The Banach-Tarski Paradox

  • Published on Aug 1, 2015
  • Q: "What's an anagram of Banach-Tarski?"
    A: "Banach-Tarski Banach-Tarski."
    Kevin’s Field Day video:
    Field Day:
    Deep dream animation by
    If you like it, you'll love this video also by Nader:
    Chocolate illusion:
    Chocolate illusion video:
    related Numberphile videos:
    sizes of infinity (includes diagonal argument):
    infinity paradoxes:
    Vi Hart on types of infinity:
    Countable & uncountable definitions:
    Banach-Tarski on wikipedia:
    Banach-Tarski proofs:
    Banach-Tarski explinations online:
    Cayley graph animated gif:
    Hilbert’s hotel on wikipedia:
    types of infinity:
    set theory and quantum physics:
    LHC gif:
    Zermelo-Fraenkel axioms of mathematics:
    Is math invented or discovered?
    more deep dream images:
    The Pea and the Sun:
    The Outer Limits of Reason:
    Why Beliefs Matter:
    Things to Make and do in the Fourth Dimension:
    Music by and
  • Science & TechnologyScience & Technology

Comments • 79 675

  • Chinmay
    Chinmay 43 minutes ago

    you misunderstood the infinity

  • Brandan McAdoo
    Brandan McAdoo Hour ago

    does that money thing work

  • a b
    a b 2 hours ago

    Why do I exist ?

  • r4ymind
    r4ymind 2 hours ago

    Can infinity fit in infinity? Like would a infinite number of... I don’t know... CARS, fit in the entire infinite universe?

  • Natalie Lemmer
    Natalie Lemmer 3 hours ago

    Did he just tear up an authentic dollar bill? I believe this is considered "destruction of government property!" Not cool, lost me at that point.

  • Note 7
    Note 7 3 hours ago

    What did that dolar do to deserve such torcher?

  • P R
    P R 4 hours ago +1

    1:15 I knew you were going to say that.

  • Javier Delgado
    Javier Delgado 5 hours ago

    Bruh the dictionary blew my fucking mind

  • Zac
    Zac 5 hours ago

    If you havent, get baked and LISTEN to vsuace, dont watch, listen. Best experience of my life

  • I’m petrified of Nipple chafing

    Wow I’m re watching this again and I’m so proud of myself because I get most of what your saying lol

  • datgamerian loquendo
    datgamerian loquendo 7 hours ago

    1:32 **Third World scream**

  • There is no God. Just think about it OK!!

    There is no God. Just think about it's brain died

  • There is no God. Just think about it OK!!

    My brain is having a heart attack 1:48

  • Tech and Games Online
    Tech and Games Online 9 hours ago


  • Lance Lote
    Lance Lote 10 hours ago’s your point?..

  • Sean O'Reilly
    Sean O'Reilly 10 hours ago

    dollar thing doesnt work tf

  • Bart de Vries
    Bart de Vries 10 hours ago

    Infinity is everything I hate about math.

  • Reverse Gamer
    Reverse Gamer 11 hours ago

    Me: You Vsauce destroyed my childhood.

    Vsauce: Or did I?
    Me:*struck by a unfilled hole*

  • Daemon Nice
    Daemon Nice 13 hours ago

    It is as Alfred North Whitehead wrote in his Introduction to Mathematics. Mathematics is an abstraction.

  • Enavoid
    Enavoid 15 hours ago


  • Namelezz
    Namelezz 16 hours ago


  • I'm An Alien
    I'm An Alien 17 hours ago

    *O R C A N Y O U?*

  • sampson lam
    sampson lam 18 hours ago

    what is the music's name start from 21:46 ?

  • bogdan dan
    bogdan dan 19 hours ago

    I am not sure mathematics does agree with this theory.
    The reason why the points defining a circle or sfere is infinit is that the volume or 2d dimensions of a point doesn't have a fixed value, so as it says at the begining of the video it is about the infinite sizes that a particle can have.
    If we asign any value, lets name it "y", which could be any of the infinite posibilities for a point we extract from a sfere and if we define the volume of the sfere at a certain density with X.
    if we substract y from x, we know y is a positive number. If Y would be smaller than zero, when we would extract it we would actualyy adding it to the sfere ( x-(-z) = x+z). So y>0.
    Acording to mathematics, if x>0 and y>o, then x-y=x is false because if x-y=x => y=0 and you cannot extract 0 from anything.
    0 = nothing.
    In conclusion, mathematicaly it is not possible.
    When you asign names to the points is similar with the cat in the box which can be dead or alive until you open the box. This only gives 2 posibilities for the cat to be dead or alive untill you open the box, but you do not have 2 cats in the box. When you open the box you get to the real world and you can state one of two theories:
    1 - "The cat is alive"
    2 - "The cat is dead"
    If you are not a doctor and you are not sure if the cat is dead or alive, just don't say anything about it.

