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Lex Fridman Podcast / episode intelligence

#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins

31 Dec 2025 28 published claims 2 attributable people
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Speakers in the public record

Lex FridmanJoel David Hamkins

Claim mix

belief 13evaluation 6commitment 3observation 2prediction 2preference 2

Evidence policy

Every row below preserves an exact excerpt. Identified speakers are linked; unresolved voices are labeled and excluded from people counts.

Claim ledger

The useful parts, with receipts.

28 published records

01 / commitment

I have been hiding from the world a bit, reading, thinking, writing, soul-searching, as we all do every once in a while. But mostly, just deeply focused on work and preparing mentally for some challenging travel I plan to take on in the new year.

“I have been hiding from the world a bit, reading, thinking, writing, soul-searching, as we all do every once in a while. But mostly, just deeply focused on work and preparing mentally for some challenging travel I plan to take on in the new year.”
Speaker
Lex Fridman
Publisher
Lex Fridman Podcast

04 / observation

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.

“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.”
Publisher
Lex Fridman Podcast

05 / belief

I was an undergrad and LaTeX was sort of unheard of, and so I was producing these beautifully typeset, you know— …problem sets, solutions, and so on. And I would print it up and submit it and so on, and the grades would come back, terrible grades, and I realized what was happening— The copy was so beautiful, mathematically typeset in this way, it looked like the kind of mathematics you find in a book.

“I was an undergrad and LaTeX was sort of unheard of, and so I was producing these beautifully typeset, you know— …problem sets, solutions, and so on. And I would print it up and submit it and so on, and the grades would come back, terrible grades, and I realized what was happening— The copy was so beautiful, mathematically typeset in this way, it looked like the kind of mathematics you find in a book.”
Publisher
Lex Fridman Podcast

06 / belief

Although set theory continues to, in my view, have an extremely successful metamathematical analysis as a foundation, I think is much more successful than set theory for any of those other foundations, but it’s much less amenable to things like computer proof and so on, which is part of the motivation to find these alternative foundations.

“Although set theory continues to, in my view, have an extremely successful metamathematical analysis as a foundation, I think is much more successful than set theory for any of those other foundations, but it’s much less amenable to things like computer proof and so on, which is part of the motivation to find these alternative foundations.”
Publisher
Lex Fridman Podcast

07 / commitment

I don’t think we understand the nature of physical reality very well at all, and I think most people aren’t even scratching the surface of the question as I intend to be asking it.

“I don’t think we understand the nature of physical reality very well at all, and I think most people aren’t even scratching the surface of the question as I intend to be asking it.”
Publisher
Lex Fridman Podcast

08 / commitment

” There’s another famous quote of Hilbert in his retirement address where he proclaims, “Wir müssen wissen, wir werden 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.

“” There’s another famous quote of Hilbert in his retirement address where he proclaims, “Wir müssen wissen, wir werden 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.”
Publisher
Lex Fridman Podcast

09 / belief

Maybe they have a good math idea, and so I want to talk to them and interact with them. And so I think the Perelman case is maybe an instance where, you know, he’s such a brilliant mind and he solved this extremely famous and difficult problem, and that is a huge achievement.

“Maybe they have a good math idea, and so I want to talk to them and interact with them. And so I think the Perelman case is maybe an instance where, you know, he’s such a brilliant mind and he solved this extremely famous and difficult problem, and that is a huge achievement.”
Publisher
Lex Fridman Podcast

11 / belief

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.

“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.”
Publisher
Lex Fridman Podcast

12 / belief

I think cellular automata is one of the most incredible, complicated, fascinating… It feels like an open door into a world we have not quite yet explored. And it’s such a beautiful illustration of that world, the Game of Life, but calling it a game… Maybe life balances it, because that’s your powerful word, but it’s not quite a game.

“I think cellular automata is one of the most incredible, complicated, fascinating… It feels like an open door into a world we have not quite yet explored. And it’s such a beautiful illustration of that world, the Game of Life, but calling it a game… Maybe life balances it, because that’s your powerful word, but it’s not quite a game.”
Speaker
Lex Fridman
Publisher
Lex Fridman Podcast

13 / belief

There’s this concept of the set-theoretic 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.

“There’s this concept of the set-theoretic 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.”
Publisher
Lex Fridman Podcast

14 / belief

Whereas I think that we do have something a little better of an understanding of the nature of mathematical existence and abstract existence.

“Whereas I think that we do have something a little better of an understanding of the nature of mathematical existence and abstract existence.”
Publisher
Lex Fridman Podcast

15 / belief

I mean, my view is that, yeah, 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, my view is that, yeah, this isn’t traumatic at all. This is rather completely eye-opening in terms of our understanding of the nature of mathematical reality.”
Publisher
Lex Fridman Podcast

16 / belief

I think his case is a reminder that some of the greatest mathematicians, some of the greatest scientists and human beings do the thing they do, take on these problems for the love of it, not for the prizes or the money or any of that.

