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The Banach–Tarski Paradox

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  • The Banach–Tarski Paradox

    24:14

    Want more brain food? Want your house to become a museum? Now you're thinking:

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

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  • Infinity shapeshifter vs. Banach-Tarski paradox

    14:58

    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.

    Enjoy :)

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  • This Math Theorem Proves that 1=1+1 | The Banach-Tarskis Paradox

    33:27

    Mathematicians are in nearly universal agreement that the strangest paradox in math is the Banach-Tarski paradox, in which you can split one ball into a finite number of pieces , then rearrange the pieces to get two balls of the same size. Interestingly, only a minority of mathematicians has ever seen the proof of this theorem. Rarely discussed, also, are the implications of this theorem, including that you can take a pea-sized ball, split it up into a finite number of pieces, and reassemble them into a ball the size of the Sun!

    Presented by David Kung
    Learn more about math paradoxes at

    0:00 What Is the Banach-Tarski Paradox?
    0:30 What Are the Implications of the Banach-Tarski Paradox?
    2:00 A Word Version of the Banach-Tarski Paradox
    04:00 Simplify by Removing Copies of Extra Letters
    04:45 What Is Concatenation?
    05:19 Every Word Has An Anti-word
    05:50 Each Word Is a Complicated Series of Rotations
    06:30 Visualize the A's Like a Triangle
    06:50 Visualize the B's As Pairs
    07:50 What Is a Cayley Diagram?
    12:15 How to Visualize the Colors in the Diagram as Circles
    14:30 Similarities with Infinity and Counting Numbers
    15:30 Banach-Tarski Goal as Pertains to Rotations
    18:57 How the Banach-Tarski Paradox Relates to Magic Tricks: Quick Conundrum
    20:39 Felix Hausdorff Proved the Decomposition of G
    21:40 A Sequence of Rotations Equals Another Single Rotation
    23:34 Creating Equivalence Classes
    24:30 Axiom of Choice
    29:00 Two Additional Complications for the Banach-Tarski Paradox
    30:35 Why the Set S is Unmeasurable
    32:00 Any Bounded Set With Non-Empty Interior Can Make Any Other Such Set

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    #banachtarskis #paradox #mathmagic

  • The Infinite Money Paradox

    10:32

    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.

    *** SOURCES ***

    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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  • The Banach-Tarski Paradox - Tales of the Forest

    45:58

    Hello everyone reading this description! Most of you that are already familiar with my channel know that I mainly post videos pertaining to genealogy (it's literally in my name), but I am also very interested in the subject of music.

    I play the keyboards, drums, and a little bass in real life, and have been in several bands. That being said, I also record (and mix) my own music, and have done so for about 3 years.

    This is my first (and hopefully not last!) attempt at a studio album. Also I chose the name The Banach-Tarski Paradox.

    Don't forget to subscribe! :

    SONGS:
    0:00 The Yellow Sun, Pt. 1: I. The Yellow Sun / II. Blue Moon Tango
    7:50 Ballade, Pt. 1 (Dance of the Purple Fairies)
    9:13 The Kuiper Belt: I. Journey / II. Encounter With a Black Hole / III. Galactic Jam
    19:01 Point of No Return
    26:44 The Yellow Sun, Pt. 2
    30:08 Tales of the Forest: I. Shelter in Place / II. Witchmother's Brew / III. Prince Ultron of Zogonia / IV. Tales From The Forest / V. Reprise
    43:31 Ballade, Pt. 2 (Battle of the Green Goblins)

    ALL MUSIC WAS MIXED AND RECORDED IN GARAGEBAND (I know I'm basic)

    #music #album #progrock #rock #spacerock #artist

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  • Infinite Chocolate Bar Trick

    2:16

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  • banach tarski paradox

    2:45

  • How the Axiom of Choice Gives Sizeless Sets | Infinite Series

    13:20

    Viewers like you help make PBS (Thank you ????) . Support your local PBS Member Station here:

    Does every set - or collection of numbers - have a size: a length or a width? In other words, is it possible for a set to be sizeless? This in an updated version of our September 8th video. We found an error in our previous video and corrected it within this version.

