The Banach-Tarski Paradox

  • Published on Aug 1, 2015
  • Q: "What's an anagram of Banach-Tarski?"
    A: "Banach-Tarski Banach-Tarski."
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Comments • 65 482

  • The GhostCodes
    The GhostCodes 3 hours ago

    1:29 R.i.p dollar 💵

  • Rorin Eaton
    Rorin Eaton 5 hours ago

    How to make the paradox work.... Fractals. Nuff sed

  • Rorin Eaton
    Rorin Eaton 5 hours ago

    You mean if I figure out a way to magically put dollars back together then I can just keep on cloning 20 dollar bills then hundreds then thousands and so on.........

  • Spaghetta 99
    Spaghetta 99 5 hours ago +1

    How bout infinity-infinity

  • AnimatedCatastrophe
    AnimatedCatastrophe 5 hours ago

    *funky music*

  • AlexBirdieBronze
    AlexBirdieBronze 6 hours ago

    Infinity x infinity!?

  • Sean Solo
    Sean Solo 9 hours ago

    Vsauce: "There is no way you can cut up a chocolate bar, and rearrange the pieces to get more than you started with."
    Me: "Don't say it.. Don't say it.. Don't s-"
    Vsauce: "Or can you?"
    Me: "FUCK."

  • Vera Night
    Vera Night 9 hours ago

    I used to have trouble sleeping. Now, I watch Vsauce videos like this one as a great nighttime paradox! great video!!

  • C Ollivier
    C Ollivier 9 hours ago

    Yes, but... if infinity + 1 = infinity. Does infinity - infinity = 1 ?

  • amethyst dawn
    amethyst dawn 10 hours ago

    The dollar part is fake. Watch the way he rips the dollar carefully then pause on the bills and you'll see that the rips don't add up. If he ripped it in half vertically first, how can the dollar bill on top be ripped in half horizontally in two solid pieces with no vertical rips?? Because it's two separate bills lol

  • Diego Castro
    Diego Castro 10 hours ago

    That relaxing brazillian music at the final dude, great vibes

  • Gannimal
    Gannimal 10 hours ago

    Vsause must be rolling in cash if he’s able to just rip a dollar into 5 pieces

  • Bledoston
    Bledoston 11 hours ago

    I bet rich people in the future will use this to clone their brains.

  • Dawit Moges
    Dawit Moges 12 hours ago


  • Ashton Vlogs
    Ashton Vlogs 13 hours ago

    I’m bout to duplicate some fucking money bro 😭😭

  • Ijin dela
    Ijin dela 13 hours ago

    For this paradox to work you first build a countable subset of points, while uncountable number if points remain unselected. You then pick next unselected point and build an additional countable subset. You the repeat this operation COUNTABLE number of times. At the end you've selected a countable*countable number of points. Which has the power of all rational numbers, which is still countable. Uncountable number of points still remain unselected after you were doing "select next unselected point and build subset" forever. Paradox shows that you can't keep selecting unused point on (0,1) until no more points remain.

  • William Cashion
    William Cashion 13 hours ago

    Awesome trippy stuff. I loved it.

  • Pandała
    Pandała 14 hours ago

    when even the perfect real world has bugs...

