The Banach-Tarski Paradox

  • Published: 01 August 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:
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Comments • 76 739

  • HiddenJuice x
    HiddenJuice x  16 minutos atrás

    *rips dollar bill* Me: FGJHFGHDYGFGFXFGCGHMVMH KyhtuyyrdddtfguknlhibftxsaYdtufTRYDUYBLUVSTSE

    • JosephFG
      JosephFG  2 horas atrás

      Between 14:15 and 14:37 is the fatal flaw. Now I will have to peruse the other Banach-Tarski Paradox vids to see if they all commit the same flaw. Here is the fallacy: "We can just pick a point we missed. . . making it a new starting point, and then run every sequence from here. After DOING THIS to an UNCOUNTABLY INFINITE NUMBER of starting points, we will indeed have named and colored every single point on the surface just once." By specifying "doing this," we are specifying a countably infinite series of tasks. There is no way to reach uncountable infinity from countable infinity. The diagonal proof (Cantor) is based on the notion of a list, and any list is by definition countable, i.e. can be enumerated by the natural numbers, and the transfinite number Aleph[null]. Since we can clearly simply keep track of each time we make a new starting point and run the sequence, its cardinality is also Aleph[null], NOT Aleph[one], which is the cardinality of the continuum. Just SAYING "after doing this an uncountably infinite number of times" does not render that possible. We know it's not physically possible, but I'm saying it is also not mathematically possible. The continuum is not merely a set of points. It is a continuum. The total length of any countably infinite number of points is still zero, whereas the continuum has actual length. Once you started naming points, you defined that set as countable. Specifying "doing this," likewise renders that series countable. I don't see a way out of this conundrum. So maybe it's not a paradox after all, but rather just a fallacy. And in any case, the physical world is clearly countable, and even finite, as far as we know. So I wouldn't get my hopes up too high for creating something from nothing.

      • OOF GOD
        OOF GOD  4 horas atrás

        Vsauce: * cuts up dollar.* Police: **angry dollar noises**

        • Qasim Ali
          Qasim Ali  8 horas atrás

          1:17 the best , can you

          • The Shiny Skuntank
            The Shiny Skuntank  12 horas atrás

            I literally have a headache because of this... but it was worth it!

            • Silver theDragon
              Silver theDragon  14 horas atrás

              Isn’t this one of the proposed models of the universe?

              • F4F D
                F4F D  17 horas atrás

                Infinity is a function, like in 000000... example is always about adding another 0 so the notion is about always adding (or deduct or ...) so infinite +1 is infinite because it will be like adding 1 to a +1 function, the function wouldn’t change

                • Ali Muhammad
                  Ali Muhammad  21 hora atrás

                  A complicated way of proving infinity is in fact infinite and therefore cannot be larger or smaller than another infinity

                  • Edgar Nackenson
                    Edgar Nackenson  2 horas atrás

                    It does not prove that at all. The infinity of the one sphere is exactly the same size as the infinity of the two spheres, but you wouldn't be able to map, say, the natural numbers with the sphere(s) on a one to one basis.

                    • isaiah trujillo
                      isaiah trujillo  21 hora atrás

                      I came for the dogs in the end

                      • rankhobo
                        rankhobo  Dia atrás

                        I got a feeling that this would be really cool if I could understand this

                        • James84x21
                          James84x21  Dia atrás

                          So is time finite? Couldn’t you break down a second into decimal and just infinitely add another zero before reaching second number two? Are we still technically with in the very first second of times existence? 🤔

                          • Sterlin Christabel
                            Sterlin Christabel  13 horas atrás

                            As far as i understand your comment, i think you can add zeros upto infinity but it stays in between the first and second second of time. It's like we could theoretically cut an object infinite times even though practically not possible. Theoretically counting from 1 to 2 is just like counting from zero to infinity as he said

                            • Sky's D limit
                              Sky's D limit  Dia atrás

                              32-1 is 32

                              • whydontiknowthat
                                whydontiknowthat  Dia atrás

                                Aren’t the points along the circumference of a circle uncountably infinite? The circumference is basically the same as the real line, which is also uncountably infinite.

                                • Edgar Nackenson
                                  Edgar Nackenson  Dia atrás

                                  They are, yeah.

                                  • Mable Calden
                                    Mable Calden  Dia atrás

                                    I- I think you broke my brain

                                    • Viktor Emil Vang Hansen 7A Ejerslykkeskolen

                                      Ain't ripping money over illegal?

