YouSaid · the spoken record

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 happen to think so. I mean, I'm on the side of realism in mathematics, and I think that these abstract objects do have a real existence in a way that we can give an account of in a way I just tried to describe.”
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 it seems to me that we don't really have any understanding of what the physical world is as opposed to the abstract world. And it's the abstract world where existence is much more clear.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“What it means to say that there's an apple on my desk and to give an account of what that physical existence really is at bottom, I think, is totally absent. Whereas we do seem to have a good A much more satisfactory account of the nature of abstract existence. I mean, I can talk about the nature of the empty set. This is the predicate which is never true or something like that. I can talk about those kind of logical properties or the singleton of the empty set and so on. I mean, of course, it's very difficult if you go very far with it, but the point is that it doesn't get more and more mysterious the more that you say it becomes only more and more clear.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Since it has a profound mystery, in fact, it becomes more and more mysterious the more physics we know. I mean back in, say, Newtonian physics, then one had a picture of the nature of physical objects as, you know, little billiard balls or something, or maybe they're infinitely divisible or something like that. Okay, but then this picture is upset with the atomic theory of matter, but then that picture is upset when we realize that the atoms actually can be split and consists of electrons and protons and neutrons and so on. But then that picture is upset when we realize that those things themselves are built out of quarks and leptons and so on. And who knows what's coming? And furthermore, all of those things, the nature of their existence is actually as wave functions in some cloud of probability and so on. And so it just becomes more and more and more mysterious the more we learn and not at all clarifying. And so the nature 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
“Imagine a certain kind of steam locomotive, and I describe the engineering of it and the weight of it and the nature of the gear linkages. And I show you schematic drawings of the whole design and so on. And we talk in detail about every single detailed aspect of this steam locomotive. But then suppose after all that conversation, I say, okay, now I would like you to tell me what would it mean for it to exist physically? I mean, as opposed to just being an imaginary steam locomotive, what could you possibly say about it? I mean, except by saying, oh, I just mean that it exists in the physical world. But what does that mean? That's the question, right? It's not an answer to the question. That is the question. So I don't think that there's anything sensible that we can say about the nature of physical.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Sometimes people find it problematic to talk about the existence of abstract objects such as numbers, and there seems to be a kind of wish that we could give an account of the existence of numbers or other mathematical objects or abstract objects that was more like the existence of tables and chairs and rocks and so on. And so there seems to be this desire to reduce mathematical exists to something that we can experience physically in the real world. But my attitude about this attempt That it's very backward, I think, because Don't think we have such a clear existence of the nature of physical objects, actually. I mean, we all have experience about existing in the physical world as we must because we do exist in the physical world. But I don't know of any Satisfactory account of what it means to exist physically. I mean, if I ask you, say”
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 excellent question. I mean, a huge part of the philosophy of mathematics is about this kind of question, that what is the nature of the existence of mathematical objects, including infinity. But I think asking about infinity specifically is... Isn't that different than asking about the number five? What does it mean for the number five to exist? What are the numbers really? This is maybe one of the fundamental questions of mathematical ontology. I mean, there's many different positions to take on the question of the nature of the existence of mathematical objects or abstract objects in general. And there's a certain kind of conversation that sometimes happens when you do that. And it goes something like 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
“But also think of them as fitting into Hubart's hotel. So just have everyone on the show. They each give one dollar to that person. So afterwards, that person has infinitely many dollars, but everyone only paid out one dollar. So it's a way of making it happen.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Ha ha ha. It's interesting to think about well, what if there were infinitely many people in your group? Then it's not true anymore. The theorem fails. In fact, you can arrange that everyone is strictly more pointed at than pointing. And also if everyone has even just $1 bill, then you can arrange that afterwards everyone has infinitely many dollar bills. Because in terms of cardinality, that's the same. It's just, say, countable infinity in each case. If you had countably many friends and everyone has $1 bill, then you can arrange a pattern of passing those dollar bills amongst each other so that afterwards everyone has infinitely many dollar bills. What you need is for each person to be attached to one of the train cars or something. So think of everyone as coming from Hilbert's train.”
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 that by itself is actually a difficult mathematical claim. I mean, if someone had to prove that, that you can't make money by trading within a group. It can't be that everyone in the group makes money just by shifting money around in the group. Maybe you think that's obvious. And it is obvious if you think about money. But if you had asked the question about mathematical functions of a certain kind and so on, then maybe it wouldn't be as clear as it is when you're talking about this money thing because of We can build on our human experience about the difficulty of getting money and, you know, or other resource. That doesn't have to be money. It could be candy, whatever. We just know that you can't easily get more things in that kind just by trading within a group.”