    Tweet at us! @pbsinfinite
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    Previous Episodes
    Your Brain as Math - Part 1


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    In this episode, we look at creating sizeless sets which we call size the Lebesgue measure - it formalizes the notion of length in one dimension, area in two dimensions and volume in three dimensions.

    Written and Hosted by Kelsey Houston-Edwards
    Produced by Rusty Ward
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  • The Infinite Hotel Paradox - Jeff Dekofsky

    6:00

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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.

  • The Banach Tarski paradox - is it nonsense? | Sociology and Pure Mathematics | N J Wildberger

    17:08

    One of the famous paradoxes of 20th century pure mathematics is the assertion that it is possible to subdivide a solid ball of radius 1 in three dimensional space into 5 disjoint pieces, take those five pieces and subject them to rigid motions, that is rotations and translations, to obtain the subdivision of a solid ball of radius 2. If it sounds crazy, no problem, we can't let reality intrude into the dreamings of us pure mathematicians. Even if our theories end up disconnected from reality and from common sense.

    Is it time that we stopped paying lip service to logical nonsense -- even if it is supported by all kinds of axiomatics - in this case the modern pure mathematician's favourite: the Axiom of Choice?

    Here is the blog from which the first part of the video is read: you can check out quite a few more math related discussions:

  • The Banach-Tarski Paradox

    3:01

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    (Science rules)
    Bill Nye the Science Guy
    (Inertia is a property of matter)
    Bill, Bill, Bill, Bill, Bill, Bill
    Bill Nye the Science Guy
    Bill, Bill, Bill,
    (T-minus seven seconds)
    Bill, Bill, Bill, Bill, Bill, Bill, Bill,
    Bill Nye, the Science Guy

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

    25

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

  • Doubling Sphere Paradox - Banach-Tarski Theorem

    1:19

    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.

  • Banach Tarski Paradox Crash Course

    20:35

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

  • The Banach-Tarski Paradox

    1:01

    Sezione video del Concorso Martemartiamo

  • The Banach-Tarski Paradox

    2:03

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  • Measure theory: Why measure theory - The Banach-Tarski Paradox

    8:38

    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.

  • 0. Banach Tarski Banach Tarski

    27:19

    A teaser introduction to measure theory, motivated by the infamous Banach-Tarski paradox.

  • The Banach-Tarski Paradox

    13



    The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.

    This Demonstration shows a constructive version of the Banach?Tarski paradox, discovered by Jan Mycielski and Stan Wagon. The three colors define congruent sets in the hyperbolic plane ?, and from the initial viewpoint the sets appear congruent to our E...

    Contributed by: Stan Wagon

    Audio created with WolframTones:

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  • The Banach-Tarski Paradox

    56

    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.

  • A taste of abstract mathematics - Banach-Tarski Paradox

    2:24

    In this video, we explore a famous paradox in mathematics that sounds absolutely absurd - the Banach-Tarski Paradox. Simply put, the paradox stays that a sphere can be chopped up into finitely many pieces and reassembled into two copies of the original sphere!

    +++
    Real analysis, measure theory, Lebesgue measure, Banach-Tarski paradox, Banach, Tarski, set theory, abstract mathematics

  • A Friendly Introduction to Transfinite Arithmetic and the Banach-Tarski Paradox | Part 1

    13:31

    Link to my paper on it if you want to read it instead of hearing an awkward physics/math kid talk about it

  • What is The Banach Tarski Paradox

    2:37

    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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  • Banach-Tarski Paradox

    1:00

    I reconstructed a model of the Banach-Tarski's Chocolate Bar paradox and I show how it is also not what it seems.
    ---------------------------------------------------------------
    Instagram: @jayson_hansen

  • The Banach-Tarski Paradox

    3:39

    Today we take a look at the fascinating Banach-Tarski Paradox, and the infinite chocolate paradox.

  • Making Infinite Copies – The Banach-Tarski paradox – A Spiritual Implication

    14:00

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  • Mathematical Marvels: the Banach-Tarski Theorem

    2:28

    In this video I look at some of the weird and wonderful things that happen because of the Axiom of Choice, including the famous Banach-Tarski Theorem. Once again, many thanks to Theo T-Rex Reynolds for doing the filming.

