Infinity, Paradoxes, Gödel Incompleteness & the Mathematical Multiverse | Lex Fridman Podcast #488
Joel David Hamkins is a mathematician and philosopher specializing in set theory, the foundations of mathematics, and the nature of infinity, and he’s the #1 highest-rated user on MathOverflow. 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 called Infinitely More.
Thank you for listening ❤ Check out our sponsors: https://lexfridman.com/sponsors/ep488-sb
See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc.
*Transcript:*
https://lexfridman.com/joel-david-hamkins-transcript
*CONTACT LEX:*
*Feedback* – give feedback to Lex: https://lexfridman.com/survey
*AMA* – submit questions, videos or call-in: https://lexfridman.com/ama
*Hiring* – join our team: https://lexfridman.com/hiring
*Other* – other ways to get in touch: https://lexfridman.com/contact
*EPISODE LINKS:*
Joel’s X: https://x.com/JDHamkins
Joel’s Website: https://jdh.hamkins.org
Joel’s Substack: https://www.infinitelymore.xyz
Joel’s MathOverflow: https://mathoverflow.net/users/1946/joel-david-hamkins
Joel’s Papers: https://jdh.hamkins.org/publications
Joel’s Books:
Lectures on the Philosophy of Mathematics: https://amzn.to/3MThaAt
Proof and the Art of Mathematics: https://amzn.to/3YACc9A
*SPONSORS:*
To support this podcast, check out our sponsors & get discounts:
*Perplexity:* AI-powered answer engine.
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*OUTLINE:*
0:00 – Introduction
2:17 – Infinity & paradoxes
49:27 – Russell’s paradox
1:02:35 – Gödel’s incompleteness theorems
1:20:06 – Truth vs proof
1:31:30 – The Halting Problem
1:47:23 – Does infinity exist?
2:04:57 – MathOverflow
2:08:49 – The Continuum Hypothesis
2:18:36 – Hardest problems in mathematics
2:28:03 – Mathematical multiverse
2:46:55 – Surreal numbers
2:57:33 – Conway’s Game of Life
2:59:49 – Computability theory
3:09:41 – P vs NP
3:12:58 – Greatest mathematicians in history
3:26:43 – Infinite chess
3:45:01 – Most beautiful idea in mathematics
*PODCAST LINKS:*
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@lexfridman
May 19, 2026 at 9:28 pm
Thank you for listening ❤ Check out our sponsors: https://lexfridman.com/sponsors/ep488-sa
See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc.
0:00 – Introduction
2:17 – Infinity & paradoxes
49:27 – Russell's paradox
1:02:35 – Gödel's incompleteness theorems
1:20:06 – Truth vs proof
1:31:30 – The Halting Problem
1:47:23 – Does infinity exist?
2:04:57 – MathOverflow
2:08:49 – The Continuum Hypothesis
2:18:36 – Hardest problems in mathematics
2:28:03 – Mathematical multiverse
2:46:55 – Surreal numbers
2:57:33 – Conway's Game of Life
2:59:49 – Computability theory
3:09:41 – P vs NP
3:12:58 – Greatest mathematicians in history
3:26:43 – Infinite chess
3:45:01 – Most beautiful idea in mathematics
Transcript:
https://lexfridman.com/joel-david-hamkins-transcript
CONTACT LEX:
Feedback – give feedback to Lex: https://lexfridman.com/survey
AMA – submit questions, videos or call-in: https://lexfridman.com/ama
Hiring – join our team: https://lexfridman.com/hiring
Other – other ways to get in touch: https://lexfridman.com/contact
EPISODE LINKS:
Joel's X: https://x.com/JDHamkins
Joel's Website: https://jdh.hamkins.org
Joel's Substack: https://www.infinitelymore.xyz
Joel's MathOverflow: https://mathoverflow.net/users/1946/joel-david-hamkins
Joel's Papers: https://jdh.hamkins.org/publications
Joel's Books:
Lectures on the Philosophy of Mathematics: https://amzn.to/3MThaAt
Proof and the Art of Mathematics: https://amzn.to/3YACc9A
SPONSORS:
To support this podcast, check out our sponsors & get discounts:
Perplexity: AI-powered answer engine.
Go to https://lexfridman.com/s/perplexity-ep488-sa
Fin: AI agent for customer service.
Go to https://lexfridman.com/s/fin-ep488-sa
Miro: Online collaborative whiteboard platform.
Go to https://lexfridman.com/s/miro-ep488-sa
CodeRabbit: AI-powered code reviews.
Go to https://lexfridman.com/s/coderabbit-ep488-sa
Chevron: Reliable energy for data centers.
Go to https://lexfridman.com/s/chevron-ep488-sa
Shopify: Sell stuff online.
Go to https://lexfridman.com/s/shopify-ep488-sa
LMNT: Zero-sugar electrolyte drink mix.
Go to https://lexfridman.com/s/lmnt-ep488-sa
MasterClass: Online classes from world-class experts.
