# The Banach-Tarski Paradox

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**Published on Aug 1, 2015**- Q: "What's an anagram of Banach-Tarski?"

A: "Banach-Tarski Banach-Tarski."

twitter: twitter.com/tweetsauce

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Kevin’s Field Day video: usclip.net/video/1zARMZ08ums/video.html

Field Day: usclip.net/channel/UCRPktNf5vnBR1J4e7t1RUVg

Deep dream animation by instagram.com/NaderMakki/

If you like it, you'll love this video also by Nader: usclip.net/video/fJ9j_z2kXI0/video.html

Chocolate illusion: mathandmultimedia.com/2014/07/22/explanation-infinite-chocolate-bars/

Chocolate illusion video: usclip.net/video/dmBsPgPu0Wc/video.html

related Numberphile videos:

sizes of infinity (includes diagonal argument): usclip.net/video/elvOZm0d4H0/video.html

infinity paradoxes: usclip.net/video/dDl7g_2x74Q/video.html

Vi Hart on types of infinity: usclip.net/video/23I5GS4JiDg/video.html

Countable & uncountable definitions:

mathinsight.org/definition/uncountable

en.wikipedia.org/wiki/Countable_set

en.wikipedia.org/wiki/Uncountable_set

Banach-Tarski on wikipedia: en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox

Banach-Tarski proofs:

math.uchicago.edu/~may/REU2014/REUPapers/Robinson.pdf

www.math.hmc.edu/~su/papers.dir/banachtarski.pdf

people.math.umass.edu/~weston/oldpapers/banach.pdf

Banach-Tarski explinations online:

www.irregularwebcomic.net/2339.html

www.kuro5hin.org/comments/2003/5/23/134430/275?pid=5#10

skepticsplay.blogspot.co.uk/2010/05/doubling-sphere.html

austinrochford.com/posts/2014-05-14-banach-tarski-paradox.html

www.math.cornell.edu/~mec/Summer2009/Whieldon/Math_Explorers_Club%3A__Lesson_Links/Entries/2009/7/28_Lesson_6%3A__Whats_an_Anagram_of_Banach-Tarski.html

rachellevanger.com/index_files/BT%20Animated%20Presentation%20Web.pdf

quibb.blogspot.co.uk/2013_03_01_archive.html

blog.computationalcomplexity.org/2011/04/what-did-banachs-wife-think-of-banach.html

geopolicraticus.wordpress.com/tag/banach-tarski-paradox/

dgleahy.com/p47.html

www.math.hmc.edu/funfacts/ffiles/30001.1-3-8.shtml

Cayley graph animated gif: twitter.com/GIFsofWikipedia/status/624202342259240960

Hilbert’s hotel on wikipedia: en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

types of infinity: www.xamuel.com/levels-of-infinity/

set theory and quantum physics: link.springer.com/article/10.1007/BF02213427#page-1

LHC gif: cms.web.cern.ch/news/lhc-data-be-made-public-open-access-initiative

Zermelo-Fraenkel axioms of mathematics: mathworld.wolfram.com/Zermelo-FraenkelAxioms.html

Is math invented or discovered?

phys.org/news/2013-09-mathematics-effective-world.html

www.scientificamerican.com/article/is-the-universe-made-of-math-excerpt/

more deep dream images: www.reddit.com/r/deepdream/

BOOKS:

The Pea and the Sun: www.amazon.com/The-Pea-Sun-Mathematical-Paradox/dp/1568813279

The Outer Limits of Reason: www.amazon.com/Outer-Limits-Reason-Science-Mathematics/dp/0262019353

Why Beliefs Matter: www.amazon.com/Why-Beliefs-Matter-Reflections-Science/dp/0198704992

Things to Make and do in the Fourth Dimension: www.amazon.com/Things-Make-Fourth-Dimension-Mathematicians/dp/0374275653

Music by usclip.net/user/JakeChudnow and

www.audionetwork.com - Science & Technology

The GhostCodes3 hours ago1:29 R.i.p dollar 💵

Rorin Eaton5 hours agoHow to make the paradox work.... Fractals. Nuff sed

Rorin Eaton5 hours agoYou 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 995 hours ago^{+1}How bout infinity-infinity

AnimatedCatastrophe5 hours ago*funky music*

AlexBirdieBronze6 hours agoInfinity x infinity!?