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 therefore it can't be possible that we're all more pointed at than pointing. And this proof illustrates something, it's one of my habits that I suggest in the book, to anthropomorphize your mathematical ideas. So you should imagine that the mathematical objects that are playing a role in your question are people or active somehow animals or something that maybe have a will and a goal and so on. This is this process of anthropomorphizing. And it often makes the problems easier to understand because we all are familiar with the fact that it's difficult to make money and the proof is totally convincing because of our knowledge that we can't make money as a group by trading dollars between us, you know, without any new money coming into the group.”
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 I was pointed at by more people than I'm pointing. So I got $10, but I only paid out seven dollars. And similarly, you got paid twenty dollars, but you only paid out fifteen dollars. So if everyone is more pointed at than pointing, then everyone makes money. But it's obviously impossible for us to make money as a group by just trading money with ourselves.”
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 don't think you can. Okay, so can you arrange it so that everyone is more pointed at than pointing? And in my book, I give a couple of different proofs of this. I think I give an induction proof and there's another proof. I think there's three different proofs in there. But why don't we just talk about my favorite proof? Suppose it were possible to arrange that we're all more pointed at than pointing. Now what we're going to do, we're going to agree. Going to give a dollar to everyone that we're pointing at. Okay. So, what happens? Everybody made money.”
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 that everyone was more pointed at than they are pointing at others. So in other words, maybe there's seven people pointing at me, but I'm only pointing at five people, and maybe there's 20 people pointing at you, but you're only pointing at 15 people or something like that, right? So I want to know, a similar question on Twitter. For a group of people on Twitter, could you arrange that everyone has more followers than following? It's the same question. Mathematically it's identically. Although, I don't know, it's not identical because I said you could point at yourself, and I think that's not. Can you follow yourself?”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Let's do a proof. There's one in the book. We can talk about it. I think it's a nice problem. It's in the discrete math, yeah, the 5.1, that one, more pointed at than pointing. Okay, so this is the following problem. Suppose you're gathered with some friends, you know, in a circle, and you can point at each other, however you want, or yourself, whatever. It doesn't matter. And you can point at more than one person, you know, use all your fingers or your feet or whatever you want. So maybe you point at three of your friends or something and they point at two or three of their friends or whatever and one person is pointing at ten people and somebody isn't pointing at anybody maybe. And various people are pointed at also, right? So the question is, Could we arrange a pattern of pointing?”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Argue for the conclusion and so on. And all of that is true and fine, and that's good to know, except if that's all that you're saying about the nature of proof, then I don't think you're really learning very much. So I felt that it was possible to have a much better kind of book, one that was much more interesting and that had interesting theorems in it that still admitted of elementary proof. So I wrote this book and tried to fill it with all of the compelling mathematical statements with very elementary proofs that exhibited lots of different proof styles in it. And so I found that the students appreciated it a lot.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Many universities have such a course, the proof writing course, which is usually taken by students who have learned some mathematics. Usually they've completed maybe the calculus sequence and are making the kind of transition to higher mathematics, which tends to involve much more proof. And it's a kind of challenging step for them. So many math departments have this kind of course on proof writing where the students would get exposed to how to write proofs. And I wasn't happy with most of the other books that exist for those kind of courses. And the reason was that they were so often so dull because they would concentrate on the totally uninteresting parts of what it's like to write, approve this kind of mechanistic procedures about how to write a proof. You know, if you're going to prove an implication, then you assume the hypothesis.”
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, this is something I find so wonderful to teach young mathematicians who are learning how to become mathematicians and learning about proof. And I wrote that book when I was teaching such a proof writing class in New York.”
