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 & TechnologyScience & Technology

Comments • 69 153

  • LORD TACHANKA
    LORD TACHANKA 3 minutes ago

    ♾-♾=0 or ♾?

  • John King'ori
    John King'ori 2 hours ago

    If you divide infinity by infinity, do you get one or is it still infinity

  • SuperNatetheawesome
    SuperNatetheawesome 3 hours ago

    how can Hilbert's hotel be full if there are an infinite amount of rooms

  • SuperNatetheawesome
    SuperNatetheawesome 3 hours ago

    what about infinity minus one?

  • Anubhav Yadav
    Anubhav Yadav 3 hours ago

    my mind is so blown away right now i cant even express it

  • elena mcelvenny
    elena mcelvenny 5 hours ago

    Wow, this was insane. Thank you.

  • Bourdin Antoine
    Bourdin Antoine 5 hours ago +1

    13:48 imagine if he made Right Right Up.

  • user name
    user name 6 hours ago

    someone translate it so i can understand

  • Jean
    Jean 10 hours ago

    Lile very much abstract mathematics. Thank you.
    Just one remark. I don't really get the need of including effects trying to be 'cool' (talking like a fool or accelerating cuts...). Sober is better.

  • Spastic fantastic
    Spastic fantastic 10 hours ago

    THERE IS NO END TO ANYTHING REEEEEEEEEEEEEEE

  • chofosho_99 cho
    chofosho_99 cho 11 hours ago

    Who's vsauce?

  • nite pipes
    nite pipes 13 hours ago

    BTW in our physical state you will never get more than you started with without introducing something more. Or at least manipulating to natural forces available to us on this planet.

  • nite pipes
    nite pipes 13 hours ago

    The catch is that in laymen s terms, the Tarski Paradox is twice as much info/material that is actually needed to complete the sphere to begin with. example- when he goes RUL the act of rotating the sphere cancels out the need to go the one step the the direction of the sphere turning. In other word the act of turning or rotating the sphere allows for enough information to be left over to build another sphere seemingly out of thin air. With out the introduction of the rotation you will not have 2 spheres out of one.

  • Lucas Blanco
    Lucas Blanco 14 hours ago

    Bossanova no final

  • Kelley Merrill
    Kelley Merrill 15 hours ago

    I keep thinking calling something a different name doesn't make it a different object. All points are poles. All poles are points. That's copying the sphere in different colors and pretending its a different object.

  • Alex Turner
    Alex Turner 15 hours ago

    I followed him until about 17:30

  • Your Best Nightmare
    Your Best Nightmare 16 hours ago

    Poor dollar

  • Ian Knowland
    Ian Knowland 17 hours ago

    Hey Mike, how does one determine the size of a "point" in this theory? Does a mathematical point even have dimensions? Because if the point you use has dimensions, then there WOULD be a finite number of them on the surface of a sphere, wouldn't there? I'd assume that there's only an infinite number of points if the size of the point is infinitely scalable. So would there be an infinite number of finite countable sets of points? Maybe the difference doesn't matter in the end, but I thought I would... point it out.
    EDIT: I don't do math, FYI. I just looked it up and for those wondering, no, a mathematical point has no dimensions. Carry on :)

    • MikeRosoftJH
      MikeRosoftJH 15 hours ago

      It's easy: a point has measure 0. Likewise, any finite or countably infinite set has measure 0. Assuming the axiom of choice, it can be proven that the Lebesgue measure is countably additive; any union of countably many disjoint measurable sets is itself measurable and has a measure equal to the sum of measures of individual sets.
      This does not apply to a union of uncountably many sets, so an uncountable set may have measure zero (such as the Cantor's discontinuum), or a non-zero measure (such as an interval), or an infinite measure (such as the whole set of real numbers), or no measure at all (assuming the axiom of choice it can be proven that there exist non-measurable sets).

  • Wyatt Denious
    Wyatt Denious 18 hours ago

    Yeah, the memes of Vsauce are great but I learn and more about things from here than in school

  • GlowstoneLove Pad
    GlowstoneLove Pad 20 hours ago

    11:20 the interesting part

  • HurricaneRider
    HurricaneRider 20 hours ago

    Yes we are strange. Bunch of atoms who managed it to understand themselves.
    It’s too obvious yet unbelievable but i think it’s not the first time that this universe happened. I think this is not only our answer why we live but for all species in universe. If it has the possibility for being self-aware and intelligent, it’ll try to understand everything; even tho it can’t it’ll find an answer. And in the end it’ll try to recreate everything and that shit starts from the beginning, somewhere else.