Go to https://lexfridman.com/s/masterclass-ep488-sa
@bassplayersayer
May 19, 2026 at 9:28 pm
Clearly this guy has not found the mathematical truth. You must study Terrence Howard and his gift to the Universe…Terryology. Open Your mind.
@openmictea
May 19, 2026 at 9:28 pm
What a great podcast you have here.
@KayAmooty433
May 19, 2026 at 9:28 pm
One breaks language by redefining the word 'infinity'. The hotel exercise is not logical. What would Wittgenstein say?
@tommac8556
May 19, 2026 at 9:28 pm
Well that cleared that up then
@michaelcary1382
May 19, 2026 at 9:28 pm
This is an amazing discussion. Advanced mathmatical concepts delivered with wit and charisma .
@lukasclark884
May 19, 2026 at 9:28 pm
Julius Ceasar, as with most humans, would have assigned himself only one number, and one number alone!
@fwill182
May 19, 2026 at 9:28 pm
Has anyone ever proved that s particular statement csn not be proven?
@peterford5408
May 19, 2026 at 9:28 pm
1:39:39 "… the proof-writing course which is usually taken by students who …"
At this point, I was provably greater than 50% confident that he was going to say something like "… thought they were signing up to a proofreading course after an administrator failed to proofread the course catalogue" 😊
Also, I just noticed that someone failed to proofread the subtitles because "proof-writing" has been substituted by "proofreading" in the above quote. 😅
@wendmagegnasfaw3928
May 19, 2026 at 9:28 pm
This absolutely wise person, even after I tried to tell as many media outlets as possible about his abuse and harassment in the comment sections of their YouTube channels until I find their correct email addresses, whether they read it or not, he keeps sending evil spirits that resemble or mimic his voice, saying stupid things like (ስም, ምታ, ትል, የሚተፋ ሰው ድምፅ, ላስ, ተው, and other stupid things):
(ስም) This means name
(ምታ) Supposed to mean masturbate
(ትል) Worm
(የሚተፋ ሰው ድምፅ) A sound a person makes when he spits
(ላስ) To leak something
(ተው) Supposed to remind me of the pornography that he made and showed to me using the woman who looks like, or is probably the same woman, whom I was talking to and had feelings for
These are supposed to be “triggers”—triggers intended to cause psychological stress or mental health problems.
I am a foolish person myself, but this level of foolishness from a middle-aged man who is supposed to be educated such disregard for human dignity and human rights.
Using evil spirits, he continues to wake me just before I go to sleep, also preventing me from staying asleep and giving me nightmares, and he continues to cause physical pain in different parts of my body. He also continues to make me hear various sounds when I am between asleep and awake, and he keeps making sounds inside my right eardrum.
He also continues to make my family members say bad things so that I will say and do bad things, and he makes them do embarrassing things just to cause mental stress.
Politicians are not this stupid. This is not politics. This person is not mentally well.
This absolutely wise person continues to accuse me of sexual sins when this person, and others like him, have been preventing me from having relationships for many years, beginning from my teenage years, preventing me from starting a family, and even going as far as sending evil spirits that affect my sexual life. It has been 11 years since I graduated from five years of university education; that is how old I am. Also, we should not forget that I am a person with no political affiliation, a civilian with an average mind and little education.
Being Ethiopian is much more than this.
I personally believe there is no such thing as confidentiality when it comes to corruption, intellectual property theft, and violations of human and democratic rights.
There is no greatness in theft.
@1290Nick
May 19, 2026 at 9:28 pm
If you want to replace each hotel visiter one room further, you are asuming that there is a beginning, infinity has no beginning.
@theodanielwollff
May 19, 2026 at 9:28 pm
I should write a paper about how you cannot go beyond infinity. Really these math guys are doing word play with facts.
@jasonsebring3983
May 19, 2026 at 9:28 pm
idk wouldn't the speed of light violate the hotel idea? you would wait for infinity for the last one to move up at some point.
@bbberliozb6691
May 19, 2026 at 9:28 pm
Sorry Hilbert's hotel is incorrect. If you are able to add 1 to infinity then it wasn't infinity. The hotel should already be filled with every possible number as infinity doesn't end. You shouldn't be able to fit another in because it's infinity. We can think of continuously adding a room but if that's the case the bottle has not reached infinity. It is just continuously growing towards infinity but if we can add a number to it it hasn't reached the infinite. We need a different word. As soon as we can add something to the infinite it wasn't infinite to begin with.
@markpriore3151
May 19, 2026 at 9:28 pm
Lex, I dare you to get someone MORE autistic
@sfischer9703
May 19, 2026 at 9:28 pm
An infinitesimally small amount of people understood what these two were going on about
@evelina7148
May 19, 2026 at 9:28 pm
66
@Juanelmajo13
May 19, 2026 at 9:28 pm
Great channel. I have watched many of your videos. The criticism is good for some laughs. 😉
@Diverse_Interests
May 19, 2026 at 9:28 pm
Is AI even constructed as nonlinear (systems thinking)? It would have to be or it’s perpetual stuck in linear thought of a certain percentage of the populations own mental software. There is no ability for linear to step up into logic leaps, skip step and multinodel thinking. If average designed the ai as themselves, it’s always going to fuck up thinking because it’s been handicapped.
