The Banach-Tarski Paradox

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  • Published on Aug 1, 2015
  • Q: "What's an anagram of Banach-Tarski?"
    A: "Banach-Tarski Banach-Tarski."
    twitter: twitter.com/tweetsauce
    Instagram: instagram.com/electricpants
    Kevin’s Field Day video: usclip.net/video/1zARMZ08ums/video.html
    Field Day: usclip.net/channel/UCRPktNf5vnBR1J4e7t1RUVg
    Deep dream animation by instagram.com/NaderMakki/
    If you like it, you'll love this video also by Nader: usclip.net/video/fJ9j_z2kXI0/video.html
    Chocolate illusion: mathandmultimedia.com/2014/07/22/explanation-infinite-chocolate-bars/
    Chocolate illusion video: usclip.net/video/dmBsPgPu0Wc/video.html
    related Numberphile videos:
    sizes of infinity (includes diagonal argument): usclip.net/video/elvOZm0d4H0/video.html
    infinity paradoxes: usclip.net/video/dDl7g_2x74Q/video.html
    Vi Hart on types of infinity: usclip.net/video/23I5GS4JiDg/video.html
    Countable & uncountable definitions:
    mathinsight.org/definition/uncountable
    en.wikipedia.org/wiki/Countable_set
    en.wikipedia.org/wiki/Uncountable_set
    Banach-Tarski on wikipedia: en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox
    Banach-Tarski proofs:
    math.uchicago.edu/~may/REU2014/REUPapers/Robinson.pdf
    www.math.hmc.edu/~su/papers.dir/banachtarski.pdf
    people.math.umass.edu/~weston/oldpapers/banach.pdf
    Banach-Tarski explinations online:
    www.irregularwebcomic.net/2339.html
    www.kuro5hin.org/comments/2003/5/23/134430/275?pid=5#10
    skepticsplay.blogspot.co.uk/2010/05/doubling-sphere.html
    austinrochford.com/posts/2014-05-14-banach-tarski-paradox.html
    www.math.cornell.edu/~mec/Summer2009/Whieldon/Math_Explorers_Club%3A__Lesson_Links/Entries/2009/7/28_Lesson_6%3A__Whats_an_Anagram_of_Banach-Tarski.html
    rachellevanger.com/index_files/BT%20Animated%20Presentation%20Web.pdf
    quibb.blogspot.co.uk/2013_03_01_archive.html
    blog.computationalcomplexity.org/2011/04/what-did-banachs-wife-think-of-banach.html
    geopolicraticus.wordpress.com/tag/banach-tarski-paradox/
    dgleahy.com/p47.html
    www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml
    Cayley graph animated gif: twitter.com/GIFsofWikipedia/status/624202342259240960
    Hilbert’s hotel on wikipedia: en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
    types of infinity: www.xamuel.com/levels-of-infinity/
    set theory and quantum physics: link.springer.com/article/10.1007/BF02213427#page-1
    LHC gif: cms.web.cern.ch/news/lhc-data-be-made-public-open-access-initiative
    Zermelo-Fraenkel axioms of mathematics: mathworld.wolfram.com/Zermelo-FraenkelAxioms.html
    Is math invented or discovered?
    phys.org/news/2013-09-mathematics-effective-world.html
    www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/
    more deep dream images: www.reddit.com/r/deepdream/
    BOOKS:
    The Pea and the Sun: www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279
    The Outer Limits of Reason: www.amazon.com/Outer-Limits-Reason-Science-Mathematics/dp/0262019353
    Why Beliefs Matter: www.amazon.com/Why-Beliefs-Matter-Reflections-Science/dp/0198704992
    Things to Make and do in the Fourth Dimension: www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374275653
    Music by usclip.net/user/JakeChudnow and
    www.audionetwork.com
  • Science & TechnologyScience & Technology

Comments • 73 556

  • Bing Boi
    Bing Boi 23 hours ago

    This is like particles 🤯

  • It’s PotatoMan58
    It’s PotatoMan58 23 hours ago

    All that work for an extra dollar

  • Rui Leite
    Rui Leite 23 hours ago

    Rip Vsauce free to see

  • xZERO
    xZERO Day ago

    Hmmmm I thought more matter being created from subatomic particles was due to bending in space-time. Then again, I’m not a scientist and I certainly wouldn’t have the answer

  • alkiter1
    alkiter1 Day ago

    Infinity - Infinity = 0

  • Wolfgang Breitenseher

    Infinity. Lol. Concepts.

