# The Banach-Tarski Paradox

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**Published: 01 August 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: brclip.com/video/1zARMZ08ums/video.html

Field Day: brclip.com/channel/UCRPktNf5vnBR1J4e7t1RUVg

Deep dream animation by instagram.com/NaderMakki/

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

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

Chocolate illusion video: brclip.com/video/dmBsPgPu0Wc/video.html

related Numberphile videos:

sizes of infinity (includes diagonal argument): brclip.com/video/elvOZm0d4H0/video.html

infinity paradoxes: brclip.com/video/dDl7g_2x74Q/video.html

Vi Hart on types of infinity: brclip.com/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 brclip.com/user/JakeChudnow and

www.audionetwork.com

*rips dollar bill* Me: FGJHFGHDYGFGFXFGCGHMVMH KyhtuyyrdddtfguknlhibftxsaYdtufTRYDUYBLUVSTSE

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.

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

1:17 the best , can you

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

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

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

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

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.

I came for the dogs in the end

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

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? 🤔

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

32-1 is 32

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.

They are, yeah.

I- I think you broke my brain

Ain't ripping money over illegal?

My brain hurts😣

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.

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

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!

@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

@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.

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 😂

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

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

Well, this was trippy

Destroying government property and counterfeiting @ 1:34

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.

1+1=1 - general relativity

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

This is total nonsense.

My mind is blown🤯

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

I cannot believe I actually learned something...

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

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

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

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

No, they know they are not dry

Twenty four minutes and thirteen seconds of me being completely confused

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

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

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

0:10 *BRAIN HURTS*

13:45 i thought michael was drawing a swastika

Apeirophobia anyone?

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

Or can you?

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

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

The hyperwebster knows where malaysia flight 370 went 🤯

12:19 Donkey Kong Country cheat codes

I have 2 brains now

I tried this with money and it actually worked lol

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

21:50 - trippy!

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?

but it doesnt tho i tried it with my money

those were two different dollars tho

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

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.

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

So what is infinity minus infinity?! :o

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

And so on...

*Math has been disabled due to an exploit*

So what's 0 times infinity?

Zero

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

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

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😂

@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🤷♂️

@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

@Sterlin Christabel ya we can

@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

@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?

bitch what

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

1+1=1

Spinnin' head

Amazing!

now you can get 2 dollars from 1

Why did you have to use BRclip premium

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

13:42 RIP John Marston

This was incredibly mindblowing

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

Wow infinity is weird Michael: no u

Aka the matrix

Real life money duplication glitch

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"

@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.

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

@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.

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

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

my brain just died