  • Banach Tarski Video

    3:29

    This is just a rough draft, pls not hate...

  • mod01lec02 - Infinite Sets and the Banach-Tarski Paradox - Part 1

    15:48

    Infinite Sets, Equinumerosity, Cantor's theorem, Cantor-Schroeder-Bernstein theorem

  • The Banach-Tarski Paradox

    13



    The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.

    This Demonstration shows a constructive version of the Banach-Tarski paradox, discovered by Jan Mycielski and Stan Wagon. The three colors define congruent sets in the hyperbolic plane ?, and from the initial viewpoint the sets appear congruent to...

    Contributed by: Stan Wagon

    Audio created with WolframTones:

  • A Friendly Introduction to Transfinite Arithmetic and the Banach-Tarski Paradox | Part 4

    4:59

    Link to my paper on it if you want to read it instead of hearing an awkward physics/math kid talk about it

  • A Friendly Introduction to Transfinite Arithmetic and the Banach-Tarski Paradox | Part 3

    15:02

    Link to my paper on it if you want to read it instead of hearing an awkward physics/math kid talk about it

  • The Banach–Tarski Paradox, but backwards

    24:14

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  • The Banach Tarski paradox

    15:07

  • The Banach-Tarski Theorem

    12:20

    We describe the Banach-Tarski Theorem as a way of providing our first hint of non-measurable sets; and a further reminder of the distinction between real numbers and physical reality.

  • A Friendly Introduction to Transfinite Arithmetic and the Banach-Tarski Paradox | Part 2

    14:35

    Link to my paper on it if you want to read it instead of hearing an awkward physics/math kid talk about it

  • The Banach-Tarski Paradox

    09



    The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily.

    This animation shows a constructive version of the Banach-Tarski paradox, discovered by Jan Mycielski and Stan Wagon. The three colors define congruent sets in the hyperbolic plane H, and from the initial viewpoint the sets appear ...

    Contributed by: Stan Wagon (Macalester College)

  • Tackling the Banach-Tarski Paradox

    3:01

    This video, created for the Breakthrough Junior challenge, explains the Banach-Tarski Paradox.
    #breakthroughjuniorchallenge

  • The breakthrough junior challenge THE BANACH TARSKI PARADOX

    3:01

    hello people, i present to you the explanation of the banakh tarski paradox hope you like it .

  • The Banach-Tarski Paradox

    12:09

  • The Banach-Tarski paradox | Andrzej Zuk | Лекториум

    33:31

    The Banach-Tarski paradox | Лектор: Andrzej Zuk | Организатор: Математическая лаборатория имени П.Л. Чебышева СПбГУ

    Смотрите это видео на Лекториуме:

    Подписывайтесь на канал:
    Следите за новостями:

  • mod01lec03 - Infinite Sets and the Banach-Tarski Paradox - Part 2

    20:51

    Problem of measuring subsets of R^n, Axiom of Choice and the Banach-Tarski Paradox, Violation of Finite-additivity

  • Banach- Tarski Paradox

    11:14

    ild

    ignore tags

  • BANACH-TARSKI PARADOX

    12:18

    it's the spheres Bois

  • Matematikrevyen 2011: Banach-Tarski

    3:43

    The famous Banach-Tarski paradox, now in real life. Watch out as oranges multiply to powers of two. And don't forget to look it up on wikipedia:

  • Banach-Tarski Paradox

    35

    In this video I will be showing you how the Banach-Tarski Paradox can actually work in real life!!!

  • Banach-Tarski

    1:18

    Algorithm for doing N jobs:

    def perform(tasks):
    if len(tasks) == 0:
    return True

    if len(tasks) == 1:
    do_work(tasks[0])
    return True

    end = len(tasks)
    mid = end / 2

    perform(tasks[0:mid])
    perform(tasks[mid:end])

    return True

  • The banach-tarski paradox #Race2space2021 #Race2space

    3:43

  • Banach-Tarski Paradox

    55

    Btw, other YouTubers also made videos on this topic, so you don't have to comment about it.

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