  • Jihyos Fan
    Jihyos Fan 19 hours ago

    no sense of understanding at all

  • Darius Z
    Darius Z 21 hour ago +1

    so if i got a girlfriend and split(sexual intercourse) her with my buddy vsauce i will get another girlfriend? and if i split again with another person i will get another girlfriend too? that makes sense, now i know how 5 babys are born every secon, it's just so much pussy

    • Darius Z
      Darius Z 21 hour ago

      pussy, ass, dick, . . . .. they are real!! and INFINITE

  • Emanuel Gonzalez
    Emanuel Gonzalez 21 hour ago

    fuck me this shit got me confused time to relax with some porn

  • Makrillol
    Makrillol 23 hours ago

    9:48 Whenever I visit Canada

  • JUAN
    JUAN Day ago +1

    i’m watching this high right now

  • kgyu993
    kgyu993 Day ago

    Wouldn't LUR and U take you to the same place if each line is the same exact distance? like the L and R are cancelling eachother out still

  • Vivek Bhana
    Vivek Bhana Day ago +1

    All this just for infinite chocolate

  • Vivek Bhana
    Vivek Bhana Day ago +1

    VSause you’ve triggered me into a seizure

  • 247 Anglers
    247 Anglers Day ago

    Infinity - Infinity = 🧐

  • Owen McDonald
    Owen McDonald Day ago

    Wow my head is spinning rn

  • Mutum
    Mutum Day ago

    RUU, DRR , LDD, ULL ;)
    ha get it?

  • Zachary Augi
    Zachary Augi Day ago


  • BenBMB
    BenBMB Day ago +1

    Rip that dollar in the intro

  • LogicalGamer
    LogicalGamer Day ago

    1:31 dude I could have used that

  • Grace With an R
    Grace With an R Day ago

    This man got me thinking all sort of science conspiracy’s

  • 500 Subs With No Videos ?

    For a heartbreak😢

  • Nova X
    Nova X Day ago

    I can’t believe I started laughing at 4:25 I need a life

  • Soursoulers
    Soursoulers Day ago

    is it that hard to flick a 3cm piece of paper? 9:23

  • David Boynton
    David Boynton Day ago

    Can u be my science teacher

  • Logan Beavers
    Logan Beavers Day ago

    So I know I may be wrong on a number of levels, but here is what I've always thought about infinity;
    There are different infinite's, hence x+1 does not equal x (follow me here)
    So infinity plus one does not equal infinity because there is an infinite amount more of possibilities because of the infinitly small decimals with the added one, (.0 barred 1) This is true because infinity is a quantity not a function
    Say 1000=infinity
    infinity is less than infinity plus one because infinity being a quantity, like 1000
    This is how 2 is greater than one
    Before if there was only one infinity and it was equal to itself, then 2 and 1 would both have the same infinite amount of possibilities between them and 0. And that begs the question, how is two greater than one?
    If infinity is not a function, but a quantity, then 2 does not have the same amount of possibilities away from zero as 1 does.
    So that leaves me with another question, how is 2 greater than 1 if infinity does not equal infinity?
    Does it just mean two is the same infinite between zero and one, just plus itself?
    And what is one? Is it just the infinite possibilities between 0 and the first whole number, or is it a question so simple a child could answer.

  • Clickachu
    Clickachu Day ago +1

    In theory sphere or circle have infinite amount of particles, but in reality they're made of countable amount of atoms. And even if atoms themselves are made of infinite amount particles inside them, the ball will still be made of countable amount particles, so even if we had technology to move ever single atom, this theory/experiment wouldnt work/be possible to do. So maybe this is the answer, that even real life ball made of infinite amount of particles is made of countable amount of atoms. So we can't do such things because of atomic structure of world, BUT... we still don't know too good what's inside the atom, maybe inside it there is infinite amount of particles and there it will be possible. But maybe we can't measure or understand atom, becasue what's inside it is just in another "dimention", like electrones can act like waves, not particles, be in superposition in 2 places or go into "quantum transition", maybe between our world and this what happenes in atom there is such "transition" and we coul make infinite amount of atoms from single atom, but in our "atomic" world maybe there is infinite amount of space, atoms and i beleive it since in theory there is even infinite amount of inifinities, but still, every ball that we could possibly observe would have limited amount of particles. I also have another conclusion, that maybe our world, even with countable amount of atoms in every item like ball/sphere i talked about, but our universe itself is infinite big and made of infinite amount of atoms but still we can just be like this infinite atom - we can't measure or observe the shape of universe becasue it's infinite, but in fact we could just be inside some atom in another universe and be a single particle in "that" outside world, but still infinite in our world and carrying information and ability to recreate every other atom, since they all contain infinite possibilities. In this theory every atom should be the same, or maybe it's just impossible to compare two infinite things to each other, maybe they have same information, but in different direction/order (or rotation like in video ;) ) So my final thought is that we are living in a "neverending matrioshka doll", that we are infinite universe, but also in every single atom there is (same?) universe like ours and so on, also in other scale direction our universe could be just a single atom of other universe or even same as ours. Maybe in all these universes everything happens exactly the same, and we are just living in a fractal like holograme. Also maybe in our world every single atom of their infinite count contain every possible other outcome, like world where i didnt write this coment. But i'm not 100% sure if every possible outcome exist in parrarel universe. Maybe there is only one "timeline", and everyting actually happened and we don't even have any impact on future, time doesnt exist and whatever we do, even on purpose to do it opposite we will do what was ment to be. But this is impossible to prove, at least in our lifes, since i beleive our soul is infini.. immortal. Sorry for using word infinity for infinite amount of times.