  • noobs dudes
    noobs dudes Day ago

    You can’t drink lava *OR CAN U?*

  • AdiMehe
    AdiMehe Day ago

    @ 13:49 just imagine a combination of URR

  • Mayank Tiwari
    Mayank Tiwari Day ago

    What the fuck was this ???

  • ko harrison
    ko harrison Day ago

    LUR and U is the same point, isn't it? 13:22

  • The Trust
    The Trust Day ago +1

    9:15 No hole will be left unfilled ( ͡° ͜ʖ ͡°)

  • die Fabe
    die Fabe Day ago

    11:31 i almost came xD

  • QbRt
    QbRt Day ago

    I'm seeing a lot of holes in this theory. For example, if there is an infinite amount of starting points on the sphere, then that SHOULD mean there are an infinite number of accompanying poles, even if some repeat.
    Also, if you rotate the lefts to the right but left then right is a back track, you should only subtract the right. You didn't add but you subtracted and it goes both ways.
    Subtracting both of them takes away an original direction that you started with and you are left with something completely different.

    To me, it didn't ADD up. It SUBTRACTED down

    • QbRt
      QbRt Day ago

      If anybody can elaborate, debate, or whatever, please do.

  • Aydin Warner
    Aydin Warner Day ago

    JUST COLOR THE CIRCLE

  • Tzar Nicolas The II

    1:38 That's a felony.

  • OsvaldoKS100
    OsvaldoKS100 Day ago

    anyone else see the dogs or animals in the outro?

  • Philosophy Of Élivágar

    Hilbert's Hotel would have been useful during the Syrian immigration crisis

  • Andrew Jatnieks
    Andrew Jatnieks Day ago

    When he attempts to plot the dots on the circle ( assuming that all directional movements are of the same length), is it inaccurate to say that the combination of right, up, left, is the same as just moving up from the center.... he seems to realize this just as he is about to draw the dot.... and then corrects this by simply drawing the dot a little below the other one....????

  • I Am
    I Am Day ago

    nooooooo not the do;;ar

  • BoomBoom
    BoomBoom Day ago

    Is nobody upset that he just tore up a dollar bill? I could’ve have that... D:

  • FireyFly RBLX
    FireyFly RBLX Day ago

    dude that’s deeeeeeeeeeeeep

  • Cam Ward
    Cam Ward Day ago

    This is all bullshit tbh

  • theHardInGame yt

    13:53
    We'll just ignore that he used Red marker for RDR

  • impendio
    impendio Day ago +1

    It's been years and I still think this is the single most genius work of digital media every presented. It literally changed my life forever.

  • Dhananjay Sawant

    What if you subtracted an infinite number from infinty?

  • Dhananjay Sawant

    But, it would take infinitely long.

  • chris boy
    chris boy Day ago

    How life lives is simple but what life if made of is compilated

  • Dementia
    Dementia Day ago +1

    Excuse me what the fuck

  • Gandjalf The Green

    This is why I don't believe in infinity. Infinity is just a mathematical idea. Its never been observed, it never will be. It only creates problems and paradoxes, but doesn't show us anything. There's no way the universe is infinite either, at some point you would surpass the last atom. And saying that the empty space following is infinite is the same as saying that a shadow can move in the speed of light. It can't, because its not materialistic. Its the opposite of existing, and will therefore be impossible to be measured.
    Infinity is just a concept constructed by humans, just like math, and does not reflect the actual world.

  • Walter Ostrowski

    I wish the dollar thing was possible

  • Crazy Time
    Crazy Time Day ago

    who also think they don't have enough brain to watch this

  • Vir Khanna
    Vir Khanna Day ago

    I'm too stupid for this

  • azder bogdan
    azder bogdan Day ago

    the one with the dollar is a scam

  • its just hard
    its just hard Day ago

    Ok.. so can i get the chocolate that i ate early back?

  • Lodevik Trijana
    Lodevik Trijana Day ago

    Why do cut money?

  • Pow
    Pow Day ago

    Hey Vsauce..


    Make a 20 minut vidio of how bad my sspeling is..

  • AL STANWOOD
    AL STANWOOD Day ago

    POLE HOLES.
    HOLLOW EARTH CONFIRMED

  • Carson Grenade
    Carson Grenade Day ago

    2:27 bruv this is trippy

  • Carson Grenade
    Carson Grenade Day ago

    16:30 I can't be the only one that sees a big puple donut. I think thats because I'm hungry.

  • E E T Z J O S H
    E E T Z J O S H Day ago +1

    I don’t know why but the concept of infinity makes me depressed.

  • JerBearRides
    JerBearRides Day ago

    Dangerously close to an arrow swastika on that circle there michael.

