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: https://www.youtube.com/watch?v=s86-Z-CbaHA&t=227s
Wikipedia article on the Axiom of Choice: https://en.wikipedia.org/wiki/Axiom_of_choice
Wikipedia article on ZFC: https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory
Music:
https://c418.bandcamp.com/album/dief
Smooth Fall by C418
Work Life Imbalance by C418
https://c418.bandcamp.com/album/life-changing-moments-seem-minor-in-pictures
In Berlin people act differently by C418
https://c418.bandcamp.com/album/seven-years-of-server-data
The first unfinished song for the Minecraft documentary by C418
https://patriciataxxon.bandcamp.com/album/aeroplane
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