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

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 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: See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc. Transcript: CONTACT LEX: Feedback – give feedback to Lex: AMA – submit questions, videos or call-in: Hiring – join our team: Other – other ways to get in touch: EPISODE LINKS: Joel’s X: Joel’s Website: Joel’s Substack: Joel’s MathOverflow: Joel’s Papers: Joel’s Books: Lectures on the Philosophy of Mathematics: Proof and the Art of Mathematics: SPONSORS: To support this podcast, check out our sponsors & get discounts: Perplexity: AI-powered answer engine. Go to Fin: AI agent for customer service. Go to Miro: Online collaborative whiteboard platform. Go to CodeRabbit: AI-powered code reviews. Go to Chevron: Reliable energy for data centers. Go to Shopify: Sell stuff online. Go to LMNT: Zero-sugar electrolyte drink mix. Go to MasterClass: Online classes from world-class experts. Go to OUTLINE: (00:00) – Introduction (01:58) – Sponsors, Comments, and Reflections (15:40) – Infinity & paradoxes (1:02:50) – Russell’s paradox (1:15:57) – Gödel’s incompleteness theorems (1:33:28) – Truth vs proof (1:44:52) – The Halting Problem (2:00:45) – Does infinity exist? (2:18:19) – MathOverflow (2:22:12) – The Continuum Hypothesis (2:31:58) – Hardest problems in mathematics (2:41:25) – Mathematical multiverse (3:00:18) – Surreal numbers (3:10:55) – Conway’s Game of Life (3:13:11) – Computability theory (3:23:04) – P vs NP (3:26:21) – Greatest mathematicians in history (3:40:05) – Infinite chess (3:58:24) – Most beautiful idea in mathematics

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#488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins is an episode from Artificial Intelligence by Lex Fridman.

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This episode was published on Dec 31, 2025.

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