2025-04-02

[public] 3.67M views, 220K likes, dislikes audio only

Veritasium

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

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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.

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

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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 - https://www.youtube.com/watch?v=X56zst79Xjg

Minutephysics - /youtube/video/A-QoutHCu4o

PBS Infinite Series - /youtube/video/hcRZadc5KpI

Vsauce - /youtube/video/s86-Z-CbaHA

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

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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

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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