kind of diet salad if it was the empty salad or it might be the universal salad which had all fruits in it if all the fruits are in 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
“So now we ask is Daniela on the committee that's named after her? Then she shouldn't be because it was the Committee of People who aren't on their own committee. And if she isn't, then she should be. So again, it's a contradiction. So when I was teaching in Oxford, one of my students came up with the following different anthropomorphization of Cantor's argument. Let's consider all possible fruit salads. We have a given collection of fruits and Know apples and oranges and grapes, whatever. And a fruit salad consists of some collection of those fruits. So there's the banana pear grape salad and so on. There's a lot of different kinds of salad. Every set of fruits makes a salad, a fruit salad. And we want to prove that for any collection of fruits, even if there are infinitely many different kinds of fruit,”
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, maybe it's always in session. I don't know. So the claim is that there's more committees than people. Okay, suppose not. Well, then we could make an association between the people and the committees. So we would have a kind of every committee could be named after a person in a one-to-one way. And I'm not saying that the person is on the committee that's named after them or not on it, whatever, maybe sometimes that happens, sometimes it doesn't. I don't know. It doesn't matter. But let's form what I call committee D. Which consists of all the people that are not on the committee that's named after them. Maybe that's everyone. Maybe it's no one. Maybe it's half the people. It doesn't matter. That's a committee. It's a set of people. And so it has to be named after someone. Let's call that person Daniela.”
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 contradiction. So therefore, the number of subsets is always greater than the number of elements for any set. The anthropomorphizing idea is the following I'd like to talk about it this way for any collection of people you can form more committees from them than there are people. Even if you have infinitely many people, Suppose you have an infinite set of people. And what's a committee? Well, a committee is just the list of who's on the committee, basically, the members of the committee. So there's all the two-person committees and there's all the one-person committees, and there's the worst committee, the one that everyone is on. The best committee is the empty committee with no members and never meets and so on. Or is the empty committee meeting all the time? I'm not sure.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Maybe nobody's like that. Maybe there's no element of X that's like that. Or maybe they're all like that. Or maybe some of them are and some of them aren't. It doesn't really matter for the argument. I defined a subset D consisting of the individuals that are not in the set that's attached to them. But that's a perfectly good subset. And so because of the equanumerosity, it would have to be attached to a particular individual. But that let's call that person, it should be a name starting with D, so Diana. And now we ask is Diana an element of D or not? But if Diana is an element of D, then she is in her set. So she shouldn't be because the set D was the set of individuals that are not in their set. So, if Diana is in D, then she shouldn't be. But if she isn't in D, then she wouldn't be in her stead, and so she should be in D.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Whether it's strictly more or not. And so candor reasoned like this. It's very simple. It's a kind of distilling the abstract diagnosization idea without encumbered by the complexity of the real numbers. So we have a set X, and we're looking at all of its subsets. That's the power set of X. Suppose that X and the power set of X have the same size. Suppose there's contradiction, they have the same size. So that means we can associate to every individual of X a subset. And so now let me define a new set. I mean another set. I'm going to define it. Let's call it D. And D is the subset of X that contains all the individuals that are not in their set Every individual Was associated with a subset of x, and I'm looking at the individuals that are not in their 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
“Right. So let's. So we talked about Cantor's proof that the real numbers, the set of real numbers is an uncountable infinity. It's a strictly larger infinity than the natural numbers. But Cantor actually proved a much more general fact, namely that for any set whatsoever, the power set of that set is a strictly larger set. So the power set is the set containing all the subsets of the original set. So if you have a set and you look at the collection of all of its subsets, then counterproved that this is a bigger set. They're not equanumerous. Of course, there's always at least as many subsets as elements because for any element you can make the singleton subset that has only that guy as a member, right? So there's always at least this many subsets as elements. But the question is whether they”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“With the result of Girdle and Cohen and so on, this consistency question specifically about the ex-mafa choice sort of falls away. We know that the axim of choice itself will never be the source of inconsistency in set theory if there's inconsistency with the aximo choice, then it's already inconsistent without the aximo choice. So it's not the cause of inconsistency. And so in that, from that point of view, the need to pay attention to whether you're using it or not from a consistency point of view is somehow less important. But still, there's this reason to pay attention to it on the grounds of these constructivist ideas that I had mentioned earlier.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Zermelo and others actually looked into the mathematical papers and so on of some of the people who had been objecting so vociferously and found in many cases that they were implicitly using the axima choice in their own arguments even though they would argue publicly against it because it's so natural to use it because it's such an obvious principle in a way. I mean it's easy to just use it by accident with if you're not critical enough and you don't even realize that you're using the axima choice. That's true now even people like to pay attention to when the axima choice is used or not used in mathematical arguments. I mean up until this day it used to be more important in the early 20th century. It was very important because people didn't know if it was a consistent theory or not and there were these antinomies arising and so there was a worry about consistency of the axioms. But then of course eventually”
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 read some historical accounts by historians about that time period specifically about Zermelus' axioms and his proof of the well order theorem, and the historians were saying never before in the history of mathematics has a mathematical theorem been argued about so publicly and so vociferously as that theorem of Sermellos. It's fascinating also because the axiom of choice was widely regarded as a kind of basic principle at first, but then but people were very suspicious of the well-order theorem because no one could imagine a well-ordering say of the real numbers. And so this was a case when Zermelo seemed to be from principles that seemed quite reasonable proving this obvious untruth. And so people were mathematicians were objecting. But then...”