Sean Solo9 hours agoVsauce: "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 Night9 hours agoI used to have trouble sleeping. Now, I watch Vsauce videos like this one as a great nighttime paradox! great video!!

C Ollivier9 hours agoYes, but... if infinity + 1 = infinity. Does infinity - infinity = 1 ?

amethyst dawn10 hours agoThe 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 Castro10 hours agoThat relaxing brazillian music at the final dude, great vibes

Gannimal10 hours agoVsause must be rolling in cash if he’s able to just rip a dollar into 5 pieces

Bledoston11 hours agoI bet rich people in the future will use this to clone their brains.

Dawit Moges12 hours agoBro

Ashton Vlogs13 hours agoI’m bout to duplicate some fucking money bro 😭😭

Ijin dela13 hours agoFor 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 Cashion13 hours agoAwesome trippy stuff. I loved it.

Pandała14 hours agowhen even the perfect real world has bugs...

usclip.net/video/_3JXCiWVCcg/video.html

Charles Leonito L. Iringan IV15 hours agoMy dollar NOOOOOOOOOO

Lucas Jarrett15 hours agoUm... 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 IV15 hours agoCan 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 K15 hours ago^{+1}*i have reached an end of my intellectual capabilities*

Ruicheng Kelvin Lu16 hours ago14:00 LUR=U

Peter Alves19 hours agoNice Brazilian jazz :3

AphidCC22 hours agoUp Up down down left right left right B A start.

The way tog land on same spot

Legend Graal23 hours agoSo, 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 jones23 hours agoThe orange and blue dot will end in the same place

Michael StevensDay agoexcept that Right Up Left would end up in the exact same place as Up

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

Tessanique WilliamsDay agoWhy can't I watch it😠😠😭

rupert888Day agoWow, infinity is infinite. Maybe I'm just not smart enough to understand why this is a paradox.

Caden :PDay ago^{+1}Or can you

**strange music plays**

VarnoxDay ago13:45 whatcha making there Michael.

Richard HeesterbeekDay agoif 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?

Brett101792Day agomy broke ass cringed when he tore the dollar 😂

Bre McCoyDay agoOnce 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 SurvivalDay agoVSauce: Or can you? *breaks bill into five pieces and makes two bills*

FBI: *breaks down door* FREEZE! YOU'RE UNDER ARREST FOR DEFILEMENT OF US CURRENCY AND CREATING ARTIFICIAL INFLATION SIMULTANEOUSLY!

Nickx13Day agoFirst vsauce I saw, tripped the shit out of me

Max NievesDay agoGuess ill just sell my brain

starless night sky lightless sunriseDay agoInfinite is a virus it makes a copy that is diffrent or the same as its self

Vanessa HayesDay agoah yes

Math

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 NotHereDay ago1:48 I could've used that ;-;

Exon VidzDay ago4:22 me writing homework

imogen taitDay agoso infinite things come and go as they need to be used?

Iris AriolaDay agoIf 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 WoganDay ago18:56 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 677890Day ago7:30 the number of unsatisfied guests it probably also infinity if they have to switch rooms every time someone new comes in lol :)

Jose D2 days agoi rather get laid! same as sex? maybe?

Muzin Kafi2 days ago^{+1}1:07

oh shit, OOOOH SHIT, HE ABOUT TO START, STRAP IN BOIS

Brett Palmer2 days agoDoes this mean we can use the infinite stones to make more infinite stones?

Casey Bade2 days agoVsauce hurts my brain

DONNAR FETT of the QUANTUM CRAZE PHAZE2 days ago^{+1}Some of this i understand.....most I do not at all...I'm to (stoned, tired, stupid, indifferent, _____fill in the blank).