                                      • Panda Pan
                                        Panda Pan  Dia atrás

                                        My brain hurts😣

                                        • sign543
                                          sign543  Dia atrás

                                          Wait...if you listed all of the real numbers between 0 and 1....wouldn’t that necessarily include the number you say you are creating by adding a 1 to it? I think I didn’t understand that part.

                                          • Edgar Nackenson
                                            Edgar Nackenson  2 horas atrás

                                            @sign543 I'm glad that one worked for you.

                                            • sign543
                                              sign543  3 horas atrás

                                              Edgar Nackenson - Okay, this finally got through my thick skull! Thank you. The way you explained it being greater than the greatest prime from the original list...that did it. I wish I was more of a math person. Thanks again!

                                              • Sterlin Christabel
                                                Sterlin Christabel  4 horas atrás

                                                @Edgar Nackenson I mean counting from an infinite lowest value. I know there is no such thing existing. Because even if we start counting from 0.0000000000000000........1(assume there's an infinite number of zeros in bw) it could be still larger than an extra zero added.That means we can never actually start counting because we're not finite about the lowest value like counting from zero to infinity in which we can never stop counting because it's infinite

                                                • Edgar Nackenson
                                                  Edgar Nackenson  6 horas atrás

                                                  @sign543 The point is that you can conclusively prove that this number does not appear on the list. You assume the list contains everything and then prove that it can't. The reason the proof is constructed this way is because you're leveraging the list itself in order to produce the new number. You're asking whether the number would have come up, and the answer is, well, where would it be? It's built such that the first digit is distinct from that of the first number, such that the second digit is distinct from that of the second number, and so on. We can identify a place where the number necessarily differs from every number previously on the list. Maybe this would be easier if you looked at a somewhat easier proof by contradiction, say the proof that there are infinitely many primes. The first step is assuming there are finitely many primes, 2, 3, 5, 7... all the way up to the last prime n. Now, consider the number (2*3*5*7*11...*n)+1. This number can't be divisible by 2 because it's one greater than a multiple of 2. It can't be divisible by 3 because it's one greater than a multiple of 3. And so on. Thus, this new number is not divisible by any prime, which means it is itself prime. We initially assumed that n was the greatest prime, but we were wrong. Assuming this greatest prime allowed us to prove the existence of a greater prime. These two proofs, you may note, function identically. You assume some list, the list of the reals and the list of the primes, assume it's complete, and then find a contradiction, either a new real or a new prime. It's kinda silly to say, "Well, what if that prime was on the original list?" It can't be because it's greater than the greatest prime from the original list. The original "complete" list didn't include this "variant" number, and so was broken.

                                                  • sign543
                                                    sign543  7 horas atrás

                                                    He says since we’ve listed every number in our set, then the new number (by adding 1) is not in our list. My thought was...wouldn’t that number have eventually come up in the initial list? Yikes...this is hard to grasp. Granted, math is NOT my strong point. I have a masters in English...not math 😂

                                                    • sign543
                                                      sign543  Dia atrás

                                                      “...or can you...” 😂 I knew it was coming (I’ve only recently discovered Vsauce, be gentle)

                                                      • SuperBoo
                                                        SuperBoo  2 dias atrás

                                                        Wow this guy is so rich he can afford to rip up money

                                                        • hope mkhonza
                                                          hope mkhonza  2 dias atrás

                                                          Well, this was trippy

                                                          • Mitchy J Watson
                                                            Mitchy J Watson  2 dias atrás

                                                            Destroying government property and counterfeiting @ 1:34

                                                            • Professor Baguette
                                                              Professor Baguette  2 dias atrás

                                                              If Banach-Tarski could apply to the real world... maybe the idea of multiplying a couple fish and bread to feed a crowd wouldn't be so absurd.

                                                              • Fuseteam
                                                                Fuseteam  2 dias atrás

                                                                1+1=1 - general relativity

                                                                • Fuseteam
                                                                  Fuseteam  2 dias atrás

                                                                  infinite: something that not finite countable infinity:you can count a subset of infinity in a finite amount of time uncountable infinity: you need an infinite amount of time to count a subset of this infinity all countable infinities are the same

                                                                  • N L
                                                                    N L  3 dias atrás

                                                                    This is total nonsense.

                                                                    • Niko Luopakka
                                                                      Niko Luopakka  3 dias atrás

                                                                      My mind is blown🤯

                                                                      • NuclearSquid527 _
                                                                        NuclearSquid527 _  3 dias atrás

                                                                        Pardon me, could you run that by me one more time?

                                                                        • Under stone
                                                                          Under stone  3 dias atrás

                                                                          I cannot believe I actually learned something...

