The Axiom Behind Math's Weirdest Paradox

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:

Visualizing the Rationals:    • Ford Circles and Farey Sequences  

What is the Measure of the Rationals vs Irrationals?    • Measuring the Rationals  

Sigma Algebras and Measures:    • The Mathematician's Measure  

Banach-Tarski Paradox Explained:    • Math's Weirdest Paradox  

Intro to Topology:    • Topological Spaces Visually Explained  

Why the Cantor Set is Perfect:    • The Topology of the Cantor Set  

Typo:
02:54 Q should be {p/q | p,q is in Z and q is not 0}

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