  • ArsenPatron
    ArsenPatron 22 hours ago

    1:36 tfw you commit a federal crime

  • Lonewolf 1129
    Lonewolf 1129 23 hours ago

    Wait that money thing works

  • Probrorule
    Probrorule 23 hours ago

    That's deep af my brain hurts

  • Music Fleen
    Music Fleen 23 hours ago

    We are finite beings in an infinite universe with infinite possibilities, with infinite thoughts ..

  • How Mull
    How Mull 23 hours ago

    There are proofs, that contain a divide by zero, which erroneously show that any number can be made equal to any other number.
    I wonder if this paradox is due to the same error of ignoring the fact that you can't perform operations on infinite quantities. In other words, you can't make a cut at an offset that is based on an irrational number because irrational numbers are infinitely long and so it would take an infinitely long time to make one cut. You would never be able to make a second cut. Once you allow operations with infinite quantities, you will get results that are undefined. If you ignore that fact that you can't do an impossible operation, then obviously you can erroneously conclude you can do anything.
    For example:
    There is a description in the video of filling a gap in a circle by shift points taken at a gap of pi but there is no way to pick a point at a irrational location. It would take infinite time to find that offset.
    There is another description of hotels with infinite rooms and infinite guests. 1st, there is no way to build that hotel because it requires infinite materials and there would be no atoms left to make hotel guests. If you move the first person out of the first room and then shifted the guess by one (room 2 to room 1, etc), there is an empty room because there will always be a room that you must shift a guest to. Since there are infinite guests and rooms, there will always be one empty room. It isn't a paradox, instead it is a mistake to think you can perform operations on infinite values.
    There are interesting twists.
    Any physically created circle is just an approximation that has errors that are too small for us to see because the perimeter of a real circle is an irrational number.
    The limit of "1/x" as x goes to infinity is "0" but "1/infinity" is not equal to "0". Note: "0 times a" is "0" but "1/infinity times a" is not a number because infinity is not a number. infinity is a description of a quantity which is not realizable.

  • Service
    Service Day ago

    @Vsauce.
    Did i miss something? Where is the missing mass added?
    Going straight to the model of the completed first stringsphere ->
    it gets broken down into 6 individual colors, giving it a 1/6 relation to mass. After rotating 2 and joining the rest you are left with 2 spheres of a 1/3 relation. - Meaning that though the products are visualy the same, they are both of a lower density than the starting point.
    Its just an illusion, enhanced by the number of strings drawn.
    So the chocolate bars would, despite appearing the same, have gone from,a "solid" bar, to basically a set of barshaped maltesers.
    I am aware that this is just a visualisation, but the principle should be the same, no matter if you "count to infinity" or simply count to 1000. As the starting/resulting mass will always be 1/1.

    • Service
      Service Day ago

      MikeRosoftJH I get that, and we are on the same page.
      I was assuming it as an answer to the question of "cloning" the bar(or bill) into 2 bars (of the same volume/mass) as questioned in the first segment. I will have watch the video again to check, but i believe he said it might be possibly at some point, thus my confusion.
      (Could easily be that i just missheard him)

    • MikeRosoftJH
      MikeRosoftJH Day ago

      No. This is not a matter of physical quantities like mass or density. For obvious reasons, this process can't be performed with physical objects (which consist of elementary particles and atoms and molecules, and therefore aren't infinitely divisible).
      What matters here is the volume; or, in mathematical terms, the measure of a set. And the crucial step is that assuming the axiom of choice it is possible to construct sets which do not have measure (volume); in particular, they don't have a zero measure, and don't have a non-zero measure, because both options lead to a contradiction. If the parts of the sphere were measurable (had a well-defined volume), then it wouldn't be possible to re-arrange them into an object of a different volume from the original; both translation and rotation preserve the measure. But this no longer holds with non-measurable sets; assuming the axiom of choice, it is possible to split a solid three-dimensional body (such as a sphere) into finitely many (non-measurable) parts, move them around by translation and rotation, and construct from them a solid body with a different volume from the original (such as two copies of the original sphere, with no gaps of course).

  • Edward Hines
    Edward Hines Day ago

    the basis of a form of encryption.