                                                                          • Bibba Nibba
                                                                            Bibba Nibba  3 dias atrás

                                                                            Do you ever just get high with your friends and just binge watch conspiracy theories and space facts?

                                                                            • Duane Finley
                                                                              Duane Finley  3 dias atrás

                                                                              Me: obviously you can't make money out of thin air Vsauce: OR CAN YOU?

                                                                              • kieran walker
                                                                                kieran walker  3 dias atrás

                                                                                4:25 decimals 1 1.5 2 2.5 3 3.5 4 4.5 5

                                                                                • Sumponeigh KnotYew
                                                                                  Sumponeigh KnotYew  3 dias atrás

                                                                                  We are very strange. Do dolphins know they are wet? We know that time passes. Does that mean we are eternal?

                                                                                  • Sterlin Christabel
                                                                                    Sterlin Christabel  13 horas atrás

                                                                                    No, they know they are not dry

                                                                                    • bts idek
                                                                                      bts idek  3 dias atrás

                                                                                      Twenty four minutes and thirteen seconds of me being completely confused

                                                                                      • Erik Dieburg
                                                                                        Erik Dieburg  3 dias atrás

                                                                                        So I draw a line here and I draw a line here and I'll shit I better stop before this turns into a swastika

                                                                                        • Cartoon Master
                                                                                          Cartoon Master  3 dias atrás

                                                                                          Michael: Has Corn MatPat and Steph: Has Skip I the only one who thinks that math and science may be somehow related to cats?

                                                                                          • Tucker Smith
                                                                                            Tucker Smith  10 horas atrás

                                                                                            only part i got was michael has corn, but yeah i agree

                                                                                            • MOAT75 MC
                                                                                              MOAT75 MC  3 dias atrás

                                                                                              0:10 *BRAIN HURTS*

                                                                                              • MAUROtele
                                                                                                MAUROtele  3 dias atrás

                                                                                                13:45 i thought michael was drawing a swastika

                                                                                                • Sunnypsyop
                                                                                                  Sunnypsyop  3 dias atrás

                                                                                                  Apeirophobia anyone?

                                                                                                  • 9sunstar9
                                                                                                    9sunstar9  3 dias atrás

                                                                                                    Why did I just watch a 24 minute video on theoretical math the night before senior year of college starts?

                                                                                                    • Orion
                                                                                                      Orion  3 dias atrás

                                                                                                      Or can you?

                                                                                                      • Hidden Ikon
                                                                                                        Hidden Ikon  4 dias atrás

                                                                                                        So theoretically, we could copy animals or even planets? Please let me know. So we could even copy endangered animal genes or dna. Vsauce?

                                                                                                        • Freki last name
                                                                                                          Freki last name  4 dias atrás

                                                                                                          I feel like this is more of an illusion than anything.

                                                                                                          • ding dong your opinion is wrong

                                                                                                            The hyperwebster knows where malaysia flight 370 went 🤯

                                                                                                            • vFrosty
                                                                                                              vFrosty  4 dias atrás

                                                                                                              12:19 Donkey Kong Country cheat codes

                                                                                                              • 1919helio1919
                                                                                                                1919helio1919  4 dias atrás

                                                                                                                I have 2 brains now

                                                                                                                • ZaAnJo Driver
                                                                                                                  ZaAnJo Driver  4 dias atrás

                                                                                                                  I tried this with money and it actually worked lol

                                                                                                                  • Imran Shishir
                                                                                                                    Imran Shishir  4 dias atrás

                                                                                                           would M or L come after A in that dictionary? Loved the video.

                                                                                                                    • Igor Gerlovin
                                                                                                                      Igor Gerlovin  4 dias atrás

                                                                                                                      21:50 - trippy!

                                                                                                                      • Libby Robertson
                                                                                                                        Libby Robertson  4 dias atrás

                                                                                                                        so this could be like a parallel universe the same but different? you take our earth, split it, but still have 2, but as you split the left right up and downs, some arent identical which is why in the common theory of a parallel universe, things are the same but slight differences? anyone know what i mean?

                                                                                                                        • witbyy
                                                                                                                          witbyy  4 dias atrás

                                                                                                                          but it doesnt tho i tried it with my money

                                                                                                                          • AWeebBoi60
                                                                                                                            AWeebBoi60  4 dias atrás

                                                                                                                            those were two different dollars tho

                                                                                                                            • Exeno
                                                                                                                              Exeno  4 dias atrás

                                                                                                                              if the universe is infinite at least the big crunch or big rip wouldnt happen.