“I think his case is a reminder that some of the greatest mathematicians, some of the greatest scientists and human beings do the thing they do, take on these problems for the love of it, not for the prizes or the money or any of that.”
Speaker
Lex Fridman
Publisher
Lex Fridman Podcast

17 / belief

Yeah, I think so, because I guess part of the point of structuralism is that it doesn’t make sense to consider mathematical objects or individuals in isolation.

“Yeah, I think so, because I guess part of the point of structuralism is that it doesn’t make sense to consider mathematical objects or individuals in isolation.”
Publisher
Lex Fridman Podcast

18 / belief

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, 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.

“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, 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.”
Publisher
Lex Fridman Podcast

19 / evaluation

The motivation is misplaced. And so I worry that this is a very dangerous source of error because it often happens in mathematics that… I mean, if I think back to when I was an undergrad, here at Caltech, I was a math major eventually, and at that time, LaTeX was a pretty new thing and I was learning LaTeX, and so I was typing up my homeworks in LaTeX and they looked beautiful.

“The motivation is misplaced. And so I worry that this is a very dangerous source of error because it often happens in mathematics that… I mean, if I think back to when I was an undergrad, here at Caltech, I was a math major eventually, and at that time, LaTeX was a pretty new thing and I was learning LaTeX, and so I was typing up my homeworks in LaTeX and they looked beautiful.”
Publisher
Lex Fridman Podcast

20 / evaluation

For example, there’s this threefold repetition rule in ordinary chess, but we just, we just get rid of this for infinite chess because, of course, threefold repetition is just a proxy for infinite play.

“For example, there’s this threefold repetition rule in ordinary chess, but we just, we just get rid of this for infinite chess because, of course, threefold repetition is just a proxy for infinite play.”
Publisher
Lex Fridman Podcast

21 / evaluation

I mean, because it’s an asymptotic property, it’s really in the limit as the size of the inputs goes to infinity, that’s the only time that polynomial or NP becomes relevant. And so maybe it’s important to keep that in mind when… Sometimes you find kind of overblown remarks made about, you know, if P equals NP, then this will be incredibly important for human civilization because it means that we’ll have feasible algorithms for solving these incredibly important… …problems in NP.

“I mean, because it’s an asymptotic property, it’s really in the limit as the size of the inputs goes to infinity, that’s the only time that polynomial or NP becomes relevant. And so maybe it’s important to keep that in mind when… Sometimes you find kind of overblown remarks made about, you know, if P equals NP, then this will be incredibly important for human civilization because it means that we’ll have feasible algorithms for solving these incredibly important… …problems in NP.”
Publisher
Lex Fridman Podcast

23 / prediction

I argue that it’s really the philosophy of structuralism that leads them to omit the ur-elements because it turns out that if you adopt ZFC axioms with ur-elements, ZFCU it’s called, or ZFA, then any structure that exists, any mathematical structure that exists in that set theoretic universe with the atoms is isomorphic to a structure that doesn’t use the atoms at all.

“I argue that it’s really the philosophy of structuralism that leads them to omit the ur-elements because it turns out that if you adopt ZFC axioms with ur-elements, ZFCU it’s called, or ZFA, then any structure that exists, any mathematical structure that exists in that set theoretic universe with the atoms is isomorphic to a structure that doesn’t use the atoms at all.”
Publisher
Lex Fridman Podcast

24 / prediction

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.

“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.”
Publisher
Lex Fridman Podcast

25 / preference

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 these totally uninteresting parts of what it’s like to write a proof, these kind of mechanistic procedures about how to write a proof.

“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 these totally uninteresting parts of what it’s like to write a proof, these kind of mechanistic procedures about how to write a proof.”
Publisher
Lex Fridman Podcast

26 / preference

I mean, before we used this 3 to the C, 5 times 5 to the S, which is a kind of, you know, overly arithmetic way to think about it. But there’s a kind of direct, you know, way to understand that it’s still a countable infinity when you have countably many countable sets because you can just start putting them on this list.

“I mean, before we used this 3 to the C, 5 times 5 to the S, which is a kind of, you know, overly arithmetic way to think about it. But there’s a kind of direct, you know, way to understand that it’s still a countable infinity when you have countably many countable sets because you can just start putting them on this list.”
Publisher
Lex Fridman Podcast

27 / evaluation

It’s 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 that 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 Burali-Forti paradox and so on.

“It’s 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 that 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 Burali-Forti paradox and so on.”
Publisher
Lex Fridman Podcast

28 / evaluation

Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new.

“Because I think, for example, if you prove a new result with a bad argument or a complicated argument, that’s great because you proved something new.”
Publisher
Lex Fridman Podcast
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