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The Banach–Tarski Paradox


The Banach–Tarski Paradox

Q: What's an anagram of Banach-Tarski?
A: Banach-Tarski Banach-Tarski.


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The Banach-Tarski Paradox

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Infinity shapeshifter vs. Banach-Tarski paradox

Take on solid ball, cut it into a couple of pieces and rearrange those pieces back together into two solid balls of exactly the same size as the original ball. Impossible? Not in mathematics!
Recently Vsauce did a brilliant video on this so-called Banach-Tarski paradox:
In this prequel to the Vsauce video the Mathologer takes you on a whirlwind tour of mathematical infinities off the beaten track. At the end of it you'll be able to shapeshift any solid into any other solid. At the same time you'll be able to appreciate like a mathematician what's really amazing about the Banach-Tarski paradox.


Burkard Polster and Giuseppe Geracitano

Magic or what ? Banach-Tarski paradox

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The Banach-Tarski Paradox

This video is an example based on the theory THE BANACH TARSKI PARADOX

Which says that a new substance can be formed by the rearrangement of substances in a object without losing anything.

Animation of the strangest paradox in math - the Banach-Tarski Paradox

from Mindbending Math: Paradoxes & Puzzles, from The Great Courses

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Eric Langlay stayed up all night with me so that this video could come out today. MUCH LOVE

Grant from 3Blue1Brown helped me wrap my head around this topic:

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Doubling Sphere Paradox - Banach-Tarski Theorem

Ever wonder how you can make two spheres out of one?

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This video was made in association with The Math Centre at Humber College, by Zack Barnes and Cheryl Yang.

The Infinite Money Paradox

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Deciding whether to play a game is usually very easy… you crunch the numbers and if they work in your favor, you play. If they don’t, you shouldn’t. Mathematical case closed.

But what happens when the math of a game tells you that you have access to infinite wealth and unlimited expected value and real life tells you not to play? Enter: The St. Petersburg Paradox.

The Bernoulli family first started corresponding about the paradox in the early 1700s with a series of letters examining the puzzling math behind the simple game. But it wasn’t until 1738 when Daniel Bernoulli realized that he could factor real life utility -- how much something actually means to you -- into the calculations.

The St. Petersburg Paradox opens up doors to how we think about what math really means to us, including modern research into Prospect Theory and everyday issues like whether we decide to buy life insurance. And in the end, one thing we know for sure: we’re all much, much more than numbers.

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Original Bernoulli family correspondence:

Play the St. Petersburg Paradox game:

“Ending the Myth of the St. Petersburg Paradox,” by Vivian Robert William:

“St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described,” by Paul Samuelson:

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(PP 1.1) Measure theory: Why measure theory - The Banach-Tarski Paradox

A playlist of the Probability Primer series is available here:

You can skip the measure theory (Section 1) if you're not interested in the rigorous underpinnings. If you choose to do this, you should start with (PP 1.S) Measure theory: Summary at:

(0:00) Intro to Probability Primer series.
(1:20) Why do we need measure theory? We illustrate the need using the remarkable Banach-Tarski Paradox.

How To Count Past Infinity

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Sources and links to learn more below!

I’m very grateful to mathematician Hugh Woodin, Professor of Philosophy and Mathematics at Harvard, for taking the time on multiple occasions to discuss this topic with me and help me wrap my (finite) head around it.

I’m also grateful to David Eisenbud, the Director of the Mathematical Sciences Research Institute (MSRI) and professor of mathematics at the University of California, Berkeley, for his help and for connecting me with Hugh Woodin.

And of course, big thanks to Brady Haran who created the “mile of pi” seen in this video and connected me with all these mathematicians in the first place. His channel, Numberphile, is superb:

BOOKS related to these topics that I used:

“The Outer Limits of Reason” by Noson S. Yanofsky:
“Infinity and The Mind” by Rudy Rucker:
“Roads to Infinity” by John C. Stilwell:
“More Precisely: The Math You Need to Do Philosophy” by Eric Steinhart:
“Satan, Cantor and Infinity: Mind-Boggling Puzzles” by Raymond M. Smullyan:

classic book that helps introduce concept of axioms: “Introduction to the Foundations of Mathematics” by Raymond L. Wilder:

Hugh Woodin speaking about infinity at the World Science Festival:

Names of large (finite) numbers:


The biggest number:

Fovant badges:

Battalion Park:

A mile of pi [VIDEO]:

Wikipedia’s great visualization of ordinals out to omega^omega:

as seen on:

this is also a good page about ordinals:


and: and



Large Cardinal game based on 2048:

Other good resources:


The Banach-Tarski Paradox

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Banach Tarski Paradox Crash Course

This is my final presentation for my Math Seminar Class (MATH 410) at Chadron State College.

The banach-tarski paradox

Riemann's paradox: pi = infinity minus infinity

With the help of a very famous mathematician the Mathologer sets out to show how you can subtract infinity from infinity in a legit way to get exactly pi.

Thank you very much to Danil Dmitriev the official Mathologer translator for Russian for his subtitles.


Burkard Polster

Thank you very much
Zacháry Dorris for contributing English subtitles for this video, Rodrigo Naranjo for contributing Spanish subtitles and Étienne Leb for his French subtitles!

Infinite Chocolate Bar Trick

Please help me to reach 100,000 subscribers! Infinite Chocolate Bar Trick EXPLAINED:
How the Infinite Chocolate Bar Trick works? Endless Chocolate Bar. How to Explain the Infinite Chocolate Trick? Never ending Chocolate Bar. Chocolate Magic. Infinite Chocolate optical illusion with an extra piece of chocolate.

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The Infinite Hotel Paradox - Jeff Dekofsky

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The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with an infinite number of rooms. Easy to comprehend, right? Wrong. What if it's completely booked but one person wants to check in? What about 40? Or an infinitely full bus of people? Jeff Dekofsky solves these heady lodging issues using Hilbert's paradox.

Lesson by Jeff Dekofsky, animation by The Moving Company Animation Studio.

Infinite chocolate bar?? How often can I do this?

Infinite chocolate bar trick... BUT: How often can I do this!?!?

Watch the video to find out!

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What is The Banach Tarski Paradox

Hey guys! Today we’re talking about the Banach Tarski Paradox. It’s a paradox of uncountable infinity. Watch more to learn more!

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