                                                                                                                              • Daniel Barka Long
                                                                                                                                Daniel Barka Long  3 dias atrás

                                                                                                                                Big crunch is preferable to infinite expansion and cooling because at least there is a possibility of restart if we go back to non-existence.

                                                                                                                                • Odil
                                                                                                                                  Odil  5 dias atrás

                                                                                                                                  I wish you were my science teacher. Maybe I would've pursued my science studies.

                                                                                                                                  • Edmundo Carvalho
                                                                                                                                    Edmundo Carvalho  5 dias atrás

                                                                                                                                    So what is infinity minus infinity?! :o

                                                                                                                                    • Sterlin Christabel
                                                                                                                                      Sterlin Christabel  13 horas atrás

                                                                                                                                      When infinity knows we are gonna subtract infinity from him, he multiplies himself with two so that he can stay infinite even if we do it. That's how it works

                                                                                                                                      • Carlos Mata
                                                                                                                                        Carlos Mata  5 dias atrás

                                                                                                                                        And so on...

                                                                                                                                        • AlexHydra2048 _
                                                                                                                                          AlexHydra2048 _  5 dias atrás

                                                                                                                                          *Math has been disabled due to an exploit*

                                                                                                                                          • Mr.Cringe Kid
                                                                                                                                            Mr.Cringe Kid  5 dias atrás

                                                                                                                                            So what's 0 times infinity?

                                                                                                                                            • Sterlin Christabel
                                                                                                                                              Sterlin Christabel  13 horas atrás


                                                                                                                                              • Wthishapping Teen
                                                                                                                                                Wthishapping Teen  5 dias atrás

                                                                                                                                                Me at 6 arguing with someone else: NOTHING IS BIGGER THAN INFINITY other kid: INFINITY AND ONE IS. me:NO WAY other kid: YES Michele out of nowhere: Actually

                                                                                                                                                • Joseph R.
                                                                                                                                                  Joseph R.  5 dias atrás

                                                                                                                                                  Him: Rips up a dollar bill and gives a really intellectual lesson designed to teach and inspire... Me:...wait....thats illegal.

                                                                                                                                                  • FerreroKeksKanone
                                                                                                                                                    FerreroKeksKanone  5 dias atrás

                                                                                                                                                    So i really Need to say: the paradox isnt paradox! At first lets get to Ur presentation. I really Like that mysterious way and was so happy that u did not make that one mistake i have seen. There were people using The Order Ruud so they did a step backwards and did not pay attention. But There's a Problem. By turning the sphere u are changing the first Letter, that means u are not using the Same color, and then we see that u only recombine the Same things as before. But There's another Thing. Each color describe the whole sphere even if u have one starting point. So if u devide the colors u would end up With copiing the sphere and not creating something more out of less. This Was Hard to See but if u have a close Look at the facts U'll See. It's Like that choclate, confusong. Ya i Just wanted to make those things clear to help people understand the mistake in the paradox not being a paradox. Sorry for my grammar mistake i am german😂

                                                                                                                                                    • FerreroKeksKanone
                                                                                                                                                      FerreroKeksKanone  11 horas atrás

                                                                                                                                                      @Sterlin Christabel but what is fact: the banack tharski paradox is meant to Show that the Mathematik way how we See Room is wrong. If u Just watch at it with Mathematic it actually works, because u cant Look at Room with using one point. It puts Stress on the Problem between math and physics. In Physik it does not work in math it does. What is wong🤷‍♂️

                                                                                                                                                      • FerreroKeksKanone
                                                                                                                                                        FerreroKeksKanone  11 horas atrás

                                                                                                                                                        @Sterlin Christabel i actually dont Know if im right. Is Just wanted to give my understanding of the paradox so that other can correct me

                                                                                                                                                        • FerreroKeksKanone
                                                                                                                                                          FerreroKeksKanone  11 horas atrás

                                                                                                                                                          @Sterlin Christabel ya we can

                                                                                                                                                          • Sterlin Christabel
                                                                                                                                                            Sterlin Christabel  12 horas atrás

                                                                                                                                                            @FerreroKeksKanone 491k ppl were mind blown over this shit. That's hilarious .Are we the one's who didn't get it well or are we the one's who's got it better. I'm totally confused

                                                                                                                                                            • Sterlin Christabel
                                                                                                                                                              Sterlin Christabel  12 horas atrás

                                                                                                                                                              @FerreroKeksKanone This is the same fucking thing that i was thinking . And there is more bullshit mistakes to this paradox that makes it a total theoretical bullshit .Can we discuss?

