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. Go to https://lexfridman.com/s/perplexity-ep488-sb *Fin:* AI agent for customer service. Go to https://lexfridman.com/s/fin-ep488-sb *Miro:* Online collaborative whiteboard platform. Go to https://lexfridman.com/s/miro-ep488-sb *CodeRabbit:* AI-powered code reviews. Go to https://lexfridman.com/s/coderabbit-ep488-sb *Chevron:* Reliable energy for data centers. Go to https://lexfridman.com/s/chevron-ep488-sb *Shopify:* Sell stuff online. Go to https://lexfridman.com/s/shopify-ep488-sb *LMNT:* Zero-sugar electrolyte drink mix. Go to https://lexfridman.com/s/lmnt-ep488-sb *MasterClass:* Online classes from world-class experts. Go to https://lexfridman.com/s/masterclass-ep488-sb *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:* - Podcast Website: https://lexfridman.com/podcast - Apple Podcasts: https://apple.co/2lwqZIr - Spotify: https://spoti.fi/2nEwCF8 - RSS: https://lexfridman.com/feed/podcast/ - Podcast Playlist: https://www.youtube.com/playlist?list=PLrAXtmErZgOdP_8GztsuKi9nrraNbKKp4 - Clips Channel: https://www.youtube.com/lexclips *SOCIAL LINKS:* - X: https://x.com/lexfridman - Instagram: https://instagram.com/lexfridman - TikTok: https://tiktok.com/@lexfridman - LinkedIn: https://linkedin.com/in/lexfridman - Facebook: https://facebook.com/lexfridman - Patreon: https://patreon.com/lexfridman - Telegram: https://t.me/lexfridman - Reddit: https://reddit.com/r/lexfridman