# 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

Chinmay43 minutes agoyou misunderstood the infinity

Brandan McAdooHour agodoes that money thing work

a b2 hours agoWhy do I exist ?

r4ymind2 hours agoCan infinity fit in infinity? Like would a infinite number of... I don’t know... CARS, fit in the entire infinite universe?

Natalie Lemmer3 hours agoDid he just tear up an authentic dollar bill? I believe this is considered "destruction of government property!" Not cool, lost me at that point.

Note 73 hours agoWhat did that dolar do to deserve such torcher?

P R4 hours ago^{+1}1:15 I knew you were going to say that.

Javier Delgado5 hours agoBruh the dictionary blew my fucking mind

Zac5 hours agoIf you havent, get baked and LISTEN to vsuace, dont watch, listen. Best experience of my life

I’m petrified of Nipple chafing5 hours agoWow I’m re watching this again and I’m so proud of myself because I get most of what your saying lol

datgamerian loquendo7 hours ago1:32 **Third World scream**

There is no God. Just think about it OK!!8 hours agoThere is no God. Just think about it's brain died

There is no God. Just think about it OK!!8 hours agoMy brain is having a heart attack 1:48

Tech and Games Online9 hours agoRURU??

Lance Lote10 hours agoOk....so....what’s your point?..

Sean O'Reilly10 hours agodollar thing doesnt work tf

Bart de Vries10 hours agoInfinity is everything I hate about math.

Reverse Gamer11 hours agoMe: You Vsauce destroyed my childhood.

Vsauce: Or did I?

Me:*struck by a unfilled hole*

Daemon Nice13 hours agoIt is as Alfred North Whitehead wrote in his Introduction to Mathematics. Mathematics is an abstraction.

Ayaan Nashid13 hours agousclip.net/video/dkf0uirw2jk/video.html

Enavoid15 hours ago21:50 OMG WHY THERE'S A BRAZILIAN SONG IN VSOS AWOJFHUSIAHDFUYSAGFDYISAGIUFGAHIHGWIUDHGAIUWHDIURAHEWimcrying

Namelezz16 hours agoTiktok

I'm An Alien17 hours ago*O R C A N Y O U?*

sampson lam18 hours agowhat is the music's name start from 21:46 ?

bogdan dan19 hours agoI am not sure mathematics does agree with this theory.

The reason why the points defining a circle or sfere is infinit is that the volume or 2d dimensions of a point doesn't have a fixed value, so as it says at the begining of the video it is about the infinite sizes that a particle can have.

If we asign any value, lets name it "y", which could be any of the infinite posibilities for a point we extract from a sfere and if we define the volume of the sfere at a certain density with X.

if we substract y from x, we know y is a positive number. If Y would be smaller than zero, when we would extract it we would actualyy adding it to the sfere ( x-(-z) = x+z). So y>0.

Acording to mathematics, if x>0 and y>o, then x-y=x is false because if x-y=x => y=0 and you cannot extract 0 from anything.

0 = nothing.

In conclusion, mathematicaly it is not possible.

When you asign names to the points is similar with the cat in the box which can be dead or alive until you open the box. This only gives 2 posibilities for the cat to be dead or alive untill you open the box, but you do not have 2 cats in the box. When you open the box you get to the real world and you can state one of two theories:

1 - "The cat is alive"

2 - "The cat is dead"

If you are not a doctor and you are not sure if the cat is dead or alive, just don't say anything about it.

Jihyos Fan19 hours agono sense of understanding at all

Darius Z21 hour ago^{+1}so if i got a girlfriend and split(sexual intercourse) her with my buddy vsauce i will get another girlfriend? and if i split again with another person i will get another girlfriend too? that makes sense, now i know how 5 babys are born every secon, it's just so much pussy

Darius Z21 hour agopussy, ass, dick, . . . .. they are real!! and INFINITE

Emanuel Gonzalez21 hour agofuck me this shit got me confused time to relax with some porn

Makrillol23 hours ago9:48 Whenever I visit Canada

JUANDay ago^{+1}i’m watching this high right now

kgyu993Day agoWouldn't LUR and U take you to the same place if each line is the same exact distance? like the L and R are cancelling eachother out still

Vivek BhanaDay ago^{+1}All this just for infinite chocolate

Vivek BhanaDay ago^{+1}VSause you’ve triggered me into a seizure

247 AnglersDay agoInfinity - Infinity = 🧐

Owen McDonaldDay agoWow my head is spinning rn

MutumDay agoRUU, DRR , LDD, ULL ;)

ha get it?

Zachary AugiDay agoWhat.