  • Charles Leonito L. Iringan IV

    My dollar NOOOOOOOOOO

  • Lucas Jarrett
    Lucas Jarrett 15 hours ago

    Um... you can't count whole or real numbers in a finite length of time... why are you saying that? What you should be saying is that countable infinity can be counted up one unit at a time, but we can't count up even one unit of an uncountable infinity. But neither kind can be counted in a finite duration. They're both still infinite. There are not "more [real] numbers" between 0 and 1 than there are whole numbers - you can add infinite 0's in either condition. Cantor's diagonal argument is bunk because its fundamental premise is impossible. You cannot construct a number made of "every" next digit (altered slightly) in an infinite series of potentially-irrational numbers, because the concept of "every" digit can't apply to a neverending set of digits.
    However, I think another argument is more valid in establishing that the infinity of numbers between 0 and 1 is greater than the infinity of whole numbers. For every number between 0 and 1 that begins .1, .2, .3, .4, .5, .6, .7, .8, or .9, the decimal point can be removed to create a whole number. However, there can also be infinite numbers beginning .0, and these cannot all be so trivially "matched" to whole numbers that aren't already matched to other reals between 0 and 1. To see this clearly, consider any number of the form .000...1. If we remove the decimal, the corresponding whole number is "1" - but we can create infinite reals between 0 and 1 before we ever need another digit that might correspond to a new whole number, because adding more 0s in front of that 1 will always create a new number between 0 and 1 that does not correspond to a new whole number when the decimal point is removed.
    Therefore, the infinity of numbers between 0 and 1 is actually infinitely bigger than the infinity of whole numbers.
    Also, I'm not sure I agree that infinities of even and whole numbers may be equally infinite if we're not going to agree that the reals and wholes are equally infinite, because they have the same relationship. We agreed that reals are "more infinite" because there are numbers in the reals that can't find correspondence with a new number in the wholes. The same could be said about the wholes in relationship to the evens. For consistency's sake, either the wholes should be considered more infinite than evens, or the reals shouldn't be considered more infinite than wholes.
    Hilbert's Hotel is dumb for the same reason that Cantor's argument is dumb. You can't logically speak of booking "every" room in an infinite series of rooms.
    Infinity doesn't care what you "subtract" from it because infinity is a concept, not a number. You don't subtract numbers from concepts, you subtract them from numbers.
    Also, abstracting sets from spheres is not the same as physically removing pieces of them. You couldn't physically remove the same piece of a sphere twice, so you could never actually make two out of one. Infinite "points" are being represented in multiple abstracted sets, but that doesn't mean we have multiple spheres of "points".

  • SuperMechaUltraNanoCyberJohnmas Mark IV

    Can someone help me understand how this isn't just an efficient shortcut to mapping an identical object?
    I mean, I know there's a degree of engaging this axiomatically, so conservation of energy in no way applies, but doesn't the thought experiment rely on the supposition that we've got an actual physical sphere that we can somehow chart to a greater degree of precision than a series of Planck lengths?
    Perhaps that's where I'm hung up, but if so, after we're finished with our categorization and we split away one category of points, what keeps them stationary as we rotate the set?
    It seems as if we're creating new material from whole cloth, which I can't see being much more impressive than simply declaring that we have an additional sphere.

  • Wojciech K
    Wojciech K 15 hours ago +1

    *i have reached an end of my intellectual capabilities*

  • Ruicheng Kelvin Lu
    Ruicheng Kelvin Lu 16 hours ago

    14:00 LUR=U

  • Peter Alves
    Peter Alves 19 hours ago

    Nice Brazilian jazz :3

  • AphidCC
    AphidCC 22 hours ago

    Up Up down down left right left right B A start.
    The way tog land on same spot

  • Legend Graal
    Legend Graal 23 hours ago

    So, for the people that did not understand the other 3/4 of the video he is saying our math system is screwed because we cant comprehend the mass amount of numbers that there are in the system he is saying we will never fully understand our mathmatical system because we cant comprehend infinity

  • Trayjen jones
    Trayjen jones 23 hours ago

    The orange and blue dot will end in the same place

  • Michael Stevens
    Michael Stevens Day ago

    except that Right Up Left would end up in the exact same place as Up

  • Aaron Martin
    Aaron Martin Day ago

    .33333*3=1 How can you take a set of infinite numbers and make a whole?

  • Tessanique Williams

    Why can't I watch it😠😠😭

  • rupert888
    rupert888 Day ago

    Wow, infinity is infinite. Maybe I'm just not smart enough to understand why this is a paradox.

  • Caden :P
    Caden :P Day ago +1

    Or can you
    **strange music plays**

  • Varnox
    Varnox Day ago

    13:45 whatcha making there Michael.

  • Richard Heesterbeek

    if Hilbert's Hotel is true, we fill EVERY room with whole numbers between 0 and infinity. Than if we add one more number, we can just put it in the first room and every body moves a room. Right the explanation works, issue however is that since we already used up EVERY whole number, we don't have any whole numbers to add anymore ... so this makes Hilbert's Hotel what?

  • Brett101792
    Brett101792 Day ago

    my broke ass cringed when he tore the dollar 😂

  • Bre McCoy
    Bre McCoy Day ago

    Once he gets to talking about the sphere points I'm like yeh..yeh I get it......I dont get it and Ive watched this 3 times. Am I that dumb!?!

