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The Axiom of Choice

Mathematics is based on a foundation of axioms, or assumptions. One of the most important and widely-used set of axioms is called Zermelo-Fraenkel set theory with the Axiom of Choice, or ZFC. These axioms define what a set is, which are fundamental objects in mathematics. And the Axiom of Choice is arguably one of the most important and interesting axioms of ZFC. But what does it really say? And how is it used? This video dives deep into the formal definition of the Axiom of Choice, as well as its important equivalences which have their own fascinating applications in various branches of mathematics. Furthermore, we look into the controversy behind AC, and why it has garnered much discussion throughout its mathematical history. 0:00 Introduction 1:28 Set Theory and ZFC 9:22 The Axiom of Choice 16:55 Zorn's Lemma 23:39 The Well-ordering Theorem 27:49 Other Equivalences of AC 29:23 Controversy & Final Thoughts Additional Resources: The Banach-Tarski Paradox by Vsauce:    • The Banach–Tarski Paradox   Wikipedia article on the Axiom of Choice: https://en.wikipedia.org/wiki/Axiom_o... Wikipedia article on ZFC: https://en.wikipedia.org/wiki/Zermelo... Music: https://c418.bandcamp.com/album/dief Smooth Fall by C418 Work Life Imbalance by C418 https://c418.bandcamp.com/album/life-... In Berlin people act differently by C418 https://c418.bandcamp.com/album/seven... The first unfinished song for the Minecraft documentary by C418 https://patriciataxxon.bandcamp.com/a... Cribwhistling by Patricia Taxxon Starboard by Patricia Taxxon Animations were made by Manim, an open-source python-based animation program by 3Blue1Brown. https://github.com/3b1b/manim