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 I have this theorem enumeration device on my desk, and I announce that I'm open for business to solve the halting problem. So you give me a program and input that you want to run that program on. And I'm going to answer the halting problem and the way I'm going to do it is I'm just going to wait for the statement coming out of the theorem enumeration device that asserts either that P does halt on that input or I wait for the statement that P does not halt on that input. But one of them is going to happen because it was the complete theory that was enumerating all the true statements of elementary mathematics. So therefore, if I had such a system, I could solve the halting problem. But we already proved that you cannot solve the halting problem. So therefore you cannot have such a complete theory of arithmetic. So that proves Girdle's theorem.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Meaning, arithmetic, and finite combinatorial things such as Turing machine computations and so on. So in fact, all those finite combinatorial processes are formalizable inside arithmetic with the standard arithmetization coding process. But let me just be a little bit informal and say, suppose we could write down a complete theory of elementary finite mathematics. So we have an axiomization of that theory. Then we could produce all possible theorems from those axioms in the way that I was describing earlier with Hilbert's program. I mean, if we had a complete theory of elementary mathematics, we could construct a theorem enumeration machine that produced all the theorems and only the theorems from that 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
“It's absolutely beautiful. Yeah, I agree. And it's following the same logic of Russell and Cantor. I mean, going back to Cantor basically, because Russell is also quoting Cantor in his letter to Frege. So therefore, the conclusion is that the halting problem is not computably decidable. And now we can immediately prove Girdles that you're using this. Actually, it's an immediate consequence. So, why don't we just do that? I view this as the simplest proof of Girdle's theorem. You don't need the Girdle sentence to prove Girdle's theorem. You can do it with the halting problem. So suppose that we could write down a computable axiomization of all of the true facts of elementary 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
“And because of this opposite behavior, Q would halt on Q if and only if Q does not halt on Q, which is a contradiction, because Q has to have the opposite behavior on Q than Q does. But that's just contradictory.”
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 I'm just grabbing program Q, and program Q takes as input P, which is itself a program. And the first thing it does is it asks the halting subroutine program, would P halt on P? And if the answer comes back from the subroutine, yeah, that would hold, then what I do in program Q is I immediately jump into an infinite loop. So I don't halt. If P halts on P, I don't halt. But if the answer came back no, P is never going to hold on P, then I halt immediately. Okay, so that's it. I've described what Q does. And the thing about Q is that Q's behavior on P was the opposite of P's behavior on P. I mean, that's how we designed Q, specifically so that Q on P had the opposite behavior as P on P. Okay, so now, of course, what do we do? Well, the same thing that Russell did and so forth and candor we ask, well, what would Q do on Q?”
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, suppose toward contradiction, I mean, all these proofs are by contradiction, and this argument is going to be a diagonal argument in the same style as the Russell argument and the candor argument and Girdle's argument that we haven't talked about yet, so many diagonal arguments come in. So suppose towards contradiction that we had a procedure for determining whether a given program halted on a given input. Now, let me describe I'm going to use that procedure as a subroutine in the following process. And my process, let's call it Q, process Q. And it takes as input a program P. And the first thing it does is it asks that subroutine, hey, would P halt if I ran it on P itself, that's the diagonal part because we're applying P to P, 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
“It seems like I would have to really understand how the program worked and what it was doing. So giving the yes answers was sort of trivial. You didn't have to understand it. You just needed to run it, which is a kind of rote task. But to give the no answers, you need to have a kind of deep insight into the nature of the program and what it's doing in such a way that you would understand it and be able to see, oh no, I can see this program is never going to hold because it's a much more difficult test to say, no, it won't halt than it is to say, yes, it halted because I ran it and it halted. And it turns out to be impossible to have a computer war procedure that gives the no answers. And the argument is not very difficult. Should we do 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
“And I could keep running it. And maybe in a week it would halt. And at that time, I could say, yes, it halted. So I can get the yes answers correctly for halting all the yes answers. But the problem is If it didn't halt yet, like maybe I waited a thousand years and it still hasn't halted. Don't seem entitled to say no, it's not going to halt. Yeah, because maybe in a thousand in one year it'll halt. And so at no point can I seem to say no. In order to say no, it won't ever 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
“For any one instance, the answer is either yes or no. That's not what we're talking about. We're talking about whether there's a computable procedure to answer all instances of this question. So it's a decision problem is given as a scheme of instances for all possible programs that you could ask about what I want to know is, is there a computable procedure that will answer those questions? And it turns out the answer is no. The halting problem is computably undecidable. There is no computable procedure that will correctly answer all instances of whether a given program will halt. And of course, we can get half the answers in the sense that. Give me a program and you say, Will this halt? And I could take that program and I could run 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
“The halting problem is expressing a fundamental property of computational processes. So given a program, or maybe we think of it as a program together with its input, but let me just call it a program. So given a program, we could run that program, but I want to pose it as a decision problem. Will this program ever complete its task? Will it ever halt? The halting problem is the question, given a program, will it halt yes or no? Course”