Arthur Grella2 days agoMPB in the end.

timwins312 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 Ocelote2 days agoIs there a F word in that Dictionary?

No better Place or time to use it.

Digital Dafydd2 days agoI wish we were shown vsauce is math class

Melvin Sandberg2 days ago🤯

Ryszard Janku2 days ago1:20 Thats buttock

Adventure of Pathan2 days agoAfter 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'vo2 days agoTo ten mądry z TVGry?

strangelf 478292 days ago8:56 you can explain this shit but you can’t count from 0 to 11

1-2-3-4-5-6-7-8-9-11

strangelf 478292 days agoInfinity - infinity

Joshua Eppinga2 days agoInfinity + 1 = infinity you spiled your other video!😂

Eobard Thawne2 days agoThe Banach tarski paradox might apply to sub atomic particles but can't act outside it ,

Eobard Thawne2 days agoWhy 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 Thawne2 days agoThe 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

Boom2 days agoWhy is it 360p

Colten Roney2 days agoImma try that dollar trick with my hundreds

Anupam Srivastava2 days ago18:00 Lost it. I believe you.

Badges of Shame2 days agoThe rendering won't allow me to load this video.

Dah Butter Asian2 days agoMy brain hurts... ima go back to watching Michael Jackson moonwalk in reverse thanks...

derrenk2 days agoGreat vid, many thanks for the headache :)

Enzoderbaecker2 days agoI'M STRAAAAAAAAAANGE!

C.I.A2 days agoMy brain hurts :(

Dima2 days agoThe 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.

MuffinsAPlentyDay agoWhat 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.

DimaDay agoMuffinsAPlenty 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.

MuffinsAPlenty2 days agoI 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 Nackenson2 days agoWhat'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 cheats2 days agoWhaaaaa?!?!?!

Gay Dinouo3 days agoThere is an infinite number of blah blah blah

-Vsause

Levi Cukas3 days agogoddamn this hurts my brain

Christopher Shelton3 days agoYou can get it only if you follow along very carefully

Giulian Zorzi3 days agoThis made my head physically hurt. Actually! Good job!

Big Braddah Bell3 days agoTrying 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 ? 😜

Christopher Shelton3 days agoLol

Christopher Shelton3 days agoYeah make that a pepperoni

Babachew Bob3 days agoEven 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.

TokyoEthan3 days agoThe 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.

Technoultimategaming3 days agook got 3/4 of video

shadow knight3 days agoSo infinity minus infinity is infinity?

RightofLeftofLeft Eddie3 days agoAt 2:03, when he says "aaaannnd ourselves, " he makes a really weird gesture

RightofLeftofLeft Eddie3 days agoAm 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 Hand3 days agoThis is how fiat currency works :)

Sean Weintz3 days agoi 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 Nackenson3 days agoI 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 Weintz3 days agoum, 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 Nackenson3 days agoI 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 Stonewall3 days agoso... since infinity minus one is infinity... then (infinity) - (infinity - 1) = 0... not 1...

rip my brain

Edgar Nackenson3 days agoInfinity minus infinity is an indeterminate, meaning it can take on any value. The same is true of infinity-infinity-1.

Wes Netmo3 days agothe diagonal argument seems bs to me. real and whole sets are equal and both infinity

Edgar Nackenson3 days agoWhat do you think is mistaken in the proof?

Κωνσταντάς Χατζηκωσταράτσος3 days agoWhy do you keep starting the videos by shouting??

Down!!

DOWN!!

Indo Raptor3 days ago1:32

Mr krabs: NOOOOOOOOOOOOOOOO

Overwhelming Conundrum3 days agoThats not possible but its fun to think about.

Donald Cameron3 days agoPoints 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 Ninja3 days agoBeing a guest at Hilbert‘s Hotel sounds like a lot of work

gxzkc3 days agoI am not from america, but isn't destroying money a federal crime?

nick g3 days agoOuch

Judy Thompson3 days agomother of god

James4wd3 days agoTake 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