  • Hugo Stiglitz
    Hugo Stiglitz Day ago

    if you had the infinite amount of numbers between 0 and 1 and made the diaganal line that number would just be a copy and was already accounted for in the list........................the 1 through 9 order of numbers never change

    • MikeRosoftJH
      MikeRosoftJH Day ago

      It can't be in the list, because it differs from the first number in the first digit after the decimal point, from the second number in the second digit after the decimal point, from the third number in the third digit, from the fourth number ... and so on. (Sure, you can add the diagonal number to the list. You can't add it to the end because there's no end, but you can insert it to the beginning or, say, to the millionth position. But by doing that you get a different list which still has countably many members, so the diagonal argument still applies.) And because I have made no assumption about the list, it follows that no countable list of real numbers can contain all reals; in other words, real numbers are uncountable.
      You can do the same with rational numbers, and indeed you get a number which also isn't on the list. What does such a construction prove? It proves that if the list covers all rational numbers (and it's not that difficult to construct a list which does), then a number constructed in this way can't be rational.

  • Panther God
    Panther God Day ago

    🤯🤯

  • Quicklyfun
    Quicklyfun Day ago

    Explaining a paradox was already hard for me to explain to my parents so im not talking ANYTHING about this in my house.

  • Orion Wilson
    Orion Wilson Day ago

    Time to stop playing God and be his children

  • Verty
    Verty Day ago

    13:59 HAH! RUL is the same point as U! Myth BUSTED! I saw you put the orange below to cover up this damning flaw!

    • MikeRosoftJH
      MikeRosoftJH 15 hours ago

      RUL is a different point from U; you can pick a die and try it.

  • Verty
    Verty Day ago

    This infinity is bigger than the other infinity! Doesn't make any sense. Not in a paradoxical way, in a stupid way.

  • chriskeenan1
    chriskeenan1 Day ago

    My mind just farted. Time for a coffee. But what if you had infinite coffees I hear you ask...and my mind just farted again. Jeez.

  • Stuart Goldsack
    Stuart Goldsack Day ago

    My head hurts

  • Speed Weed
    Speed Weed Day ago

    Hold up guys Isn’t it illegal to destroy the US dollar without permission from the Federal Government? Does this mean Michael just committed a federal crime on camera?
    I won’t tell if you don’t! 😂

    • Scoota Hoo
      Scoota Hoo 21 hour ago

      He didnt destroy it, he ripped it

  • Sesky U
    Sesky U Day ago

    why is LUR not same as U?

  • Alfredo Caraveo
    Alfredo Caraveo Day ago

    i like turtles.

  • Kristen Winstead

    I though that's what splicing was

  • Priyankar Agrawal

    up upleft right down down right left left right up doyn letf htjhjhn...... I'M OUTTA HERE

  • Joelle Wong
    Joelle Wong Day ago

    I'M SO CONFUSED

  • The King Of Cringe

    Cool

  • James Laupan
    James Laupan Day ago

    There’s an infinite number of strange comments about this video, all you have to do is add 1!

  • B.W Squad
    B.W Squad Day ago

    Holy shit

  • HipO
    HipO Day ago

    Pause a 3:32😂

  • Vertex TB
    Vertex TB Day ago

    I don’t even understand.

  • Lauren Youtube
    Lauren Youtube Day ago

    Don’t get it

  • AlphaM 3000
    AlphaM 3000 Day ago

    i'm learning about infinity in my geometry class.

  • Young Bryce
    Young Bryce Day ago

    Similar to my mathematical proof that between any two points there is either 0 or infinite distance between them.

  • Chamelfo Ropatras

    Please do a Spanish channel..

  • ACJT Clan
    ACJT Clan Day ago +1

    9:16

  • James lester Penuliar

    I thought this was a magic vid when he cut that choc bar and that dollar bill i was amazed but when that weird number thing came in im bored man just do more illusion magic hehe kidding

  • Sam Scherbak
    Sam Scherbak Day ago

    The video never institutionalizes a "z" axis. He starts with talking about tracing a coordinate in a spherical object but relies on 2 dimensional directions. If you took an leftward direction (In respective to x/y/z, would be -1/0/0) and rotated 90 degrees towards the face (0/0/1) that technically has a coordinate of "one value" towards the perspective used to make the direction in the first place. Another by which should stay the same (the perspective). If keeping the same perspective, rotating the leftmost coordinates 90 degrees towards the face would not simply add a right direction to every vector, it would CHANGE THE VECTOR overall. He adds that they added nothing yet the purple sphere miraculously adds colors to perceive the idea that the sphere is regenerating the colors simply through the change of perspective. The coordinates would change in themselves and the COLORS RESPECTIVE would change not be added. This is the same idea as the chocolate bar mentioned in the beginning of the video: An illusion.