@Diverse_Interests
May 19, 2026 at 9:28 pm
The misplaced argument and flawed logic is a reflection of other people’s minds and AI is largely designing itself from interactions with average Joe and not other mind types, so it will mirror the mentality and logic of average.
@Diverse_Interests
May 19, 2026 at 9:28 pm
Also dump the information online so there is access in the data set the AI is drawing on
@Diverse_Interests
May 19, 2026 at 9:28 pm
You can can essentially discuss rubbish answers from AI with AI and hope it learns from the interactions. If you see a flaw speak of it and give the sample information. It’s fascinating but it can be frustrating if a programmer dumped their own limitations into the program.
@Diverse_Interests
May 19, 2026 at 9:28 pm
This man’s mind is refreshing. It’s true, high complexity does reduce to elegance when you understand the complexity. I love that he wants the beauty (elegant flow) of the systems. Still has curiosity, how delightful.
@Diverse_Interests
May 19, 2026 at 9:28 pm
So there is the difference of mental abilities right there. To do highly complex thinking that creates elegant solutions requires that playfulness and speed of mind for instant computation and draw on multiple nodes of information. It’s always a game because of systems thinking without limitations and it doesn’t make it not any more real than “serious” as thought by people unable to understand but caught up on image, conformity and social heirarchy behaviours. That perception is linear thinkers using one node as the brains default and having to move step by step in absolutely everything ever thought of. Linear NT minds can’t see the humour or game like playfulness in highly complex work because they have different hardware and software. Physiological and cognitive changes to the brain occur over the 130 IQ in gifted people. A bright person can go to uni and be fine but they never break through the linear processing system in their head and miss more than 80% of the information of a system they are looking at and because of the filtering out of information they simply are not aware that anything exists beyond their linear perception.
Enjoying the work, playing with concepts, info, ideas, that’s not a bad thing, that’s ease with complexity and ability to see all the pieces of the puzzle and manipulate them in your head instantly. If those people where stuck on needing a social performance show of hierarchy to feel good , then that’s old ape behaviour taking over and has nothing to do with knowledge and the application of knowledge. On the light side, how boring is that, please stop enjoying the work and having fun with it and make it as dull and grey as we are. Lame.
@Diverse_Interests
May 19, 2026 at 9:28 pm
Delightful. Math has built in playfulness in the right hands and can be approached as a game.
@Diverse_Interests
May 19, 2026 at 9:28 pm
I love that in a discussion of infinity there was the insistence on going further back to the Ancient Greek. He gets bonus points from me in the first few minutes. It’s upped my excitement level for what can be expected from him.
@rogerparkhurst5796
May 19, 2026 at 9:28 pm
Is the value of Pi an example of a defined ‘infinity?’
@rogerparkhurst5796
May 19, 2026 at 9:28 pm
The term ‘longer’ limits the concepts of infinity by attempting to define a difference. We are limited by the terms we use to describe infinity…I would suggest infinity cannot have limits or we have to develop a new term.
@andreykovtun5304
May 19, 2026 at 9:28 pm
Is it possible to perfectly order the set of bosons in a Bose-Einstein condensate?
@hebejeebee
May 19, 2026 at 9:28 pm
Remind me never to stay at Hilbert's Hotel. It sounds like I'd be getting woken up infinite times to change rooms..!
@AshishGupta-xy6ie
May 19, 2026 at 9:28 pm
Please Include Hindi Audio Track also❤❤❤
@Phyzm1
May 19, 2026 at 9:28 pm
I really wanted to enjoy this but they failed at making this interesting.
@1life530
May 19, 2026 at 9:28 pm
A proof is the simplest form of falsifiability.
@1life530
May 19, 2026 at 9:28 pm
The axiom of choice: disregarding it is a choice, and so you exercise it by disregarding it
@1life530
May 19, 2026 at 9:28 pm
If you've ever grappled with rebelion against mathenatical rigor, this is a MUST WATCH.
@1life530
May 19, 2026 at 9:28 pm
This is one of the best interviews ever… I love the emphasis on construction to demystify all notions of abstract existences
@1life530
May 19, 2026 at 9:28 pm
Bachelor in Math here, how does an infinitely countable set of countable infinites violate Euclid's Principle?
The principle is that THE whole is greater than the parts. Without a proof and an identity for a "whole,"
how can one conclude that there is no "whole" above the aforementioned part?
@sheilagibson982
May 19, 2026 at 9:28 pm
I am loving your videos.
@catfishshickenchit174
May 19, 2026 at 9:28 pm
I'm still trying to figure out what Lex brings to the table and why people watch his channel.
@iGortaur
May 19, 2026 at 9:28 pm
JDH: natural numbers start with 0, those who start them with 1 are wrong. Also him: let’s have r1, r2, r3,…
@davidcpearson
May 19, 2026 at 9:28 pm
I didn't understand one thing from start to finish. Great to fall asleep to though