  • laika noosh
    laika noosh Day ago

    The minute you catergorize infinity is when everything gets fucked which is why the sphere part is so confusing. Technically you can just fill the sphere of Right points because its infinite. Also the fact the distance of the turns and the size of the dots isn't accounted for makes the idea of "filling" the sphere impossible.

  • Mask Guy
    Mask Guy Day ago

    Hollllllyyyyy shit!

  • Beep Beep I'm A Sheep

    song at 0:00

  • Why Me
    Why Me Day ago

    I'm too high for this shit

  • Kuyaki
    Kuyaki Day ago

    Then...infinity makes everything .... insignificant?

  • SC4RCH
    SC4RCH Day ago

    my brain always fucked when i see him explaining paradox / other vid, like rly...

  • y - Animator
    y - Animator Day ago

    - -= +

  • Unnamed
    Unnamed Day ago

    Wtf

  • Christopher Crum

    How many of you actually tried the trick with a dollar bill? 3/5 of a whole cannot have the same mass as 5/5. It doesn't work. Just the pieces that he had shown required three, one dollar bills to create. They all had different serial numbers. None of the pieces from the first bill that he tore up were used to create the other two Spending $3 to make two. I wouldn't want you to manage any of my money. How did he come up with this. Now I'm supposed to believe anything else he says? A lot of things that seem to work in Theory don't work in practice. Ha ha.

  • Dominique J
    Dominique J Day ago

    Don't LUR and U end up at the same point if the distances between them are the same and they start from the same point?...

  • BigTark69
    BigTark69 Day ago

    Well I was about to fall asleep.

  • Caleb Powell
    Caleb Powell Day ago

    Mind literally blown

  • Joseph Menniti
    Joseph Menniti Day ago

    Thanks I just lost a fucking dollar

  • Midnght Eclipse
    Midnght Eclipse Day ago

    I really hope you didn't actually tear up a dollar, because that's incredibly illegal.

  • krokart
    krokart Day ago

    Size only exists through measurement. Measurement , only exists through the comparison of abstract or concrete objects, and the accord ( the act of agreement), between one or more observers. Since infinity is not observable, it cannot exist. It is an erroneous idea, which attempts to be both the largest and smallest circles in a Venn diagram , at the same time. The only way that you can prove that infinity exists , would be something like a Russian nesting doll , that eats its smallest self , while simultaneously growing large enough to eat its smallest self again and again and again forever while never gaining or losing any weight or physical matter. SO infinity must be a fluctuation, or frequency. Im guessing that humans are infinity. life is infinity. Both "Donnie Darko" and "Dinner with Andre" are worth some your infinity to watch. The "princess bride" too.... "As you Wishhhhhhhhhhhhhhhhhh. "
    I like tacos.

  • Garrett Bice
    Garrett Bice Day ago

    Umm how did you even think of this. In all of my life I’ve never thought about a fraction of this

  • Emma
    Emma Day ago

    screams while he rips a dollar bill

  • MrPaceTv
    MrPaceTv Day ago

    I thought , infinity is a gauntlet ...

  • I win 25 8 Boi
    I win 25 8 Boi Day ago

    I think that's how cells make more of them selves

  • Trumpwall666
    Trumpwall666 Day ago

    also theoretically it halpens all the time, it’s just a nine month process.

  • Spencer_m1 Matijak

    Da faq

  • Trumpwall666
    Trumpwall666 Day ago

    poor people at the hotel, they have to move to a different room every time someone enters, and leaves

  • # n0mE Squad
    # n0mE Squad Day ago

    1:49 my reaction, I must do this and become rich. Got dollar did so got a ripped dollar worth nothing such 😔

  • Iamalex.j Potatoes

    Science is a helluva drug

  • Neil Rosenbaum
    Neil Rosenbaum Day ago

    I used the Banach-Tarski Paradox to double my salary. I am selling my book that will tell you how to do this yourself. It normally costs $79.98 but I will use the Banach-Tarski Paradox so you can get my book for the discounted price of $39.99.

  • Pinkpatty 76
    Pinkpatty 76 Day ago

    18:58 made me question my life’s own existence

  • Simonplayer 26
    Simonplayer 26 Day ago

    I’m i the only one that is confused?

  • Marc Moncada
    Marc Moncada 2 days ago

    The reason that this theorem remains a paradox and not a fact is because you cannot extract the abstract (0th dimension) from the practical (3rd dimension).

  • Dersu42
    Dersu42 2 days ago +3

    Once you start conceptualizing something in the infinite things get weird. The illusion of the paradox is that you have a starting point to begin with. Once one end of your list of potential moves becomes unbound there is no longer an effective beginning anymore.