BenBMBDay ago^{+1}Rip that dollar in the intro

LogicalGamerDay ago1:31 dude I could have used that

Grace With an RDay agoThis man got me thinking all sort of science conspiracy’s

500 Subs With No Videos ?Day ago^{+1}1:30

For a heartbreak😢

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

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

David BoyntonDay agoCan u be my science teacher

Logan BeaversDay agoSo I know I may be wrong on a number of levels, but here is what I've always thought about infinity;

There are different infinite's, hence x+1 does not equal x (follow me here)

So infinity plus one does not equal infinity because there is an infinite amount more of possibilities because of the infinitly small decimals with the added one, (.0 barred 1) This is true because infinity is a quantity not a function

Say 1000=infinity

infinity is less than infinity plus one because infinity being a quantity, like 1000

This is how 2 is greater than one

Before if there was only one infinity and it was equal to itself, then 2 and 1 would both have the same infinite amount of possibilities between them and 0. And that begs the question, how is two greater than one?

If infinity is not a function, but a quantity, then 2 does not have the same amount of possibilities away from zero as 1 does.

So that leaves me with another question, how is 2 greater than 1 if infinity does not equal infinity?

Does it just mean two is the same infinite between zero and one, just plus itself?

And what is one? Is it just the infinite possibilities between 0 and the first whole number, or is it a question so simple a child could answer.

Logan BeaversDay agoI am lonely

Logan BeaversDay agoWhy thank you

Logan BeaversDay agoYou're so smart

ClickachuDay ago^{+1}In theory sphere or circle have infinite amount of particles, but in reality they're made of countable amount of atoms. And even if atoms themselves are made of infinite amount particles inside them, the ball will still be made of countable amount particles, so even if we had technology to move ever single atom, this theory/experiment wouldnt work/be possible to do. So maybe this is the answer, that even real life ball made of infinite amount of particles is made of countable amount of atoms. So we can't do such things because of atomic structure of world, BUT... we still don't know too good what's inside the atom, maybe inside it there is infinite amount of particles and there it will be possible. But maybe we can't measure or understand atom, becasue what's inside it is just in another "dimention", like electrones can act like waves, not particles, be in superposition in 2 places or go into "quantum transition", maybe between our world and this what happenes in atom there is such "transition" and we coul make infinite amount of atoms from single atom, but in our "atomic" world maybe there is infinite amount of space, atoms and i beleive it since in theory there is even infinite amount of inifinities, but still, every ball that we could possibly observe would have limited amount of particles. I also have another conclusion, that maybe our world, even with countable amount of atoms in every item like ball/sphere i talked about, but our universe itself is infinite big and made of infinite amount of atoms but still we can just be like this infinite atom - we can't measure or observe the shape of universe becasue it's infinite, but in fact we could just be inside some atom in another universe and be a single particle in "that" outside world, but still infinite in our world and carrying information and ability to recreate every other atom, since they all contain infinite possibilities. In this theory every atom should be the same, or maybe it's just impossible to compare two infinite things to each other, maybe they have same information, but in different direction/order (or rotation like in video ;) ) So my final thought is that we are living in a "neverending matrioshka doll", that we are infinite universe, but also in every single atom there is (same?) universe like ours and so on, also in other scale direction our universe could be just a single atom of other universe or even same as ours. Maybe in all these universes everything happens exactly the same, and we are just living in a fractal like holograme. Also maybe in our world every single atom of their infinite count contain every possible other outcome, like world where i didnt write this coment. But i'm not 100% sure if every possible outcome exist in parrarel universe. Maybe there is only one "timeline", and everyting actually happened and we don't even have any impact on future, time doesnt exist and whatever we do, even on purpose to do it opposite we will do what was ment to be. But this is impossible to prove, at least in our lifes, since i beleive our soul is infini.. immortal. Sorry for using word infinity for infinite amount of times.

Clickachu16 hours agoMe too, I'd like to write a book one day, but it's hard for me to concentrate and write a long text but i can do it unconciously while writing a comment hah +Darius Z

Darius Z21 hour agou love writing do you?

Haseeb ZarrarDay agoHi ,

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

ElenaDay agoWhat the actual fuck I’m too dumb for this

Henry ChavezDay agoeveryoine else: fortnite top 10 plays

me, an intellectual: this

later plebs

Emerald BrosDay ago^{+1}michal: obviously you can not cut up a chocolate bar and rearrange the pieces to more than your started with.

me: that's quite obvious

michal: or can you

TRIGGERED

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

TheMiner1501Day agoWhy does everyone forget he destroyed money?

ShaggyDay ago^{+1}What happens if you take infinity from infinity -from someone called sparrow gaming -so this gets more views

z. tuğçe özcanDay agomuhteşem magnificant ♥️

Sheep GameDay agoThat hotel paradox hurts my brain

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

*1) Propoganda, by Edward Bernays*

*2) The Crowd; A Study Of The popular Mind, by Gustav Le Bon*

*3) The Secret Teachings Of All Ages, by Manly P. Hall*

*4) The Creature From Jekyll Island, by G. Edward Griffin*

#UnibomberManifesto

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

Jonny Grow LatelyDay ago*LOL, no...common sense says...* #FlatEarth

dat personDay ago...what?