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. In general, the provability problem, we can formulate it as a decision problem. Given a theory and given a statement, is that statement a consequence of that theory? This is one of the most famous decision problems. In fact, the very first one, because it's equivalent to the Hilbert Ackerman and Cheiden's program, which is also appearing in the title of Turing's 1936 paper that was so important for computability theory. So it's a formulation of the Inchardung's problem. Does a given theory have a given statement as a logical consequence, which because of Girdle's completeness theorem, not his incompleteness theorem, but his earlier completeness theorem, Girdle had proved that the proof systems that they studied did have this completeness property that I mentioned. So provability is the same as logical consequence. And this is an undecidable decision 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
“I mean, my view is that this isn't traumatic at all. This is rather. Completely eye-opening in terms of our understanding of the nature of mathematical reality. I mean, we're not We understand this profound fact about our situation with regard to mathematical truth. The incompleteness theorem tells us, look, we just can't write down a list of axioms that is going to be consistent and is going to answer all the questions. It's impossible. And so I don't think of it as trauma. I just think, look, this is the nature of mathematical reality and it's good that we know it. And so now we need to move on from that and do what we can in light of 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 girdle proof that this is impossible. You cannot write down a computable list of axioms that is complete in that sense. There will always be statements if the theory is consistent, there will always be statements that you cannot prove and you cannot refute. So they are independent of that 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
“And discussing. So the incompleteness theorem is the question whether we could say write down a theory for arithmetic, say for the standard model of arithmetic where we have the natural numbers and plus and times and zero one and less than and so on in that formal language we can express an enormous number of statements about the nature not only of arithmetic but actually by various coding methods we can express essentially all of finite mathematics in that structure. So the question would be can we write down a computable list of axioms that will answer all those questions by proof? In other words, we want to have a complete theory. A theory of arithmetic that proves all and only the true statements. That would be the goal. Hilbert would love that. I mean that would be supportive of Hilbert's program to have such a complete theory of 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
“So that doesn't count as a proof. So generally, all the classical proof systems that are used are sound and complete and also computably decidable in the sense that we can decide whether something is a proof or 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
“And the proof systems generally have both of those properties. They're sound and complete. There's a third property a lot of logicians talk about sound and complete sound and complete this, sound and complete that. But actually, there's a hidden third adjective that they should always be talking about in any such case, which is that you should be able to recognize whether or not something is a proof or not. So there's a computable aspect to the proof systems. We want to be able to recognize whether something is a proof. It should be computably decidable whether a given sequence of statements is a proof or not. So we don't want a proof system in which someone claims to have a proof, but we can't check that fact. Whether it's a proof or not. We want to be able to correctly adjudicate all claims to having a proof.”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Systems, there's a lot of different formal proof systems that exist that are studied by the proof theorists. And all of them have the property that they're sound, which means that if the premises of the argument are all true in a structure and you have a proof to get a conclusion, then the conclusion is also true in that structure. So that's what it means to be sound. Proofs preserve truth. Their truth preserving arguments. But also... The proof systems are also generally complete. They're both sound and complete, and complete means that whenever a statement is a consequence, a logical consequence of some other statements, which means that whenever the assumptions are true, then the consequence is also true in the structure. Whenever you have a logical consequence, then there is a proof of 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
“Okay, that's truth. Proof, on the other hand, is in this Hilbert way of thinking, we can develop proof theory. What is a proof for a mathematician, for a mathematical logician, a proof is a certain sequence or arrangement of sentences in the formal language that accord with the logical rules of a proof system. So the certain modes of reasoning that are allowed. So if you know A and you know A implies B in the proof, then at a later step you're allowed to write B as a consequence. So if you know A and you know A implies B, those are both two statements that are known, then you can deduce B as a consequence according to the rule of modus ponents. This is the rule modus ponents. And, you know, there's a lot of other rules. Some people would call this implication elimination, this different kinds of proof.”
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 allows you to define by induction the truth of any assertion in a formal language inside any mathematical structure. And so to And so maybe we have in mind the standard model of arithmetic or something with the natural numbers and the arithmetic structure. And I want to know, is a given statement true in that structure, then we have a formal definition of what that means according to the Tarski recursive definition of truth.”
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, it has the discotation. And so this idea can be for all the logical connectors and quantifiers and everything. You're applying Taraski's idea of discotation.”
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 you can use this idea of disquotation to give a formal definition of truth in a mathematical structure of a statement in a formal language. So for example, if I have a formal language that allows me to make atomic statements about the objects and relations of the structure, and I can build up a formal language with the logical connectives of AND and or and implies and not and so on. And maybe I have quantifiers, right? For example, to say that the structure satisfies phi and c, that single statement, phi and c, I'm thinking of that as one statement, just means that it satisfies phi and it satisfies c. And if you notice what happened there, At first, the and was part of the sentence inside the sentence, but then in the second part I was using the word an to refer to the conjunction of the two conditions so.”