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 going on in this set. So it's just if two sets have the same embrace, then they are the same set. So it's maybe the most primitive axiom in some respect.”
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 history of it is really quite fascinating. So Zumelo introduced most of these axioms as part of what's now called Zermelo set theory to formalize his proof from the axiomatoice to the well-order principle, which was an extremely controversial result. So in 1904, he gave the proof without the theory, and then it was challenged to provide the theory. And so in 1908, he produced the Zermello set theory and gave the proof that in that theory you can prove that every set admits a well-ordering. And so the axioms on the list, these things like extensionality express the most fundamental principles of the understanding of sets that he wanted to be talking about. For example, extensionality says if two sets have the same members, then they're equal. So it's this idea that the sets consists of the collection of their members, and that's it. There's nothing else.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“An argument and you appeal to the maximum choice, then maybe you're admitting that the objects that you're producing in the proof are not going to be constructive. You're not going to be able to necessarily say specific things about them. But if you're just claiming to make an existence claim, that's totally fine. Whereas if you have a constructive attitude about the nature of mathematics and you think that mathematical claims maybe are only warranted when you can provide an explicit procedure for producing the mathematical objects that you're dealing with, then you're probably going to want to deny the maximum choice and maybe much more.”
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, well, look, I mean, I don't, not every way of choosing the Sox has to be defined by a rule. Why should everything that exists in mathematical reality follow a rule or a procedure of that sort if I have the idea that my mathematical ontology is rich with objects, then I think that there are all kinds of functions and ways of choosing. Those are all part of the mathematical reality that I want to be talking about. And so I don't have any problem asserting the aximo choice. Yes, there is a way of choosing, but I can't necessarily tell you what it is. But in a mathematical argument, I can assume that I fix the choice function because I know that there is one. So the philosophical difference between working when you have the axim of choice and when you don't is the question of this constructive nature of the argument. So if you make”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“When you think about the infinite collection of socks that the person has in their closet. And if we assume that socks are sort of indistinguishable within each pair, you know, they match each other, but this sort of, you know, indiscernible, then the butler wouldn't have any kind of rule for which sock in each pair to pick. And so it's not so clear that he has a way of producing one sock from each pair because that's what's at stake is the question of whether you can specify a rule by which the choice function, you know, a rule that it obeys that defines the choice function or whether there's sort of this arbitrary choosing aspect to it. That's when you need the axiom of choice to know that there is such a function. But of course, as a matter of mathematical ontology, we might find attractive the idea.”
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 he tells his butler please go and give me one shoe from each pair. And the butler can do this easily because he can for any pair of shoes he can just always pick the left shoe. I mean, there's a way of picking that we can describe. We always take the left one or always take the right one or take the left one if it's a red shoe and the right one if it's a brown shoe or, you know, we can invent rules that would result in these kind of choice functions. We can describe explicit choice functions. And for those cases, you don't need the axiom of choice to know that there's a choice function. When you can describe a specific way of choosing, then you don't need to appeal to the axiom to know that there's a choice function. But the problematic case occurs.”
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. So one should be aware also that there's huge process of mathematics that pay attention to whether the External Choice is being used and they don't want to use the exmo choice that they work out the consequences that are possible without the eczema choice or with weakened forms of Sermella-Frankl set theory and so on. And this is quite a vibrant amount of work in that area. I mean, but going back to the Exma choice for a bit, it's maybe interesting to To give Russell's description of how to think about the axiom of choice. So Russell describes this rich person as a An infinite closet, and in that closet he has infinitely many pairs of shoes.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Exists. You want to have the view that, well, there is a way of choosing. I don't have an easy way to say what the function is, but there definitely is one. This is the way of thinking about the eczema choice.”
