# The Banach–Tarski Paradox

Q: “What’s an anagram of Banach-Tarski?”

A: “Banach-Tarski Banach-Tarski.”

twitter: https://www.twitter.com/tweetsauce

Instagram: http://www.instagram.com/electricpants

Kevin’s Field Day video: https://www.youtube.com/watch?v=1zARMZ08ums

Field Day: https://www.youtube.com/channel/UCRPktNf5vnBR1J4e7t1RUVg

Deep dream animation by http://instagram.com/NaderMakki/

If you like it, you’ll love this video also by Nader: https://www.youtube.com/watch?v=fJ9j_z2kXI0

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

Chocolate illusion video: https://www.youtube.com/watch?v=dmBsPgPu0Wc

related Numberphile videos:

sizes of infinity (includes diagonal argument): https://www.youtube.com/watch?v=elvOZm0d4H0

infinity paradoxes: https://www.youtube.com/watch?v=dDl7g_2x74Q

Vi Hart on types of infinity: https://www.youtube.com/watch?v=23I5GS4JiDg

Countable & uncountable definitions:

http://mathinsight.org/definition/uncountable

https://en.wikipedia.org/wiki/Countable_set

https://en.wikipedia.org/wiki/Uncountable_set

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

Banach-Tarski proofs:

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

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

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

Banach-Tarski explinations online:

http://www.irregularwebcomic.net/2339.html

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

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

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

http://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

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

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

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

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

http://dgleahy.com/p47.html

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

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

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

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

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

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

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

Is math invented or discovered?

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

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

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

BOOKS:

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

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

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

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

Music by http://www.youtube.com/JakeChudnow and

http://www.audionetwork.com

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My nigga lost 3 dollars making this video

The dollar thing was a joke, right? You tryna make people rip up money?

Life is fucking weird

Things can grow to infinity but infinity itself doesn't exist in this universe.

What about Quantum Theory?

Makes me think of how we exist…our universe replicating itself from one moment in time to the next…in an infinite amount of ways…leaving our previous copy in the past…continually…for an infinite amount of time………………Huh…Everything exists in a single moment of time…all focusing in with my perception of my moment in reality….

this should be named "how to go insane"

You spelled explanations wrong in the description.

Did anyone notice how he drew his eight?

what is the difference between RUL and U?

4:23 😂

can they get any worse

I don't think our language is able to even explain these things

This paradox explains exactly how parallel universes could exist.

The sphere is a 3 dimensional object, but When the points move, they are experiencing time. You can use any set of points to create an entire sphere separate from the original by introducing them to time.

If you apply this to life, each point represents a person — no two are exactly alike and the reproduction is infinite. There are infinite possibilities to how a single point could navigate the sphere through time, just like there are infinite ways one person could experience their life.

Then, to take it farther, if you look at each point on the sphere as being a galaxy in the universe, the entire universe can exist in a different sequence an infinite amount of times all at once.

Good day.

The serial numbers are different from each other…

Wait, I have watched this video like 10 times, and the one thing I never understand is that, while you are creating matter, you are still using energy to create it. I don't know if some has done the math on it, but it seems that to separate the spheres, rotate them, and put them back together would use more energy than the spheres would contain. It is still a net-loss of energy. You may be able to create matter, but what good does that do if it uses up energy? This is not my territory by any means and I might not be understanding how energy is used. Like, would the energy used to create the second sphere be altered in some way? I know that the first law of thermodynamics is that matter cannot be created and destroyed, which obviously the Banach-Tarski paradox breaks this rule, but that's the point of the video, that is why it is a paradox.

So I guess question is this, can the paradox be used to stop the heat-death of the universe? Can you use this paradox to create an object with more potential energy than it took to create it? I know that question is impossible to answer right now, but it is just a thought that crossed my mind.

What if you take infinity away from infinity? Is it 0?

I just saw a man rip up 2 fresh dollars right infront of my eyes…

1:37 different pieces

I thought that made sense, but then I realized "whaaaaaaat"

What song is 21:46 and 22:05?

For this mathematical theorie the dots you name have to be infinitly small right?

Because not only our measurements, but also the particles that make up any object, have a certain size and thus are larger than infinitly small, making this theorie unapplicable in any practical application.

mind got blown infinte amount of times during this video

It's just an argument is not a prove.

So everything is ….. everything? Our reality?

man I really love your long videos

"I really appreciate who all of you are"

how can something so simple make me feel so much..

I was doing well up to when he said vsauce michael here, it all went downhill from there.

Alright guys. Let's duplicate earth and we're done with over populations

And the point is ? Infinity means infinity ,no beginning,no ending, no mathematical solution period.

great video and all, but is there still a way to make more chocolate out of the chocolate you have?

ORRR CANN YOUU????

Hey Vsauce! Micheal here… or am I?

I love this guy and this channel. But the word infinite does not contain a third letter N after the final I and before the letter T. As such and by general consensus, at least here in the UK, the word is not pronounced IN-FIN-NINT.

It's IN-FI-NIT or IN-FY-NITE (depending on context).

kids in africa couldve lived a day with that money wow

"Q: "What's an anagram of Banach-Tarski?"

A: "Banach-Tarski Banach-Tarski.""

This is absolutely brilliant.

it's all mathematical!