  • C- Dub
    C- Dub Day ago

    This video helped me connect the westernized way of thinking with science and spirituality, which are not separated in eastern sciences and philosophies. Thank you! I totally see Fibonacci sequencing in this description. The flower of life begins! 🤔👁😳🤯😱🤓🤩👍🔀🌀✳️⚫️⚪️

  • HannesTM
    HannesTM Day ago +1

    13:48 I see where this is going...

  • Udayon Sen
    Udayon Sen Day ago

    I cried when he tore the dollar

  • Toby Barnwell
    Toby Barnwell Day ago

    A real life sphere woulnt be infinite

  • Julian Kern
    Julian Kern 2 days ago

    360p?

  • Eoseo wa bangtaneun cheoeumiji?

    this video made me feel so frustrated...how many things are out there that we don't know...i mean i obviously knew there are a lot of things that we don't know..but there are A LOT...and i'm a person with a normal level of knowledge and when i watch stuff like that it makes me wonder..now imagine the researchers and the scientists that they know way more details...how do they feel about knowing more than an average pesron but at the same time not being able t know as much as they want t..

  • yungmaz3 13
    yungmaz3 13 2 days ago

    378k likes... it deserves it

  • yungmaz3 13
    yungmaz3 13 2 days ago

    This video just changed how i look at the world

  • Drewcifer Christ
    Drewcifer Christ 2 days ago

    My guy Hilbert is making so much money from that hotel

  • 8obbyLP
    8obbyLP 2 days ago

    I am watching this late at night and i just don't know where or what i am #%#@$%}~#/ XD

  • Sembo / Gaming
    Sembo / Gaming 2 days ago

    That still fucks my brain lol

  • Absolute Zero
    Absolute Zero 2 days ago

    Maybe it already has happened in our world. Identical twins could come from the same sperm.

  • 박지영
    박지영 2 days ago

    im lost

  • NicerDiceR
    NicerDiceR 2 days ago

    3:20 how can you count an infinite amount of numbers in a finite amount of time?

    • NicerDiceR
      NicerDiceR 2 days ago

      actually didn´t even expect that someone explains it to me since the vid is 3 years old, and then in a way that I understand it. Thanks!

    • MikeRosoftJH
      MikeRosoftJH 2 days ago

      The formulation is that you can reach any number from another in a finite amount of time; that is, in a finite number of steps. For example, to reach the number googolplex from number 1 obviously takes a finite number of steps. (Infinity itself is not a natural number.) For example, what if we add "infinity" to the set of natural numbers, getting the set {0, 1, 2, 3, ..., ω}? Such a set can be re-arranged in such a way that every number is reachable in a finite amount of steps: {ω, 0, 1, 2, 3, ...}, so it is countable. (Formally, a set is countably infinite if it can be mapped one-to-one with natural numbers.)

  • polina chupina
    polina chupina 2 days ago

    Mind blowing 🌋 Thank you so much for this amazing video!!!

  • Ghalaghor McAllistor

    The FBI or Illuminati is going to knock on your door any minute now.

  • PootisDuspunsurHarambe4967

    Mr.Beast uploads a video
    *_Couting through Infinity_*

  • Maddy The Witch
    Maddy The Witch 2 days ago

    So... Is the infinity between 0 and 1 smaller than the infinity between 0 and 2?

    • MikeRosoftJH
      MikeRosoftJH 2 days ago

      Both the intervals have the same cardinality (number of elements), because they can be mapped one-to-one using the function f: x -> 2*x. They differ by measure (interval length): the first set has measure 1, the second has measure 2. (Likewise, we can define a measure on 2-dimensional, 3-dimensional, etc. space, corresponding to area, volume, and so on.)

  • hm?
    hm? 2 days ago

    11:14 huh, tits

  • Karyuu
    Karyuu 2 days ago

    That Hilbert guy must be rolling in money with all those guests in his hotel

  • SpydersByte
    SpydersByte 2 days ago

    seems kind of stupid that "countable" and "uncountable" infinity only differ by which side of the decimal you're placing the 0's on. He says that the space between 0 and 1 is uncountably infinite because you can always add another 0 *after* the decimal place, making your number smaller.... but isn't the exact same thing true for the set of all whole numbers? The space between 0 and infinity on the whole number line is never complete because you can always add another 0 *before* the decimal place, making your number bigger. They are the exact same, no?