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 on the one hand, I mean, the eczema choice principle is. Completely obvious that we want this to be true, that it is true. I mean, a lot of people take it as a law of logic. If you have a bunch of sets, then there's a way of picking an element from each of them. There's a function. If I have a bunch of sets, then there's a function that when you apply it to any one of those sets gives you an element of that set. It's a completely natural principle. I mean, it's called the Exmo choice, which is a way of sort of anthropomorphizing the mathematical idea. It's not like the function is choosing something. I mean, it's just that if you were to make such choices, there would be a function that consisted of the choices that you made. And the difficulty is that when you can't specify a rule or a procedure by which you're making choices, then it's difficult to say what the function is that you're asserting.”
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, I think that's right. So, I mean, the history of the current set theory axioms known as the Zermelo-Franco axioms came out in the early 20th century with Zermelo's idea. I mean, the history is quite fascinating because Zermelo in 1904 offered a proof that what's called the axiom of choice implies the well-order principle. So he described his proof. And that was extremely controversial at the time. There was no theory, there weren't any axioms there. Cantor was not working in an axiomatic framework. He didn't have a list of axioms in the way that we have for set theory now. And Zermello didn't either. And his ideas were challenged so much with regard to the well-order theorem that he was pressed to produce the theory that in which his argument could be formalized, and that was the origin of what Known as their mellow set theory”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Way to think of a collection of things as one thing. That's the central idea of set theory. A set is a collection of things But you think of the set itself as one abstract thing, so when you form the set of real numbers, then that is a set, it's one thing, it's a set, and it has elements inside of it, so it's sort of like a bag of objects. A set is kind of like a bag of objects. And so we have a lot of different axioms that describe the nature of this idea of thinking of a collection of things as one thing itself, one abstract thing.”
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 great question. Cetheri really has two roles that it's serving. This kind of two ways that set theory emerges. On the one hand, set theory is its own subject of mathematics with its own problems and questions and answers and proof methods. And so really, from this point of view, set theory is about the transfinite recursive constructions or well-founded definitions and constructions. And those ideas have been enormously fruitful and setheists have looked into them and developed so many ideas coming out of that. But set theory has also happened to serve in this other foundational role. It's very commented here things said about set theory that really aren't taking account of this distinction between the two roles that it's serving. It's its own subject, but it's also serving as a foundation of mathematics. So in its foundational role, set theory provides”
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 was really the start of so many other observations that were made, including Russell's paradox and the halting problem and the recursion theorem and so many other principles are using diagnosization at their core.”
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. Because the only kind of case where that phenomenon occurs is when the number is eventually zero or eventually nine. And so since our numbers E never had any zeros or nines in it, it wasn't one of those numbers. And so actually in those cases, we didn't need to do anything special to diagnose just the mere fact that our number has a unique representation already means that it's not equal to those numbers. So maybe it was controversial in Cantor's day more than 100 years ago, but I think it's most commonly looked at today as one of the initial main results in set theory and its profound and amazing and insightful and the beginning point of so many later arguments. And this diagnosization idea has proved to be an extremely fruitful proof method and almost every major result in mathematical logic is using in an abstract way the idea of diagnosis.”
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 given a list of numbers, Cantor is proving that it's interesting that you say that actually because there's a kind of philosophical controversy that occurs as in connection with this observation about whether Cantor's construction is constructive or not. Given a list of numbers, Cantor gives us a specific means of constructing a real number that's not on the list. This one aspect which I alluded to earlier, but some real numbers have more than one decimal representation and it causes this slight problem in the argument. For example, the number one, you can write it as 1.000 forever, but you can also write it as 0.999 forever. Those are two different decimal representations of exactly the same 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
“We've made our numbers E so that the nth digit of z is different from the nth digit of the nth num But now it follows that Z is not on the list because Z is different from R one because well, the first digit after the decimal point of Z is different from the first digit of R1 after the decimal point. That's exactly how we built it. And the second digit of Z is different from the second digit of R2 and so on. The nth digit of Z is different from the nth digit of R sub n. For every n. So therefore Z is not equal to any of these numbers R sub n. But that's a contradiction because we had assumed that we had every real number on the list, but yet here is a real number z that's not on the list. And so that's the main contradiction.”
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 then I want to do something a little bit more, and that is I'm going to make it different in a way that I'm never using the digits zero or nine. I'm just always using the other digits and not zero or nine. There's a certain technical reason to do that. But the main thing is that I make the digits of Z different in the nth place from the nth digit of the nth number. If you had drawn out the numbers on the original list R one, R two, R three, and so on, and you made it, you know, and they were each filling a whole row, and you thought about the nth digit of the nth number, it would form a kind of diagonal going down and to the right. And for that reason, this argument is called the diagonal argument because we're looking at the nth digit of the nth number, and those exist on a kind of diagonal going down”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source