  • Zombie Survival
    Zombie Survival Day ago

    VSauce: Or can you? *breaks bill into five pieces and makes two bills*

  • Nickx13
    Nickx13 Day ago

    First vsauce I saw, tripped the shit out of me

  • Max Nieves
    Max Nieves Day ago

    Guess ill just sell my brain

  • starless night sky lightless sunrise

    Infinite is a virus it makes a copy that is diffrent or the same as its self

  • Vanessa Hayes
    Vanessa Hayes Day ago

    ah yes
    It has so much potential to create magic
    but all we use it for is witchcraft and 2+2 = 4 -1 that's 3 quick maffs

  • KittCat NotHere
    KittCat NotHere Day ago

    1:48 I could've used that ;-;

  • Exon Vidz
    Exon Vidz Day ago

    4:22 me writing homework

  • imogen tait
    imogen tait Day ago

    so infinite things come and go as they need to be used?

  • Iris Ariola
    Iris Ariola Day ago

    If you rotate a directional piece in the opposite direction, then it will be congruent to the original direction points plus the parallel direction points and starting points.

  • Cole Wogan
    Cole Wogan Day ago

    18:56 this is the only bit i didnt get can someone explain it to me. Your video was about how 1 can = 2 right? Or infinite expansion so how does 1+1=1 work for the sphere and quark examples?

  • Travis 677890
    Travis 677890 Day ago

    7:30 the number of unsatisfied guests it probably also infinity if they have to switch rooms every time someone new comes in lol :)

  • Jose D
    Jose D 2 days ago

    i rather get laid! same as sex? maybe?

  • Muzin Kafi
    Muzin Kafi 2 days ago +1


  • Brett Palmer
    Brett Palmer 2 days ago

    Does this mean we can use the infinite stones to make more infinite stones?

  • Casey Bade
    Casey Bade 2 days ago

    Vsauce hurts my brain


    Some of this i understand.....most I do not at all...I'm to (stoned, tired, stupid, indifferent, _____fill in the blank).

  • Arthur Grella
    Arthur Grella 2 days ago

    MPB in the end.

  • timwins31
    timwins31 2 days ago +1

    I like the comments on here. Its refreshing, people admitting they just don't understand something rather than arguing with it to make it fit their own reality. Nicely done folks. You deserve a pat on the back, we need more people like you.
    . . .and don't feel discouraged. . .lol. 25 years of educating has taught me that the human mind just does not like the concept of infinity, it rejects it violently, no matte how intelligent the person, how educated. Some people will be lucky and eventually it will click, but most of us just have to accept it and stop trying to grasp it. I know that answer sucks but . . . that's life. :-D

  • Ser Ocelote
    Ser Ocelote 2 days ago

    Is there a F word in that Dictionary?
    No better Place or time to use it.

  • Digital Dafydd
    Digital Dafydd 2 days ago

    I wish we were shown vsauce is math class

  • Melvin Sandberg
    Melvin Sandberg 2 days ago


  • Ryszard Janku
    Ryszard Janku 2 days ago

    1:20 Thats buttock

  • Adventure of Pathan
    Adventure of Pathan 2 days ago

    After watching the whole Concept... I Ate a Big Mac. and Coke and Goes for Sleep for 10 hours... Thats the life bro I realised 😅

  • Nihill'vo
    Nihill'vo 2 days ago

    To ten mądry z TVGry?

  • strangelf 47829
    strangelf 47829 2 days ago

    8:56 you can explain this shit but you can’t count from 0 to 11

  • strangelf 47829
    strangelf 47829 2 days ago

    Infinity - infinity

  • Joshua Eppinga
    Joshua Eppinga 2 days ago

    Infinity + 1 = infinity you spiled your other video!😂

  • Eobard Thawne
    Eobard Thawne 2 days ago

    The Banach tarski paradox might apply to sub atomic particles but can't act outside it ,

  • Eobard Thawne
    Eobard Thawne 2 days ago

    Why don't u do the direction combination with respect to the surface area and distance between each direction change , u might still get a paradox the one similar to the uncountable infinity like there will a point between these two points and so on

  • Eobard Thawne
    Eobard Thawne 2 days ago

    The circumference of the circle paradox explanation is wrong , how come der is a hotel room where der is a break in the circumference , when other rooms are associated with a number , number one cant move to place where there is a break in circumference it's contradictory

  • Boom
    Boom 2 days ago

    Why is it 360p

  • Colten Roney
    Colten Roney 2 days ago

    Imma try that dollar trick with my hundreds

  • Anupam Srivastava
    Anupam Srivastava 2 days ago

    18:00 Lost it. I believe you.

  • Badges of Shame
    Badges of Shame 2 days ago

    The rendering won't allow me to load this video.

  • Dah Butter Asian
    Dah Butter Asian 2 days ago

    My brain hurts... ima go back to watching Michael Jackson moonwalk in reverse thanks...