    All of the maneuvers and separations are simply more complex examples of not being able to subtract from infinity and have it be less than infinity.

    Countable infinity and uncountable are the same. Countable infinity gives the illusion of progressing through a block of numbers by incrementally adding one. But even after having counted in this manner for any imaginable amount of time, you are still no closer to finishing than when you started. Uncountable infinity simply shows this more plainly as you never get to progress beyond 0.00000000...

    There is no paradox other than trying to place limits on the limitless. Once you bring infinity into the picture all bounds and parameters are no longer applicable.

  • FlatusOhlfahrt
    FlatusOhlfahrt 2 days ago

    great vid!!

  • Billy Wheeler
    Billy Wheeler 2 days ago

    So this is how Jesus fed the multitude with only 5 pieces of bread and 2 fish. Now how to figure out how to use this theory for my finances. lol

  • Goldenempire
    Goldenempire 2 days ago

    i -dont- hate myself

  • Marcus Henry
    Marcus Henry 2 days ago

    Imma be rich

  • Goldenempire
    Goldenempire 2 days ago

    *wow*

  • Ayaan Chowdhury
    Ayaan Chowdhury 2 days ago

    These videos makes me feel smarter lmao

  • alfie Barton
    alfie Barton 2 days ago

    Wow

  • DIEHARD CH1EF
    DIEHARD CH1EF 2 days ago

    Does infinity • 0= 0 or infinity

  • Duncan Alba
    Duncan Alba 2 days ago

    HWTOBASIC

  • Riley Weaver
    Riley Weaver 2 days ago +1

    Micheals favorite term, 'or can you?'

  • SAMUEL Karlsson
    SAMUEL Karlsson 2 days ago

    If I’d get a cent for every time you mind fuck me id be a millionaire

  • MrElf 05
    MrElf 05 2 days ago

    These videos make me think about everything that cant be explained or known

  • damnryn
    damnryn 2 days ago

    *No hole will be left unfilled.*

  • Isabella Cobb
    Isabella Cobb 2 days ago +1

    What

  • Dyl59909
    Dyl59909 2 days ago +1

    Are we just gonna ignore how he drew that 8??

  • jolie dubreze
    jolie dubreze 2 days ago

    Micheal: *or can you?*
    Mr: uh oh.

  • Infamous weeb
    Infamous weeb 2 days ago +2

    lol is lol backwards

    • YungNite
      YungNite Day ago

      Infamous weeb wow is wow backwards wow

  • Nuggetz
    Nuggetz 2 days ago

    Now i see infinite from a whole new perspective.

  • Sean Beaudoin
    Sean Beaudoin 2 days ago

    V sauce is a liar when he ripped the dollar, the letter large letter left of George in the circle is different, the first one is k, the second is f

  • Werefox alpha
    Werefox alpha 2 days ago

    I can't believe they turned Michael's intro into a meme. His videos are AMAZING !

  • Avantika Rao
    Avantika Rao 2 days ago

    when he ripped the money i winced

  • InnocentOnPaper
    InnocentOnPaper 2 days ago

    Opium is one hell of a drug.

  • hobi’s sunshine
    hobi’s sunshine 2 days ago

    Michael, MY BRAIN

  • scrappythenell
    scrappythenell 2 days ago

    I actually disagree with this video. A line has zero mass and no width or depth, only length. So no matter how many lines you pull out of the initial ball into a new "ball", the second "ball" will always be either a single invisible line if each is set side by side, or and invisible, weightless mess of lines. 1(L) x 0(W) x 0(D) = 0. Zero times any number is always zero. So the second "ball" will never have any mass at all. If you pull out an infinite number of planes, you would theoretically be able to see it (assuming a zero mass plane was capable of somehow reflecting light), but it would still have zero mass. 1(L) x 1(W) x 0(D) = 0. The only way to create a second ball with mass would be to take a sliver of the original ball that has Length, Width, AND Depth. Once you do that however, every non-zero piece you take reduces the mass of the original ball. You would simply end up with two balls that are each half the weight of the original.

  • Rusty Roberts
    Rusty Roberts 2 days ago

    I feel sick and my brain hurts.

  • Lewis Goodwin
    Lewis Goodwin 2 days ago +1

    I'm going to cry man this video hurt my head

  • GameGod account
    GameGod account 2 days ago

    Ofc it can't be real. The number of atoms on that sphere are finite!!!

  • Holy Cucumber
    Holy Cucumber 2 days ago

    Yeaaaaaaaaaaaaa wtf

  • InspiredKangroo
    InspiredKangroo 2 days ago

    1:39 TEACH ME

  • ANUP SHARMA
    ANUP SHARMA 2 days ago

    Electrons