The Most Controversial Rule In Math

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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.

▀▀▀ 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

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