The Axiom Behind Math's Weirdest Paradox

Join my Patreon community:   / abidebyreason   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:    • What is a Measure?   Banach-Tarski Paradox Explained:    • Math's Weirdest Paradox   Intro to Topology:    • The Hierarchy of Math Spaces   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} _____ Animations created using Manim: https://www.manim.community/ Music by Vincent Rubinetti Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/a... Stream the music on Spotify: https://open.spotify.com/playlist/3zN...