#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 #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-sc 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://www.perplexity.ai/ **Fin:** AI agent for customer service. Go to https://fin.ai/lex **Miro:** Online collaborative whiteboard platform. Go to https://miro.com/ **CodeRabbit:** AI-powered code reviews. Go to https://coderabbit.ai/lex **Chevron:** Reliable energy for data centers. Go to https://chevron.com/power **Shopify:** Sell stuff online. Go to https://shopify.com/lex **LMNT:** Zero-sugar electrolyte drink mix. Go to https://drinkLMNT.com/lex **MasterClass:** Online classes from world-class experts. Go to https://masterclass.com/lexpod **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