How do you make infinite choices? 👉 To try everything Brilliant has to offer for free for a full 30 days, visit http://brilliant.org/veritasium. You’ll also get 20% off an annual premium subscription. Try Snatoms! A molecular modelling kit I invented where the atoms snap together. https://ve42.co/SnatomsV Sign up to the Veritasium newsletter for weekly science updates - https://ve42.co/Newsletter ▀▀▀ 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. Head over and sign up to our Patreon for some exclusive behind the scenes footage, showing how the animations and illustrations for this video were made - https://www.patreon.com/posts/patreon-behind-126443880?utm_medium=clipboard_copy&utm_source=copyLink&utm_campaign=postshare_creator&utm_content=join_link ▀▀▀ 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 ▀▀▀ References: Up and Atom - https://www.youtube.com/watch?v=X56zst79Xjg Minutephysics - https://www.youtube.com/watch?v=A-QoutHCu4o PBS Infinite Series - https://www.youtube.com/watch?v=hcRZadc5KpI&t=125s Vsauce - https://www.youtube.com/watch?v=s86-Z-CbaHA&t=474s 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 - https://www.youtube.com/watch?v=llnNaRzuvd4&t=834s Simons Foundation. Fields Medal: James Maynard. Youtube https://www.youtube.com/watch?v=un-z8kgOrV0&t=8s ▀▀▀ Special thanks to our Patreon supporters: Adam Foreman, Albert Wenger, Alex Porter, Alexander Tamas, Anton Ragin, Autodidactic Studios, Balkrishna Heroor, Bertrand Serlet, Blake Byers, Bruce, Dave Kircher, David Johnston, David Tseng, Evgeny Skvortsov, Garrett Mueller, Gnare, gpoly, Greg Scopel, HydrochloRick, Jon Jamison, Juan Benet, Keith England, KeyWestr, Kyi, Lee Redden, Marinus Kuivenhoven, Matthias Wrobel, Meekay, meg noah, Michael Krugman, Orlando Bassotto, Paul Peijzel, Richard Sundvall, Sam Lutfi, Tj Steyn, TTST, Ubiquity Ventures, wolfee ▀▀▀ 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