# The Banach-Tarski Paradox

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

A: "Banach-Tarski Banach-Tarski."

twitter: twitter.com/tweetsauce

Instagram: instagram.com/electricpants

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

Field Day: usclip.net/channel/UCRPktNf5vnBR1J4e7t1RUVg

Deep dream animation by instagram.com/NaderMakki/

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

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

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

related Numberphile videos:

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

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

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

Countable & uncountable definitions:

mathinsight.org/definition/uncountable

en.wikipedia.org/wiki/Countable_set

en.wikipedia.org/wiki/Uncountable_set

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

Banach-Tarski proofs:

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

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

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

Banach-Tarski explinations online:

www.irregularwebcomic.net/2339.html

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

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

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

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

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

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

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

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

dgleahy.com/p47.html

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

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

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

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

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

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

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

Is math invented or discovered?

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

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

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

BOOKS:

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

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

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

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

Music by usclip.net/user/JakeChudnow and

www.audionetwork.com - Science & Technology

Bing Boi23 hours agoThis is like particles 🤯

It’s PotatoMan5823 hours agoAll that work for an extra dollar

Rui Leite23 hours agoRip Vsauce free to see

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

alkiter1Day agoInfinity - Infinity = 0

Rui Leite23 hours ago= 0

Wolfgang BreitenseherDay agoInfinity. Lol. Concepts.

noobs dudesDay agoYou can’t drink lava *OR CAN U?*

Rui Leite23 hours agonoobs dudes you can.

AdiMeheDay ago@ 13:49 just imagine a combination of URR

Mayank TiwariDay agoWhat the fuck was this ???

ko harrisonDay agoLUR and U is the same point, isn't it? 13:22

The TrustDay ago^{+1}9:15 No hole will be left unfilled ( ͡° ͜ʖ ͡°)

die FabeDay ago11:31 i almost came xD

QbRtDay agoI'm seeing a lot of holes in this theory. For example, if there is an infinite amount of starting points on the sphere, then that SHOULD mean there are an infinite number of accompanying poles, even if some repeat.

Also, if you rotate the lefts to the right but left then right is a back track, you should only subtract the right. You didn't add but you subtracted and it goes both ways.

Subtracting both of them takes away an original direction that you started with and you are left with something completely different.

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

QbRtDay agoIf anybody can elaborate, debate, or whatever, please do.

Aydin WarnerDay agoJUST COLOR THE CIRCLE

Tzar Nicolas The IIDay ago1:38 That's a felony.

OsvaldoKS100Day agoanyone else see the dogs or animals in the outro?

Philosophy Of ÉlivágarDay agoHilbert's Hotel would have been useful during the Syrian immigration crisis

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

I AmDay agonooooooo not the do;;ar

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

FireyFly RBLXDay agodude that’s deeeeeeeeeeeeep

Cam WardDay agoThis is all bullshit tbh

theHardInGame ytDay ago13:53

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

impendioDay ago^{+1}It's been years and I still think this is the single most genius work of digital media every presented. It literally changed my life forever.

Dhananjay SawantDay agoWhat if you subtracted an infinite number from infinty?

Dhananjay SawantDay agoBut, it would take infinitely long.

chris boyDay agoHow life lives is simple but what life if made of is compilated

DementiaDay ago^{+1}Excuse me what the fuck

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

Infinity is just a concept constructed by humans, just like math, and does not reflect the actual world.

Walter OstrowskiDay agoI wish the dollar thing was possible

Crazy TimeDay agowho also think they don't have enough brain to watch this

Vir KhannaDay agoI'm too stupid for this

azder bogdanDay agothe one with the dollar is a scam

its just hardDay agoOk.. so can i get the chocolate that i ate early back?

Lodevik TrijanaDay agoWhy do cut money?

PowDay agoHey Vsauce..

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

AL STANWOODDay agoPOLE HOLES.

HOLLOW EARTH CONFIRMED

Carson GrenadeDay ago2:27 bruv this is trippy

Carson GrenadeDay ago16:30 I can't be the only one that sees a big puple donut. I think thats because I'm hungry.

E E T Z J O S HDay ago^{+1}I don’t know why but the concept of infinity makes me depressed.

JerBearRidesDay agoDangerously close to an arrow swastika on that circle there michael.

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

Mask GuyDay agoHollllllyyyyy shit!

Beep Beep I'm A SheepDay agosong at 0:00

Why MeDay agoI'm too high for this shit

KuyakiDay agoThen...infinity makes everything .... insignificant?