Lovro Planinšek /. Project 5421Day agoThe thing that doesn't make sense to me is how can you turn an object to change what it physically is?

Patato KeftesDay ago^{+1}Wow, total mess.

Numbers express fraction, quantity or value.

Infinity cannot be quantified therefore is not a number, cannot be broken into fractions and has no absolute value.

Trying to engage in math including infinity is like trying to add dollars to celcius degrees and come up with a quantifiable result.

In essence infinity is a sum where ANY number has a value of nothing at all.

Logan BeaversDay agoI don't know, you can't quantify pie, since it goes on forever...

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

Shivang GulatiDay agoIs he an alien?

Hello! My name is LOLDay agoI *LITERALLY* got a headache after watching this

Davino Sayaza2 days agoNot gonna lie, after finished the video, I felt something in my head and I don't know what...

cammiecookies2 days ago1:48 what the infinity pasta

Timothy Top_Notch_Tim_72 days agoHow to get rich 101

Lawrence Joiner2 days ago12:00 the beginning of the Swastika Paradox.

v_killer72 days agoI get this after viewing several times, even with my level ov unintelligence.

R Herman2 days agoThis was really confusing. I can understand why there were so many thumbs down. they probably didn't understand it. However, the fact that you were able to articulate this with so much passion & enthusiasm was incredible. Like I said, I will have to watch it again. Hey thanks!

Liam Neilson2 days agoIf we were able to go faster than the universe expanding would be bump into something? How would that work???

nathan glover sr.2 days agoThis is the tip of the iceberg to antimatter production

Hill Top2 days agoI'm interested in this but after watching this video I realized I'm too stupid.

Shaun Ambrose2 days agoMy head hurts...

James Armstrong2 days agoU melted my mind

Jackson Kelly2 days agothat dollar shit got me hurt. i tried it and just ruined a dollar

Thomas Tang2 days agoWow

Vincent Snijder2 days agoThis is mind-blowing on so many levels

Siddl •2 days agoCost of video : $3

Janko Djuric2 days agoIf you think closely

Present is infinite. (hope so)

LightSpirit242 days agoThose points have no size to them. They are not wide or thick or tall. So you can never fill up the surface of the sphere.

Drunk Muffin2 days agoHow small is a dot?

Amelia2 days agoMind: blown

Dynamo2 days ago1:50 yall saw the k and the f right yeah my homeboy u thought u was slik

Nicholas Soutter2 days agoMy fucking brain broke

Jude Yovichin2 days ago^{+2}This video is great, but there is a logical fallacy when he starts talking about the circle. There are an infinite amount of points on a circle because a circle has dimension while a point does not. Therefore, the poor he “told off the circle” would have taken up no space on the circle in teh first place and he wouldn’t have had to fill the space left behind. There was no space left behind. All this means that he didn’t create more by taking out of the circle, he just didn’t take off of the circle.

Jude Yovichin2 days ago^{+1}Banach-Tarski’s Paradox isn’t supported by mathematical theory and, by extension, can never be possible in the real world as he suggests at the end of the video.

Jude Yovichin2 days ago^{+1}This logic can also be applied to the spheres, you can’t make a sphere out of points, because points have no dimension, but the sphere has to. You can’t make something out of nothing.

Ivar Melker2 days agoThe point would end up in the same location if you did Left Up Right Down...

Thescott162 days agoPerhaps Joseph and Mary should have gone to Hilbert's Hotel then...

Zach C2 days agofavorite video by far return to it every year just to get my brain going again

jervee saclolo2 days agoWhy am I watching this? I dont even understand a single thing

Yogscast Minecraft_Lewis03 days ago14:00 Isn't the red point and the purple point same? Assuming that you move the same distance in each direction, L and R being opposite would cancel out and left with U. This makes LUR and U the same point. (Unless the distance you move is different)

TenemaesLament3 days ago^{+2}All the dislikes are from physicists

Cici Bracken3 days agoEarth is flat don’t @ me

Papa Beasticle3 days agoI'm not sure if I gained a brain cell or lost one at 7:19

BringThePizza3 days ago“The universe isn’t strange, we are”

-Vsauce

Ricardo Lopez3 days agoI want to get high and watch this so I can be fully Amazed by this..!!

Talo B983 days agoFor The infinity is equal to one and sphere and stuff watch past this point 11:23

SpyCommander133 days agoScientists have discovered a duplication glitch.

Joey O'Sullivan3 days agoMy fucking brain hurts

BuhVuhGuh3 days agoYou know, you could just go by the law of conservation of mass for that chocolate bar?

darthkarl993 days agoAnd this is why a good ;portion of extremly capable mathematicians through the centuries have suffered from various mental disorders. You kinda have to be a little crazy to get your head around math above a certain point.