Lex Fridman Podcast
#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins
31 Dec 2025
Chapter 1: What is the significance of Joel David Hamkins in mathematics?
The following is a conversation with Joel David Hamkins, a mathematician and philosopher specializing in set theory, the foundation of mathematics, and the nature of infinity. He is the number one highest rated user on Math Overflow, which I think is a legendary accomplishment. Math Overflow, by the way, is like Stack Overflow, but for research mathematicians.
He is also the author of several books, including Proof and the Art of Mathematics, and Lectures on the Philosophy of Mathematics. And he has a great blog, infinitelymore.xyz.
This is a super technical and super fun conversation about the foundation of modern mathematics and some mind-bending ideas about infinity, nature of reality, truth, and the mathematical paradoxes that challenged some of the greatest minds of the 20th century. 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. Through all of it, a recurring thought comes to me. How damn lucky I am to be alive and to get to experience so much love from folks across the world.
I want to take this moment to say thank you from the bottom of my heart for everything, for your support, for the many amazing conversations I've had with people across the world. I got a little bit of hate and a whole lot of love, and I wouldn't have it any other way. I'm grateful for all of it. And now, a quick few second mention of a sponsor.
Check them out in the description or at lexfriedman.com slash sponsors. It is, in fact, the best way to support this podcast. We got Perplexity for curiosity-driven knowledge exploration. Finn for customer service AI agents. Miro for brainstorming ideas with your team. CodeRabbit for code review.
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And I'm just really grateful for the patience and the support of the sponsors. The companies and the humans behind those companies have been really amazing over the years. So I don't think I would be able to do many of the crazy and the difficult things I'm doing with this podcast without the support of the sponsors. So please go check out their stuff. Please go support them.
Please buy whatever they're selling, really. It helps a lot. It is the best way to support the podcast. I'll do the full ad reads now. I try to make them interesting, but if you skip, please still do check out the sponsors. I enjoy their stuff. Maybe you will too. To get in touch with me for whatever reason, go to lexfriedman.com slash contact.
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Chapter 2: What themes are explored in the conversation about infinity?
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And a company that makes a large number of people happy is going to be a great company. Anyway, go to fin.ai.com to learn more about transforming your customer service and scaling your support team. That's fin.ai.com. This episode is also brought to you by Miro, an online collaborative platform.
They have this innovation workspace that blends AI and human creativity to turn ideas into real things, into results. One of the things I love the most, Recently getting back into the research environment, working on a lot of fun robotics projects with a lot of brilliant mechanical engineers, software engineers, machine learning people, robotics people. It's just the conversations we have.
Sometimes the aimless exploration of ideas, sometimes banter, sometimes humor, sometimes real rigor over mathematical models of a particular phenomena, whether it's the controllers, whether it's the perception of the robots, whether it's the different stages of the ML process, whether it's the different layers of the stack,
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Help your teams develop great ideas into results with Miro. Go to miro.com to find out how. That's M-I-R-O dot com. This episode is also brought to you by CodeRabbit, a platform that provides AI-powered code reviews directly within your terminal. As more and more code is generated, developers end up spending more and more time reading and reviewing code.
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Chapter 3: What challenges did Cantor's ideas present to mathematics?
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All of us together, every living organism, collaborating, leveraging whatever energy we get into creating something incredible. Anyway, visit chevron.com slash power to learn more. That's chevron.com forward slash power. This episode is brought to you by Shopify. I like how I'm getting more and more intense. A platform designed for anyone to sell anywhere with a great looking online store.
If you want to understand why Shopify is awesome on the engineering side, you want to go listen to the conversation I had with DHH, who espoused the beauty, the power, the elegance of Ruby on Rails that Shopify was built on. On another note, I went to NeurIPS and hung around in the booth, I guess you could say, of Shopify engineering.
It's just a bunch of great engineers talking about the various aspects of what it took to bring Shopify to life. I think it's Shopify.engineering, if you're curious, actually. If this is your kind of thing, if you want to understand why Shopify as a machine, as a service, is incredible, you go there. Anyway, that's not the point. Engineering is just awesome.
So it's always nice to know there's great engineering behind a thing. And the thing is a way to sell stuff online. That's Shopify.com. And you can sign up for a $1 per month trial period at Shopify.com slash Lex. That's all lower case. Go to Shopify.com slash Lex to take your business to the next level today. I'm doing the announcer voice more and more, and doing so poorly.
This episode is also brought to you by Element. My daily zero sugar and delicious electrolyte mix that I'm currently drinking, that I'm currently enjoying, enjoying a little too much, but really never enough because it's always good for you. Really good balance of electrolytes, sodium, potassium, and magnesium. Always the same flavor.
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Chapter 4: How does the discussion transition to Gödel's incompleteness theorems?
I'm going to show up at the beginning, and I'm going to go to the end and beyond, which means potentially an hour and a half, maybe two hours of training. One must celebrate the end of the year properly, my friends, and replenish properly after battle. With some electrolytes. Get a free 8-count sample pack with any purchase. Try it at drinkelement.com slash lex.
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And I fell in love with Texas, and I intend to keep it that way. Carlos Santana on guitar, of course. Europa, one of my favorite instrumental songs. It's a way to make the guitar cry. We could sing. Now, you could also be like Tom Morello, also on guitar. Also has a master class. Now, he can make a guitar the instrument of rebellion. Now, since we're talking about mathematics here with Joel,
we must mention that Terence Tao, the great Terence Tao, also has a masterclass on mathematical thinking. And finally, Martin Scorsese, A person I absolutely must talk to. Figure out a way to talk to him. But in the meantime, he also has a master class on filmmaking. One of the greatest directors in history. One of the greatest storytellers in history.
