The Axiom Behind Math's Weirdest Paradox

Video Description

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. _____ Join my Patreon community: https://www.patreon.com/abidebyreason Support the channel with a one-time donation: https://ko-fi.com/abidebyreason _____ Related Videos: Visualizing the Rationals: https://youtu.be/wE1iJkXs1Xg 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 Why the Cantor Set is Perfect: https://youtu.be/c4oKdtUtJOk 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/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/playlist/3zNK20qC96mVSww60lVi1k