  • derrenk
    derrenk 2 days ago

    Great vid, many thanks for the headache :)

  • Enzoderbaecker
    Enzoderbaecker 2 days ago


  • C.I.A
    C.I.A 2 days ago

    My brain hurts :(

  • Dima
    Dima 2 days ago

    The method by which the sphere is decomposed and composed back is smart, but why is the theorem result surprising? A sphere surface has an uncountably infine number of points. 4 spheres also have an uncountably infinite number of points. By the very definition of 'uncountble infinity' there exist a bijective mapping between the 2 sets. Ie all points of the 4 spheres can be assigned a unique point on the one sphere such that all points of the one sphere are covered. So that theorem result is quite obvious from the definition alone. The exact method of mapping points is just a detail.

    • MuffinsAPlenty
      MuffinsAPlenty Day ago

      What you say is the resolution of the paradox. The fact that at least one of the pieces has no defined volume explains the weird phenomenon I mentioned about volume additivity.
      It also gives some more insight into general measure theory. It had been known that you can construct non-measurable sets in the Lebesgue measure. But Banach-Tarski implies that there does not exist a non-trivial, isometry-invariant measure which gives bounded sets finite measure and for which all subsets of R^3 are measurable. So it's not like there is some nice hidden non-Lebesgue measure out there that does everything we want it to.

    • Dima
      Dima Day ago

      MuffinsAPlenty there is something missing in the analogy. The volumes of the sets the sphere is decomposed do not have volume at all because they are so porous. It’s not zero, not any other number, it’s undefined. The existence of bounded sets in 3D space without volume was a known and unsurprising fact. The only surprising thing is that you can split a shape with defined volume into a few sets of undefined volume, then reasamble the shapes with undefined volume into a shape of defined higher volume than the original. This is indeed very Interestingly, but shapes with undefined volume can do a lot of things. It’s not like in physical reality you can have objects with undefined volume. So it should not come as a shock that you can make volume bigger out of nothing mathematically if you are allowed to go into shapes of undefined volume. Undefined + undefined can equal anything, including a defined value. I find it interesting, but hardly shocking. I find it more interesting that the number of sets is finite. You can’t do this in 2d. In 2d you can do this if you are allowed to have countably many splits, but with a finite number of splits works only in 3D.

    • MuffinsAPlenty
      MuffinsAPlenty 2 days ago

      I want to build off what Edgar Nackenson said.
      Think about it this way: How do you find the volume of a "silo" shape (a cylinder with a hemisphere on top of it)? Well, we split it up into the two pieces: find the volume of the cylinder, find the volume of the hemisphere, and add them both together.
      This is our intuition: that splitting up solids into finitely many pieces preserves total volume.
      But as Edgar Nackenson pointed out, you can split a sphere into _finitely many_ pieces, and after performing a couple _geometry preserving actions,_ you get two spheres which are identical to the first. In this instance, total volume is not preserved.
      That is what is surprising about the Banach-Tarski result.

    • Edgar Nackenson
      Edgar Nackenson 2 days ago

      What's surprising isn't that there's a mapping at all. What's surprising is that you can pull it off with a finite quantity of sets combined with no stretching or warping. It's just like six or seven sets and a bit of rotation.

  • A guy Who cheats
    A guy Who cheats 2 days ago


  • Gay Dinouo
    Gay Dinouo 3 days ago

    There is an infinite number of blah blah blah

  • Levi Cukas
    Levi Cukas 3 days ago

    goddamn this hurts my brain

  • Christopher Shelton
    Christopher Shelton 3 days ago

    You can get it only if you follow along very carefully

  • Giulian Zorzi
    Giulian Zorzi 3 days ago

    This made my head physically hurt. Actually! Good job!

  • Big Braddah Bell
    Big Braddah Bell 3 days ago

    Trying to figure out a way to use this to get a raise on my paycheck or a pizza . Ummm, 'nope', I've got bollocks ! Anybody wana chip in and get a pizza ? 😜

  • Babachew Bob
    Babachew Bob 3 days ago

    Even a casual observer will notice that your dollar bills have different serial numbers on them. So your "exact duplicates" evaporate into a puff of logic. Carry on.

  • TokyoEthan
    TokyoEthan 3 days ago

    The problem with infinity plus 1 is you cant move it foward because there is always someone in the room after them so they cant move foward.

  • Technoultimategaming

    ok got 3/4 of video

  • shadow knight
    shadow knight 3 days ago

    So infinity minus infinity is infinity?