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 true if and only if snow is white. And what he means by that is look to say truth is a property of an assertion so we can think of the assertion as it syntactically. So the sentence is true if and only if the content of the sentence is the case. So the sentence snow is white in quotations is true. That just means that snow is white. And that's why it's called the disquotational theory because we remove the quotation marks from the assertion, 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
“Truth is on the semantic side of the syntax semantics dichotomy. Truth has to do with the nature of reality. I mean, okay, when I talk about reality, I'm not talking about physical reality. I'm talking about mathematical reality. So we have a concept of something being true in a structure, a statement being true in a mathematical structure. Like maybe you have the real field or something and you want to know, does it satisfy this statement or that statement, or you have a group of some kind, or maybe you have a graph, this is a particular kind of mathematical structure that has a bunch of vertices and edges, and you want to know, you know, does this graph satisfy that state? And Tarski gave this absolutely wonderful account of the nature of truth in what's now known as the disquotational theory of truth. And what Tarski says is, the sentence, quote, snow is white.”
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 mathematical investigation or analysis. Maybe it was already taken to be fully clear. But because of the incompleteness theorem, we realized that actually those quite subtle things happening. So why don't we talk about this distinction a bit? To me, it's absolutely core and fundamental to our understanding of mathematical logic now. This distinction between truth and proof.”
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, this is a really core distinction that. It's fascinating to me to go back and read even the early 20th century people before Girdle and Tarski. And they were totally sloppy about this distinction between truth and proof. It wasn't clear at all until Girdle, basically. Although even as late as Bourbaki has a kind of confusion in these foundational works, so this standard graduate level textbooks used in France in the presentation of logic, are conflating truth and proof to be true for them means to be provable. So in the early days, maybe it wasn't clear enough that the concept of truth needed”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source
“Usually, you only get to decide the positive instances. If something is a theorem, you will eventually come to recognize that. But if something is a theorem, maybe at no point will you be able to say no, that's not a theorem.”
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, they're so closely connected. Although all the features aren't the same, so. If you have a computable list of axioms for a theory, then you can start enumerating the consequences of the axioms, but you won't be able to computably decide whether a given statement is a consequence or not. You can enumerate the consequences so you can semi-decide the consequences, but you won't be able to decide yes or no whether a given statement is a consequence or not. So it's the distinction between a problem being computably decidable and a problem being computably enumerable, which was made clear following the work of Turing and others that came from that. So that's one difference between the list of axioms of the theory and the theory itself. The axioms could be, you can decide maybe computably whether something is an axiom or not, but that doesn't mean that you can decide computably whether or not something is a theorem or 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
“Right. So in mathematical logic theory is a technical term, and it means any set of sentences in a formal language. And so if you say axiomatic system, it's basically synonymous to my usage with theory. So a theory means the consequences of a set of axioms, or people are sometimes unclear on whether they just mean the axioms or the consequences of the axioms.”
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 mean, would you trust a theory that proves of itself that it's consistent? I mean, that's like the used car salesman telling you, oh, I'm trustworthy. I mean, it's not a reason to trust the used car salesman. Is it just because he says that? So similarly, if you have a theory that proves its own consistency, well, I mean, even an inconsistent theory would prove its own consistency. And so it doesn't seem to be a logical reason to believe in the consistency if you have a theory that proves itself 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
“Consistency of the strong infinitary theory, but even the infinitary theory can't prove its own consistency, right? That's the second incompleteness theorem. And so it's in that sense a decisive takedown of the Hilbert program, which is His theorem just really answered that whole puzzle. It's quite amazing. But there's another aspect. Kind of easy to think about. I mean, if you're wondering about theories that prove their own consistency, 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
“Would have questions that they stumble with and are unable to answer, independence would occur. But then also because of the failure of the second goal, we would also have to be constantly worrying about whether our theories were consistent or not, and we wouldn't have any truly convincing means of saying that they were free from contradiction. And the fact of Girdle's incompleteness theorem shows that that is exactly the nature of mathematical reality, actually. Those are the two incompleteness theorems. So the first incompleteness theorem says you cannot write down a computably axematizable theory that answers all the questions. Every such theory will be incomplete, assuming it includes a certain amount of arithmetic. And secondly, no such theory can ever prove its own consistency. So not only is it the case that the finitary theory can't prove the”
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 okay, so to talk about the alternative to the Hilbert point of view, I mean, if he's wrong, then what is the nature of mathematical reality? Well, it would mean that we couldn't ever maybe for the first goal, we couldn't ever write down a theory that answered all the questions. So we would always be in a situation where our best theory, even the infinitary theories”
2025-12-31 · Lex Fridman Podcast · #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins · IDENTIFIED FROM THE TRANSCRIPT · source