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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

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@bjohang From YouTube ↗ 22 Feb 2026 1
Thank you for having a pure Mathematician on your podcast.
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Lex you are spoiling us with these recent podcasts man
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Thank you for this. Speaking as a numberphile, I feel like this was the kind of conversation I needed to hear during my first brush with set theory as an undergrad. There was just never enough time in class to have the fun discussions, and back then, we couldn't just look up YouTube videos to do a deep dive. I always ended up feeling like there were things we covered that I was supposed to have learned elsewhere but didn't, and I didn't have a very deep understanding of them. Beyond the calculations and the proofs, it seems very few professors actually took the time to help students conceptualizing what it meant. This is good.
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As a fellow engineer, I thank you for the intellectual conversations you have with your guests
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Infinity discussions are fascinating, but Olovka simplifies essay citations, making the writing process much smoother.
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Very good choice that you invited him to your show
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This is the best episode I have seen in my lifetime and I can prove it within the model by using the logic embeded in the formal system.
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Just what I wanted while hiding from social obligations. Happy new year 🎉
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Happy New Year, Lex Gang!
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Happy New Year. How damn lucky we all are to have Lex Fridman's interviews online for free!
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@MJLDN12 From YouTube ↗ 22 Jan 2026 108
Happy New Year everyone. So glad Lex is sticking to maths, science, tech these days.
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Happy New Year Everyone 🥳
Thanks for all the likes may your New Year be amazing
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@EoleoV From YouTube ↗ 22 Jan 2026 16
Wow, such a wonderful guest, I am blown away !
Thank you so much for inviting him and for having such an insightful and brilliant conversation
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@sbastos01 From YouTube ↗ 22 Jan 2026 124
What a joy to be alive. Happy New Year for all infinities
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Math, physics, and biology are by far my favorite topics on this cast. Please keep them coming Lex.
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This is awesome Lex. I know it's much less popular to have someone talk at length about something so involved but for the people who watch and are interested in this topic area its a brilliant resource and refreshing to go in depth on a topic, rather than keeping it pop sci.
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I was telling my husband how pondering infinity is slowly making me go insane and then this pops up not long after to make it worse 😅 send thoughts and prayers to my husband cause he’s about to be annoyed af for the next few years, as there’s no way I’m shutting up about this
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@BBKiD11-11 From YouTube ↗ 22 Jan 2026 154
Yep I’m understanding all of this 🙃
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The remark Lex made about professor Hamkins' status as a user of mathoverflow really is no joke: I have lost the count of how many times his answers have been useful for my own curiosity and self-study. The level of detail and clarity is beyond belief, such a gem of a contributor
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@lexfridman From YouTube ↗ 22 Jan 2026 50
Thank you for listening ❤ Check out our sponsors: 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:
lexfridman.com/joel-david-hamkins-transcript

CONTACT LEX:
Feedback - give feedback to Lex: lexfridman.com/survey
AMA - submit questions, videos or call-in: lexfridman.com/ama
Hiring - join our team: lexfridman.com/hiring
Other - other ways to get in touch: lexfridman.com/contact

EPISODE LINKS:
Joel's X: x.com/JDHamkins
Joel's Website: jdh.hamkins.org/
Joel's Substack: www.infinitelymore.xyz/
Joel's MathOverflow: mathoverflow.net/users/1946/joel