Lex Fridman Podcast #488 – Infinity, Paradoxes that Broke Mathematics, Gödel Incompleteness & the Multiverse – Joel David Hamkins
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Dec 31, 2025 
Joel David Hamkins, a prominent mathematician and philosopher, explores the complexities of set theory and the nature of infinity. He delves into Cantor's revolutionary work on different sizes of infinity, using captivating examples like Hilbert's Hotel. Discussing Gödel's incompleteness theorems, Joel clarifies the profound distinctions between truth and provability in mathematics. He also invites listeners into the fascinating realm of surreal numbers and the enigmas of the multiverse, all while emphasizing collaborative problem-solving in mathematical creativity.
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Some Infinities Are Bigger
- Cantor showed infinities have sizes and some infinities are strictly larger than others by using one-to-one correspondences.
- This breaks Euclid’s whole-is-greater-than-part intuition and reshaped modern set theory.
Galileo’s Paradox Explained
- Galileo noticed paradoxes comparing infinite sets, like squares vs. naturals, revealing tension between one-to-one correspondence and 'whole greater than part.'
- Cantor resolved this by formalizing equinumerosity via bijections (Cantor-Hume principle).
Hilbert's Hotel Paradox
- Hilbert's Hotel illustrates countable infinity: even when full it can accommodate more guests by shifting occupants.
- This vivid example helps internalize how countable sets can remain the same size after adding elements.