Deep in the foundations of mathematics lies a simple axiom that produces one of the strangest paradoxes in history. And a direct consequence of this axiom is that not only are there mathematical sets with zero volume but there are also sets for which it is impossible to assign a meaningful sense of volume.
Can all mathematical sets be assigned a meaningful volume? In this video, I will show you how this simple question plays a crucial role in the Banach-Tarski Paradox and use it to motivate the study of a fascinating subject known as Measure Theory.
Related Videos:
What is the Measure of the Rationals vs Irrationals? https://youtu.be/1qbjjgesp-c
Sigma Algebras and Measures: https://youtu.be/1BhSQiHTNbg
Banach-Tarski Paradox Explained: https://youtu.be/R--iM5KbDEg
Intro to Topology: https://youtu.be/B-Y3-XpAdMU
Intro to Group Theory: https://youtu.be/5qmsqwxrSLc
Typo:
02:54 Q should be {p/q | p,q is in Z and q is not 0}
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