Morpheus

From Lex Fridman Podcast

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

Published
December 31, 2025
Starts at
1:44:52

Audio is streamed from the publisher.

Chapters

Read from the show notes

  1. 0:00 Introduction
  2. 1:58 Sponsors, Comments, and Reflections
  3. 15:40 Infinity & paradoxes
  4. 1:02:50 Russell’s paradox
  5. 1:15:57 Gödel’s incompleteness theorems
  6. 1:33:28 Truth vs proof
  7. 1:44:52 The Halting Problem
  8. 2:00:45 Does infinity exist?
  9. 2:18:19 MathOverflow
  10. 2:22:12 The Continuum Hypothesis
  11. 2:31:58 Hardest problems in mathematics
  12. 2:41:25 Mathematical multiverse
  13. 3:00:18 Surreal numbers
  14. 3:10:55 Conway’s Game of Life
  15. 3:13:11 Computability theory
  16. 3:23:04 P vs NP
  17. 3:26:21 Greatest mathematicians in history
  18. 3:40:05 Infinite chess
  19. 3:58:24 Most beautiful idea in mathematics

Not from the publisher — times may be approximate.

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]( https://lexfridman.com/sponsors/ep488-sc">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]( https://lexfridman.com/joel-david-hamkins-transcript">https://lexfridman.com/joel-david-hamkins-transcript **CONTACT LEX:** **Feedback** – give feedback to Lex: [https://lexfridman.com/survey]( https://lexfridman.com/survey">https://lexfridman.com/survey **AMA** – submit questions, videos or call-in: [https://lexfridman.com/ama]( https://lexfridman.com/ama">https://lexfridman.com/ama **Hiring** – join our team: [https://lexfridman.com/hiring]( https://lexfridman.com/hiring">https://lexfridman.com/hiring **Other** – other ways to get in touch: [https://lexfridman.com/contact]( https://lexfridman.com/contact">https://lexfridman.com/contact **EPISODE LINKS:** Joel’s X: [https://x.com/JDHamkins]( https://x.com/JDHamkins">https://x.com/JDHamkins Joel’s Website: [https://jdh.hamkins.org]( https://jdh.hamkins.org">https://jdh.hamkins.org Joel’s Substack: [https://www.infinitelymore.xyz]( https://www.infinitelymore.xyz">https://www.infinitelymore.xyz Joel’s MathOverflow: [https://mathoverflow.net/users/1946/joel-david-hamkins]( https://mathoverflow.net/users/1946/joel-david-hamkins">https://mathoverflow.net/users/1946/joel-david-hamkins Joel’s Papers: [https://jdh.hamkins.org/publications]( https://jdh.hamkins.org/publications">https://jdh.hamkins.org/publications Joel’s Books: Lectures on the Philosophy of Mathematics: [https://amzn.to/3MThaAt]( https://amzn.to/3MThaAt">https://amzn.to/3MThaAt Proof and the Art of Mathematics: [https://amzn.to/3YACc9A]( https://amzn.to/3YACc9A">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/]( https://lexfridman.com/s/perplexity-ep488-sc">https://www.perplexity.ai/ **Fin:** AI agent for customer service. Go to [https://fin.ai/lex]( https://lexfridman.com/s/fin-ep488-sc">https://fin.ai/lex **Miro:** Online collaborative whiteboard platform. Go to [https://miro.com/]( https://lexfridman.com/s/miro-ep488-sc">https://miro.com/ **CodeRabbit:** AI-powered code reviews. Go to [https://coderabbit.ai/lex]( https://lexfridman.com/s/coderabbit-ep488-sc">https://coderabbit.ai/lex **Chevron:** Reliable energy for data centers. Go to [https://chevron.com/power]( https://lexfridman.com/s/chevron-ep488-sc">https://chevron.com/power **Shopify:** Sell stuff online. Go to [https://shopify.com/lex]( https://lexfridman.com/s/shopify-ep488-sc">https://shopify.com/lex **LMNT:** Zero-sugar electrolyte drink mix. Go to [https://drinkLMNT.com/lex]( https://lexfridman.com/s/lmnt-ep488-sc">https://drinkLMNT.com/lex **MasterClass:** Online classes from world-class experts. Go to [https://masterclass.com/lexpod]( https://lexfridman.com/s/masterclass-ep488-sc">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