The Man Who Almost Broke Math (And Himself...)

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A huge thank you to Dr Asaf Karagila, Prof. Alex Kontorovich, Prof. Joel David Hamkins, Prof. Andrew Marks, Prof. Gabriel Goldberg and Prof. Elliot Glazer for their invaluable expertise and contributions to this video.

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0:00 What comes after one?
2:42 Some infinities are bigger than others
6:17 The Well Ordering Principle
10:32 Zermelo And The Axiom Of Choice
17:22 Why is the axiom of choice controversial?
23:16 The Banach–Tarski Paradox
27:53 Obviously True, Obviously False
29:58 Your Proof Your Choice

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References:

Up and Atom -    • Cantor's Infinity Paradox | Set Theory  
Minutephysics -    • How to Count Infinity  
PBS Infinite Series -    • How the Axiom of Choice Gives Sizeles...  
Vsauce -    • The Banach–Tarski Paradox  
Ernst Zermelo via Wikipedia - https://ve42.co/zermeloBio
Axiom of choice via Wikipedia - https://ve42.co/choiceAxiom
Georg Cantor via Wikipedia - https://ve42.co/cantorMath
Gregory H. Moore (2013). Consequences of the Axiom of Choice. Dover Publications - https://ve42.co/choiceBook
Georg Cantor (1874). On a property of the class of all real algebraic numbers. Journal für die Reine und Angewandte Mathematik - https://ve42.co/MeyerCantor1874
Heinz-Dieter Ebbinghaus (Dec 2012). Zermelo and the Heidelberg Congress 1904. Historia Mathematica - https://ve42.co/SciDirect1904
Herbert B. Enderton (1977). Elements of Set Theory. - https://ve42.co/SciDirectGCH

Additional References - https://ve42.co/AoCAdRefs

Images & Video:

Foundations of a general theory of sets by Georg Cantor via ViaLibri - https://ve42.co/grundlagen
Alfred Tarski by George Bergman via Wikimedia Commons - https://ve42.co/tarski
Alfred Tarski Offprint Group by Alfred Tarski via Bonhams - https://ve42.co/tarskipaper
La mission strasbourgeoise de Maurice Fréchet by Laurent Mazliak via Images des mathematiques - https://ve42.co/frechet
Kurt Gödel by Alfred Eisenstaedt via New Yorker - https://ve42.co/godel
Leopold Kronecker by Granger via Fine Art America - https://ve42.co/kronecker
Lashi Bandara (2006). Zermelo-Frankel Set Theory and Well Orderings. ResearchGate - https://ve42.co/zermelofrankel
Heidelberg, Germany 1936 by Wagner & Debes via Ward Maps - https://ve42.co/heidelberg
Pythagoras by J. Augustus Knapp via the marginalian - https://ve42.co/pythag
Paul Cohen by C. J. Mozzochi via C. J. Mozzochi - https://ve42.co/paulcohen
Instituto de Matemática Pura e Aplicada. Lecture 01: Introduction: a non-measurable set via Youtube -    • Lecture 01: Introduction: a non-measu...  
Simons Foundation. Fields Medal: James Maynard. Youtube    • Fields Medal: James Maynard  

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Directed by Kaela Albert
Written by Kaela Albert and Emily Zhang
Edited by Jack Saxon and Luke Molloy
Assistant Edited by James Stuart
Animated by Fabio Albertelli, Andrew Neet, Alex Zepharin, Mike Radjabov, Emma Wright and Ivy Tello
Illustrations by Jakub Misiek, Maria Gusakovich, Cainejan Esperanza, Tommy A. Steven and Emma Wright
Additional research by Emilia Gyles, Gabe Bean, Geeta Thakur and Vincent Cheng
Produced by Kaela Albert, Casper Mebius, Derek Muller, Emily Zhang, Zoe Heron, Rob Beasley Spence, and Tori Brittain
Additional Editing by Luke Molloy and James Stuart

Thumbnail contributions by Ben Powell, Peter Sheppard and Ren Hurley

Additional video/photos supplied by Getty Images and Storyblocks
Music from Epidemic Sound