    • Clickachu
      Clickachu 16 hours ago

      Me too, I'd like to write a book one day, but it's hard for me to concentrate and write a long text but i can do it unconciously while writing a comment hah +Darius Z

    • Darius Z
      Darius Z 21 hour ago

      u love writing do you?

  • Haseeb Zarrar
    Haseeb Zarrar Day ago

    Hi ,
    Would you be able to answer the question of conservation ? Did the paradox imply that the laws of conservation doesn’t exist ? Or are we assuming this paradox outside the scope of that law? Thanks. Hope some light can be shed on this point.

  • Elena
    Elena Day ago

    What the actual fuck I’m too dumb for this

  • Henry Chavez
    Henry Chavez Day ago

    everyoine else: fortnite top 10 plays
    me, an intellectual: this
    later plebs

  • Emerald Bros
    Emerald Bros Day ago +1

    michal: obviously you can not cut up a chocolate bar and rearrange the pieces to more than your started with.
    me: that's quite obvious
    michal: or can you

  • Alvaro Jimenez
    Alvaro Jimenez Day ago

    This is the most beautiful (hidden) explanation of the Axiom of Choice I've ever seen.

  • TheMiner1501
    TheMiner1501 Day ago

    Why does everyone forget he destroyed money?

  • Shaggy
    Shaggy Day ago +1

    What happens if you take infinity from infinity -from someone called sparrow gaming -so this gets more views

  • z. tuğçe özcan

    muhteşem magnificant ♥️

  • Sheep Game
    Sheep Game Day ago

    That hotel paradox hurts my brain

  • Jonny Grow Lately

    *Also, instead of the Pea & The Sun book, may I suggest more relevant and urgent reading in .....*

    *1) Propoganda, by Edward Bernays*
    *2) The Crowd; A Study Of The popular Mind, by Gustav Le Bon*
    *3) The Secret Teachings Of All Ages, by Manly P. Hall*
    *4) The Creature From Jekyll Island, by G. Edward Griffin*


    *...thats just the start of learning the **_REEEEAAAL TRUTH_** ... you maybe could search for Rothschilds "25 Point Plan For World Domination" and read that as well...*

  • Jonny Grow Lately

    *LOL, no...common sense says...* #FlatEarth

  • dat person
    dat person Day ago


  • Lovro Planinšek /. Project 5421

    The thing that doesn't make sense to me is how can you turn an object to change what it physically is?

  • Patato Keftes
    Patato Keftes Day ago +1

    Wow, total mess.
    Numbers express fraction, quantity or value.
    Infinity cannot be quantified therefore is not a number, cannot be broken into fractions and has no absolute value.
    Trying to engage in math including infinity is like trying to add dollars to celcius degrees and come up with a quantifiable result.
    In essence infinity is a sum where ANY number has a value of nothing at all.

    • Logan Beavers
      Logan Beavers Day ago

      I don't know, you can't quantify pie, since it goes on forever...

  • Sumit Raut
    Sumit Raut Day ago

    Just consider if you can exponentially break those reference poles it means you have countably infinite number of references and hence infinite possibility of countable combinations.

  • Shivang Gulati
    Shivang Gulati Day ago

    Is he an alien?

  • Hello! My name is LOL

    I *LITERALLY* got a headache after watching this

  • Davino Sayaza
    Davino Sayaza 2 days ago

    Not gonna lie, after finished the video, I felt something in my head and I don't know what...

  • cammiecookies
    cammiecookies 2 days ago

    1:48 what the infinity pasta

  • Timothy Top_Notch_Tim_7

    How to get rich 101

  • Lawrence Joiner
    Lawrence Joiner 2 days ago

    12:00 the beginning of the Swastika Paradox.

  • v_killer7
    v_killer7 2 days ago

    I get this after viewing several times, even with my level ov unintelligence.