  • RightofLeftofLeft Eddie

    At 2:03, when he says "aaaannnd ourselves, " he makes a really weird gesture

  • RightofLeftofLeft Eddie

    Am I the only one who found this not that profound? Like, yea, infinity is profound, but the 50 different "paradoxes..." like, yea dude, I get it.. infinite.

  • Learned Hand
    Learned Hand 3 days ago

    This is how fiat currency works :)

  • Sean Weintz
    Sean Weintz 3 days ago

    i call bullshit on the assertion that a countable infinity is the smallest defined infinity. Take, for instance, the set of integers, vs. the set of positive integers. BOTH are countable infinite sets, but i can prove mathematically via correspondence that the set of positive integers is exactly half the size of the set of all integers.

    • Edgar Nackenson
      Edgar Nackenson 3 days ago

      I haven't split the set of integers into two sets at all. 0, 1, -1, 2, -2... is exactly one set, unsplit, which can map as I've described to the positive integers on a one to one basis. You say this is a supposed "two to one" mapping, but can you name a single positive integer that I've mapped more than one integer to, or vice versa?

    • Sean Weintz
      Sean Weintz 3 days ago

      um, what you describe is the generally accepted proof of a two to one, not one to one mapping. Since you split the set of integers in half into two sets, each of which has a one to one mapping with the set of negatives.
      if half of set A is equal to set b in size, then set A is exactly twice the size of set B.
      You actually prove the exact opposite of what you claim.

    • Edgar Nackenson
      Edgar Nackenson 3 days ago

      I can prove mathematically via correspondence that the set of positive integers is exactly the same size as the set of all integers. The process is pretty easy. Just create a list of the first set, and then a list of the second. So, for the positive integers, it'd be 1, 2, 3, 4, 5... For all integers, it'd be 0, 1, -1, 2, -2, 3, -3... Now, take the first element of the positive integers, and pair it with the first element of all integers. Take the second element of the positives and pair it with the second element of the integers. And so on. Every positive integer will be paired with exactly one integer.
      It's trivial to create, for any pair of infinite sets, a mapping that fails to account for any quantity of one of the two sets. Even infinitely many elements. This is even true when comparing, say, the set of all integers to the set of all integers. The question is whether it is possible to create a mapping that succeeds to account for the entirety of both sets. For any two countably infinite sets, and that includes both the positive integers and the set of all integers, it will always be possible to create such a mapping. For a countable infinity and an uncountable infinity, doing so will be impossible.

  • Hugo Stonewall
    Hugo Stonewall 3 days ago

    so... since infinity minus one is infinity... then (infinity) - (infinity - 1) = 0... not 1...

    rip my brain

    • Edgar Nackenson
      Edgar Nackenson 3 days ago

      Infinity minus infinity is an indeterminate, meaning it can take on any value. The same is true of infinity-infinity-1.

  • Wes Netmo
    Wes Netmo 3 days ago

    the diagonal argument seems bs to me. real and whole sets are equal and both infinity

  • Κωνσταντάς Χατζηκωσταράτσος

    Why do you keep starting the videos by shouting??

  • Indo Raptor
    Indo Raptor 3 days ago


  • Overwhelming Conundrum

    Thats not possible but its fun to think about.

  • Donald Cameron
    Donald Cameron 3 days ago

    Points cannot move; they cannot touch; they are discrete locations of mass and time. The universe is analog rather than digital. All that actually moves is the density of points. In this analog universe the are two infinities. One is the outwardly infinite (the ever expanding universe) which can only be truly infinite if it is bidirectional, Infinity is unbounded. There can be no origin as that would be a boundary, and infinity is unbounded. The complement that solves the boundary issue is an inwardly infinite universe.
    This disturbing assertion lies outside of existing disciplines giving way to Complexity. Not Complexity Theory nor Complexity science - just complexity.
    This does not diminish the good video above.

  • Diamond Ninja
    Diamond Ninja 3 days ago

    Being a guest at Hilbert‘s Hotel sounds like a lot of work

  • gxzkc
    gxzkc 3 days ago

    I am not from america, but isn't destroying money a federal crime?

  • nick g
    nick g 3 days ago


  • Judy Thompson
    Judy Thompson 3 days ago

    mother of god

  • James4wd
    James4wd 3 days ago

    Take idea that Ive never thought that much in depth about.
    Then make me question my existence and the purpose of life.
    I give you a Vsauce video