                                                                                                                                                              • luna
                                                                                                                                                                luna  5 dias atrás

                                                                                                                                                                bitch what

                                                                                                                                                                • Ansh Aryan
                                                                                                                                                                  Ansh Aryan  6 dias atrás

                                                                                                                                                                  michael: ... or can you? *ah shit, here we go again*

                                                                                                                                                                  • Reind playz
                                                                                                                                                                    Reind playz  6 dias atrás


                                                                                                                                                                    • Like for No Reason
                                                                                                                                                                      Like for No Reason  6 dias atrás

                                                                                                                                                                      Spinnin' head

                                                                                                                                                                      • Marcio Jorge
                                                                                                                                                                        Marcio Jorge  6 dias atrás


                                                                                                                                                                        • Potentially Famous
                                                                                                                                                                          Potentially Famous  6 dias atrás

                                                                                                                                                                          now you can get 2 dollars from 1

                                                                                                                                                                          • Roblox Scamming
                                                                                                                                                                            Roblox Scamming  6 dias atrás

                                                                                                                                                                            Why did you have to use BRclip premium

                                                                                                                                                                            • lachie Bosman
                                                                                                                                                                              lachie Bosman  6 dias atrás

                                                                                                                                                                              People: the chocolate thing is fake Michael: *infinity -1 is still infinity*

                                                                                                                                                                              • Penguin1098
                                                                                                                                                                                Penguin1098  6 dias atrás

                                                                                                                                                                                13:42 RIP John Marston

                                                                                                                                                                                • markus salling
                                                                                                                                                                                  markus salling  6 dias atrás

                                                                                                                                                                                  This was incredibly mindblowing

                                                                                                                                                                                  • Sterlin Christabel
                                                                                                                                                                                    Sterlin Christabel  12 horas atrás

                                                                                                                                                                                    Pay close attention to the facts and think over. You'll understand the whole thing is theoretical and practical bullshit. This paradox have a lot of loopholes that makes the whole paradox impossible

                                                                                                                                                                                    • Zachary Denomme
                                                                                                                                                                                      Zachary Denomme  6 dias atrás

                                                                                                                                                                                      Wow infinity is weird Michael: no u

                                                                                                                                                                                      • M
                                                                                                                                                                                        M  6 dias atrás

                                                                                                                                                                                        Aka the matrix

                                                                                                                                                                                        • WyKills 9898
                                                                                                                                                                                          WyKills 9898  6 dias atrás

                                                                                                                                                                                          Real life money duplication glitch

                                                                                                                                                                                          • Marcus Engel
                                                                                                                                                                                            Marcus Engel  6 dias atrás

                                                                                                                                                                                            min(13:55) "U" would land on "LUR" correct???? Unless I'm the only one looking at this as a grid. where a unit is measured by "L,R,U,D"

                                                                                                                                                                                            • Edgar Nackenson
                                                                                                                                                                                              Edgar Nackenson  4 horas atrás

                                                                                                                                                                                              @Sterlin Christabel It's not remotely random. You actually mark every single solitary point that's reachable from a given starting point after finite steps. All these points are well defined in location, so not at all "anywhere". And, given the structure of the motions, they never land you in the same place twice. You actually miss an uncountable infinity of sphere points through this mechanism, but then you pick an unlabeled point and start again.

                                                                                                                                                                                              • Sterlin Christabel
                                                                                                                                                                                                Sterlin Christabel  4 horas atrás

                                                                                                                                                                                                @Edgar Nackenson That's randomness according to BT Paradox. The points could end up anywhere

                                                                                                                                                                                                • Edgar Nackenson
                                                                                                                                                                                                  Edgar Nackenson  10 horas atrás

                                                                                                                                                                                                  @Sterlin Christabel The units are exactly the same. They just yield different destinations. You can try this out yourself, if you like. Take a sphere or something, and trace a line up, right, down, and left, using a single large unit to exaggerate the effect. You will not land where you started.

                                                                                                                                                                                                  • Sterlin Christabel
                                                                                                                                                                                                    Sterlin Christabel  12 horas atrás

                                                                                                                                                                                                    @Edgar Nackenson No, they won't. The units are the same. Other wise they're not accurate

                                                                                                                                                                                                    • Edgar Nackenson
                                                                                                                                                                                                      Edgar Nackenson  3 dias atrás

                                                                                                                                                                                                      They would be the same on a grid drawn on a plane. On a sphere though, they land on different places.

                                                                                                                                                                                                      • The Colinizer
                                                                                                                                                                                                        The Colinizer  6 dias atrás

                                                                                                                                                                                                        my brain just died