  • R Herman
    R Herman 2 days ago

    This was really confusing. I can understand why there were so many thumbs down. they probably didn't understand it. However, the fact that you were able to articulate this with so much passion & enthusiasm was incredible. Like I said, I will have to watch it again. Hey thanks!

  • Liam Neilson
    Liam Neilson 2 days ago

    If we were able to go faster than the universe expanding would be bump into something? How would that work???

  • nathan glover sr.
    nathan glover sr. 2 days ago

    This is the tip of the iceberg to antimatter production

  • Hill Top
    Hill Top 2 days ago

    I'm interested in this but after watching this video I realized I'm too stupid.

  • Shaun  Ambrose
    Shaun Ambrose 2 days ago

    My head hurts...

  • James Armstrong
    James Armstrong 2 days ago

    U melted my mind

  • Jackson Kelly
    Jackson Kelly 2 days ago

    that dollar shit got me hurt. i tried it and just ruined a dollar

  • Thomas Tang
    Thomas Tang 2 days ago


  • Vincent Snijder
    Vincent Snijder 2 days ago

    This is mind-blowing on so many levels

  • Siddl •
    Siddl • 2 days ago

    Cost of video : $3

  • Janko Djuric
    Janko Djuric 2 days ago

    If you think closely
    Present is infinite. (hope so)

  • LightSpirit24
    LightSpirit24 2 days ago

    Those points have no size to them. They are not wide or thick or tall. So you can never fill up the surface of the sphere.

  • Drunk Muffin
    Drunk Muffin 2 days ago

    How small is a dot?

  • Amelia
    Amelia 2 days ago

    Mind: blown

  • Dynamo
    Dynamo 2 days ago

    1:50 yall saw the k and the f right yeah my homeboy u thought u was slik

  • Nicholas Soutter
    Nicholas Soutter 2 days ago

    My fucking brain broke

  • Jude Yovichin
    Jude Yovichin 2 days ago +2

    This video is great, but there is a logical fallacy when he starts talking about the circle. There are an infinite amount of points on a circle because a circle has dimension while a point does not. Therefore, the poor he “told off the circle” would have taken up no space on the circle in teh first place and he wouldn’t have had to fill the space left behind. There was no space left behind. All this means that he didn’t create more by taking out of the circle, he just didn’t take off of the circle.

    • Jude Yovichin
      Jude Yovichin 2 days ago +1

      Banach-Tarski’s Paradox isn’t supported by mathematical theory and, by extension, can never be possible in the real world as he suggests at the end of the video.

    • Jude Yovichin
      Jude Yovichin 2 days ago +1

      This logic can also be applied to the spheres, you can’t make a sphere out of points, because points have no dimension, but the sphere has to. You can’t make something out of nothing.

  • Ivar Melker
    Ivar Melker 2 days ago

    The point would end up in the same location if you did Left Up Right Down...

  • Thescott16
    Thescott16 2 days ago

    Perhaps Joseph and Mary should have gone to Hilbert's Hotel then...

  • Zach C
    Zach C 2 days ago

    favorite video by far return to it every year just to get my brain going again

  • jervee saclolo
    jervee saclolo 2 days ago

    Why am I watching this? I dont even understand a single thing

  • Yogscast Minecraft_Lewis0

    14:00 Isn't the red point and the purple point same? Assuming that you move the same distance in each direction, L and R being opposite would cancel out and left with U. This makes LUR and U the same point. (Unless the distance you move is different)

  • TenemaesLament
    TenemaesLament 3 days ago +2

    All the dislikes are from physicists

  • Cici Bracken
    Cici Bracken 3 days ago

    Earth is flat don’t @ me

  • Papa Beasticle
    Papa Beasticle 3 days ago

    I'm not sure if I gained a brain cell or lost one at 7:19

  • BringThePizza
    BringThePizza 3 days ago

    “The universe isn’t strange, we are”

  • Ricardo Lopez
    Ricardo Lopez 3 days ago

    I want to get high and watch this so I can be fully Amazed by this..!!

  • Talo B98
    Talo B98 3 days ago

    For The infinity is equal to one and sphere and stuff watch past this point 11:23

  • SpyCommander13
    SpyCommander13 3 days ago

    Scientists have discovered a duplication glitch.

  • Joey O'Sullivan
    Joey O'Sullivan 3 days ago

    My fucking brain hurts

  • BuhVuhGuh
    BuhVuhGuh 3 days ago

    You know, you could just go by the law of conservation of mass for that chocolate bar?

  • darthkarl99
    darthkarl99 3 days ago

    And this is why a good ;portion of extremly capable mathematicians through the centuries have suffered from various mental disorders. You kinda have to be a little crazy to get your head around math above a certain point.