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

About this episode

<p>Joel David Hamkins is a mathematician and philosopher specializing in set theory, the foundations of mathematics, and the nature of infinity, and he&#8217;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.<br /> Thank you for listening ❤ Check out our sponsors: <a href="https://lexfridman.com/sponsors/ep488-sc">https://lexfridman.com/sponsors/ep488-sc</a><br /> See below for timestamps, transcript, and to give feedback, submit questions, contact Lex, etc.</p> <p><b>Transcript:</b><br /> <a href="https://lexfridman.com/joel-david-hamkins-transcript">https://lexfridman.com/joel-david-hamkins-transcript</a></p> <p><b>CONTACT LEX:</b><br /> <b>Feedback</b> &#8211; give feedback to Lex: <a href="https://lexfridman.com/survey">https://lexfridman.com/survey</a><br /> <b>AMA</b> &#8211; submit questions, videos or call-in: <a href="https://lexfridman.com/ama">https://lexfridman.com/ama</a><br /> <b>Hiring</b> &#8211; join our team: <a href="https://lexfridman.com/hiring">https://lexfridman.com/hiring</a><br /> <b>Other</b> &#8211; other ways to get in touch: <a href="https://lexfridman.com/contact">https://lexfridman.com/contact</a></p> <p><b>EPISODE LINKS:</b><br /> Joel&#8217;s X: <a href="https://x.com/JDHamkins">https://x.com/JDHamkins</a><br /> Joel&#8217;s Website: <a href="https://jdh.hamkins.org">https://jdh.hamkins.org</a><br /> Joel&#8217;s Substack: <a href="https://www.infinitelymore.xyz">https://www.infinitelymore.xyz</a><br /> Joel&#8217;s MathOverflow: <a href="https://mathoverflow.net/users/1946/joel-david-hamkins">https://mathoverflow.net/users/1946/joel-david-hamkins</a><br /> Joel&#8217;s Papers: <a href="https://jdh.hamkins.org/publications">https://jdh.hamkins.org/publications</a><br /> Joel&#8217;s Books:<br /> Lectures on the Philosophy of Mathematics: <a href="https://amzn.to/3MThaAt">https://amzn.to/3MThaAt</a><br /> Proof and the Art of Mathematics: <a href="https://amzn.to/3YACc9A">https://amzn.to/3YACc9A</a></p> <p><b>SPONSORS:</b><br /> To support this podcast, check out our sponsors &#38; get discounts:<br /> <b>Perplexity:</b> AI-powered answer engine.<br /> Go to <a href="https://lexfridman.com/s/perplexity-ep488-sc">https://www.perplexity.ai/</a><br /> <b>Fin:</b> AI agent for customer service.<br /> Go to <a href="https://lexfridman.com/s/fin-ep488-sc">https://fin.ai/lex</a><br /> <b>Miro:</b> Online collaborative whiteboard platform.<br /> Go to <a href="https://lexfridman.com/s/miro-ep488-sc">https://miro.com/</a><br /> <b>CodeRabbit:</b> AI-powered code reviews.<br /> Go to <a href="https://lexfridman.com/s/coderabbit-ep488-sc">https://coderabbit.ai/lex</a><br /> <b>Chevron:</b> Reliable energy for data centers.<br /> Go to <a href="https://lexfridman.com/s/chevron-ep488-sc">https://chevron.com/power</a><br /> <b>Shopify:</b> Sell stuff online.<br /> Go to <a href="https://lexfridman.com/s/shopify-ep488-sc">https://shopify.com/lex</a><br /> <b>LMNT:</b> Zero-sugar electrolyte drink mix.<br /> Go to <a href="https://lexfridman.com/s/lmnt-ep488-sc">https://drinkLMNT.com/lex</a><br /> <b>MasterClass:</b> Online classes from world-class experts.<br /> Go to <a href="https://lexfridman.com/s/masterclass-ep488-sc">https://masterclass.com/lexpod</a></p> <p><b>OUTLINE:</b><br /> (00:00) &#8211; Introduction<br /> (01:58) &#8211; Sponsors, Comments, and Reflections<br /> (15:40) &#8211; Infinity &#38; paradoxes<br /> (1:02:50) &#8211; Russell&#8217;s paradox<br /> (1:15:57) &#8211; Gödel&#8217;s incompleteness theorems<br /> (1:33:28) &#8211; Truth vs proof<br /> (1:44:52) &#8211; The Halting Problem<br /> (2:00:45) &#8211; Does infinity exist?<br /> (2:18:19) &#8211; MathOverflow<br /> (2:22:12) &#8211; The Continuum Hypothesis<br /> (2:31:58) &#8211; Hardest problems in mathematics<br /> (2:41:25) &#8211; Mathematical multiverse<br /> (3:00:18) &#8211; Surreal numbers<br /> (3:10:55) &#8211; Conway&#8217;s Game of Life<br /> (3:13:11) &#8211; Computability theory<br /> (3:23:04) &#8211; P vs NP<br /> (3:26:21) &#8211; Greatest mathematicians in history<br /> (3:40:05) &#8211; Infinite chess<br /> (3:58:24) &#8211; Most beautiful idea in mathematics</p>