SC4RCHDay agomy brain always fucked when i see him explaining paradox / other vid, like rly...

y - AnimatorDay ago- -= +

UnnamedDay agoWtf

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

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

BigTark69Day agoWell I was about to fall asleep.

Caleb PowellDay agoMind literally blown

Joseph MennitiDay agoThanks I just lost a fucking dollar

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

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

I like tacos.

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

EmmaDay agoscreams while he rips a dollar bill

MrPaceTvDay agoI thought , infinity is a gauntlet ...

I win 25 8 BoiDay agoI think that's how cells make more of them selves

Trumpwall666Day agoalso theoretically it halpens all the time, it’s just a nine month process.

Spencer_m1 MatijakDay agoDa faq

Trumpwall666Day agopoor people at the hotel, they have to move to a different room every time someone enters, and leaves

# n0mE SquadDay ago1:49 my reaction, I must do this and become rich. Got dollar did so got a ripped dollar worth nothing such 😔

Iamalex.j PotatoesDay agoScience is a helluva drug

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

Pinkpatty 76Day ago18:58 made me question my life’s own existence

Simonplayer 26Day agoI’m i the only one that is confused?

Marc Moncada2 days agoThe reason that this theorem remains a paradox and not a fact is because you cannot extract the abstract (0th dimension) from the practical (3rd dimension).

Dersu422 days ago^{+3}Once you start conceptualizing something in the infinite things get weird. The illusion of the paradox is that you have a starting point to begin with. Once one end of your list of potential moves becomes unbound there is no longer an effective beginning anymore.

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

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

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

FlatusOhlfahrt2 days agogreat vid!!

Billy Wheeler2 days agoSo this is how Jesus fed the multitude with only 5 pieces of bread and 2 fish. Now how to figure out how to use this theory for my finances. lol

Goldenempire2 days agoi -dont- hate myself

Marcus Henry2 days agoImma be rich

Goldenempire2 days ago*wow*

Ayaan Chowdhury2 days agoThese videos makes me feel smarter lmao

alfie Barton2 days agoWow

DIEHARD CH1EF2 days agoDoes infinity • 0= 0 or infinity

Duncan Alba2 days agoHWTOBASIC

Riley Weaver2 days ago^{+1}Micheals favorite term, 'or can you?'

SAMUEL Karlsson2 days agoIf I’d get a cent for every time you mind fuck me id be a millionaire

MrElf 052 days agoThese videos make me think about everything that cant be explained or known

damnryn2 days ago*No hole will be left unfilled.*

Isabella Cobb2 days ago^{+1}What

Dyl599092 days ago^{+1}Are we just gonna ignore how he drew that 8??

jolie dubreze2 days agoMicheal: *or can you?*

Mr: uh oh.

Infamous weeb2 days ago^{+2}lol is lol backwards

YungNiteDay agoInfamous weeb wow is wow backwards wow

Nuggetz2 days agoNow i see infinite from a whole new perspective.

Sean Beaudoin2 days agoV sauce is a liar when he ripped the dollar, the letter large letter left of George in the circle is different, the first one is k, the second is f

Werefox alpha2 days agoI can't believe they turned Michael's intro into a meme. His videos are AMAZING !

Avantika Rao2 days agowhen he ripped the money i winced

InnocentOnPaper2 days agoOpium is one hell of a drug.

hobi’s sunshine2 days agoMichael, MY BRAIN

scrappythenell2 days agoI actually disagree with this video. A line has zero mass and no width or depth, only length. So no matter how many lines you pull out of the initial ball into a new "ball", the second "ball" will always be either a single invisible line if each is set side by side, or and invisible, weightless mess of lines. 1(L) x 0(W) x 0(D) = 0. Zero times any number is always zero. So the second "ball" will never have any mass at all. If you pull out an infinite number of planes, you would theoretically be able to see it (assuming a zero mass plane was capable of somehow reflecting light), but it would still have zero mass. 1(L) x 1(W) x 0(D) = 0. The only way to create a second ball with mass would be to take a sliver of the original ball that has Length, Width, AND Depth. Once you do that however, every non-zero piece you take reduces the mass of the original ball. You would simply end up with two balls that are each half the weight of the original.

Rusty Roberts2 days agoI feel sick and my brain hurts.

Lewis Goodwin2 days ago^{+1}I'm going to cry man this video hurt my head

GameGod account2 days agoOfc it can't be real. The number of atoms on that sphere are finite!!!

Holy Cucumber2 days agoYeaaaaaaaaaaaaa wtf

InspiredKangroo2 days ago1:39 TEACH ME

ANUP SHARMA2 days agoElectrons