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This is the Lex Friedman Podcast. To support it, please check out our sponsors in the description, where you can also find ways to contact me, ask questions, give feedback, and so on. And now, dear friends, here's Joe David Hempkins. Some infinities are bigger than others. This idea from Cantor at the end of the 19th century, I think it's fair to say broke mathematics before rebuilding it.
And I also read that this was a devastating and transformative discovery for several reasons. So one, it created a theological crisis because infinity is associated with God. How could there be multiple infinities? and also Cantor was deeply religious himself. Second, there's a mathematical civil war.
The leading German mathematician, Kronecker, called Cantor a corrupter of youth and tried to block his career. Third, many fascinating paradoxes emerged from this. like Russell's paradox about the set of all sets that don't contain themselves, and those threatened to make all of mathematics inconsistent. And finally, on the psychological side, on the personal side, Cantor's own breakdown.
He literally went mad, spending his final years in and out of sanatoriums, obsessed with proving the continuum hypothesis. So laying that all out on the table, Can you explain the idea of infinity, that some infinities are larger than others, and why was this so transformative to mathematics?
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Chapter 5: What is the nature of mathematical existence?
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.
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 existence to something that we can experience physically in the real world. But my attitude about this attempt is that it's very backward, I think, because I 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—
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. 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 existence. It is 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 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.
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Chapter 6: How does structuralism influence our understanding of numbers?
The only thing that matters is, what are the properties of the number 4 in a given mathematical system? Recognizing that there are other isomorphic copies of that system and the properties of that other system's number 4 are going to be identical to the properties of this system's number 4. with regard to any question that's important about the number four.
But those questions won't be about essence. So in a sense, structuralism is kind of anti-essentialism in mathematics.
So is it fair to think of numbers as a kind of pointer to a deep underlying structure?
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. What's interesting and important about mathematical objects is how they interact with each other and how they behave in a system.
And so maybe one wants to think about the structural role that the objects play in a larger system, a larger structure. There's a famous question that Frege had asked, actually, when he was looking into the nature of numbers. Because in his logistic program, he was trying to reduce all mathematics to logic.
And in that process, he was referring to the Cantor-Hume principle that whenever two sets are equinumerous, then they have the same number of elements, if and only if. And he founded his theory of number on this principle.
But he recognized that, well, there was something that dissatisfied him about that situation, which is that the Kanner-Hume principle does not seem to give you a criteria for which things are numbers. It only tells you a kind of identity criteria for when are two numbers equal to each other. Well, Two numbers are equal just in case the sets of those sizes are equinumerous.
So that's the criteria for number identity, but it's not a criteria for what is a number. And so this problem has become known as the Julius Caesar problem because Frege said, we don't seem to have any way of telling from the Hume principle whether Julius Caesar is a number or not.
So he's asking about the essence of number and whether, of course, one has the sense that he picked maybe what he was trying to present as a ridiculous example, because maybe you have the idea that, well, obviously, Julius Caesar is not a number. And there's a lot of philosophical writing that seems to take that line also, that obviously the answer is that Julius Caesar is not a number.
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Chapter 7: What are the implications of Gödel's and Cohen's results?
Yeah, so... So what's more real, physics or the mathematical platonic space?
Well, the mathematical platonic realm is... I'm not sure I would say it's more real, but I'm saying we understand the reality of it in a much deeper and more convincing way. 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.
So, you know, obviously we understand physical reality. I mean, I knock on the table and so on and we know all about what it's like to, you know, have a birthday party or to drink a martini or whatever. And so we have a deep understanding of existing in the physical world. But maybe understanding is the wrong word.
We have an experience of living in the world and riding bicycles and all those things. But I don't think we actually have an understanding at all. I mean, very, very little of the nature of physical existence. I think it's a profound mystery. Whereas I think that we do have something a little better of an understanding of the nature of mathematical existence and abstract existence.
So that's how I would describe the point.
Somehow it feels like we're approaching some deep truth from different directions. And we just haven't traveled as far in the physics world as we have in the mathematical world.
Maybe I could hope that someone will give the convincing account. But it seems to be a profound mystery to me. I can't even imagine what it would be like to give an account of physical existence.
Yeah, I wonder, like a thousand years from now, as physics progresses, what this same conversation would look like.
Right, that would be quite interesting.
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Chapter 8: How do surreal numbers unify different number systems?
So I think it is possible to have this kind of progress, even when the subject kind of shifts away from the earlier concerns as a result of the progress, basically.
To take a tangent on a tangent, since you mentioned philosophy, maybe potentially more about the questions and maybe mathematics is about the answers, I have to say you are a legend, right? on Math Overflow, which is like Stack Overflow, but for math. You're ranked number one all time on there with currently over 246,000 reputation points.
How do you approach answering difficult questions on there?
Well, Math Overflow has really been one of the great pleasures of my life. I've really enjoyed it. And I've learned so much from interacting on Math Overflow. I've been on there since 2009, which was shortly after it started. I mean, it wasn't exactly at the start, but a little bit later. And...
I think it gives you the stats for how many characters I typed, and I don't know how many million it is, but this enormous amount of time that I've spent thinking about those questions, and it has really just been amazing to me.
How do you find the questions that grab you and how do you go about answering them?
So I'm interested in any question that I find interesting. And it's not all questions. Sometimes certain kinds of questions just don't appeal to me that much.
So you go outside of set theory as well.
So I think when I first joined Math Overflow, I was basically one of the few people in logic who was answering. I mean, there were other people who know some logic, particularly from category theory and other parts of mathematics that aren't in the most traditional parts of logic, but they were answering some of the logic questions.
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