The Most Controversial Idea In Math

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 20