    • MikeRosoftJH
      MikeRosoftJH 2 days ago

      No, you're still missing the point; what you say is still true for rational numbers. That is, the set of rational numbers between 0 and 1 is countably infinite - even though there is no smallest rational number greater than 0. Likewise, there is no smallest real number greater than 0, but the set of real numbers in the interval from 0 to 1 is uncountably infinite. (Indeed all numbers you have mentioned so far are rational.)
      Just because a particular function from natural numbers to set X doesn't cover all elements, it doesn't mean that set X is not countable. To prove that a set is countable, it suffices to find a single function from natural numbers to X which covers all the elements; conversely, to prove that it is uncountable requires proving that no possible function from natural numbers to X can cover all elements.
      And you also have it backwards. First you need to define natural numbers (either using axioms, or by an explicit construction in set theory); afterwards, it can be proven that every natural number has a decimal representation with finitely many non-zero digits. (Here "finite" has a very specific definition. In set theory, natural numbers are defined as the following: 0 is the empty set, 1 is the set {0} - set containing the empty set as its only element, 2 is {0,1}, 3 is {0,1,2}, and so on. Then by definition, a set is finite if it has the same number of elements as some natural number; that is, if it can be mapped one-to-one with it.)

    • SpydersByte
      SpydersByte 2 days ago

      @MikeRosoftJH there's no reason why every integer could only have a finite number of digits before the decimal point, you're not defining an infinity at that point. You could always add another zero making the number larger by a power of ten. However, after I wrote the initial post I realized what the actual difference was. The difference between countable and uncountable infinity is simply the fact that you know where to start with countable infinity. You start at 1 and you count upward, never ending. However, if you wanted to count the infinity between 0 and 1 you would need a place to start counting and such a number simply doesn't exist. You couldn't start at .1, you couldn't start at .01, and you couldn't start at .001, ect, ect, meaning that this particular infinity is uncountable.

    • MikeRosoftJH
      MikeRosoftJH 2 days ago

      The crucial difference is that every integer (and likewise every real number) has finitely many digits before the decimal point; but a real number may have infinitely many digits after the decimal point. (All numbers with finitely many non-zero digits after the decimal points are rational - they are fractions with the denominator of the form 2^x*5^y, where x and y are non-negative integers - and there are countably many rational numbers.)

  • JOhn Stuart
    JOhn Stuart 2 days ago

    Paradoxes are never real. They are almost always a play on words. Just listen to the video and question each word. Our grammar and language concepts make us assume things that are not real.

  • BrandtChicken GD
    BrandtChicken GD 2 days ago

    Wait, wait, hold up, so if i were thanos, would bepis?

  • Jurwaffe
    Jurwaffe 2 days ago

    After you did the money thing, I just said "what" for 2 minutes straight.

  • The Pepito Channel
    The Pepito Channel 2 days ago

    Hilbert's hotel didn't convince me; as a hotel manager, before moving my clients in order to free a room, I think I would have first to identify a room being free at the endless end of the corridor(which I can't), then could I give each client a room, all rooms being freed instantanously.

    • The Pepito Channel
      The Pepito Channel Day ago

      +MikeRosoftJH No offense sir, but I feel like you just keep on repeating what he said already in his video, without resolving the issue. I have the feeling that logistics is a major issue of that logic, for you can't magically build up any free room either at the end or between or in paralel of occupied rooms and... Buh, I am repeating myself now. Well, whatever. Thanks for the debate anyway.

    • MikeRosoftJH
      MikeRosoftJH Day ago

      No, this still misses the point. Don't care about the logistics. Let's get the set of all natural numbers: {0,1,2,3,4,...}. Apply the function f: x->x+1 to it. You get the set {1,2,3,4,5,...}. That is: you get a one-to-one mapping between the set of natural numbers and its strict subset. Likewise, you can map the set of natural numbers with two copies of the same set (this corresponds to fitting a bus with infinitely many guests into the hotel when all rooms are already full). You can even fit (countably) infinitely many buses, or infinitely many copies of the set of natural numbers (represented in set theory as a set of pairs of natural numbers; in other words, a Cartesian product of the set of natural numbers with itself); similar to that is the proof that there are countably many rational numbers.

    • The Pepito Channel
      The Pepito Channel Day ago

      +MikeRosoftJH Meaning you need a corridor, in order to: 1) first move everybody out of their rooms, 2) and then, once everybody's out in the corridor, to move them in their new rooms. ...which is, I think, precisely the issue of that theory.
      For instance, speaking of an infinite list we know well: Integers. It is a one dimensional list of entities with no intermediate rooms between them. You could eventually build some free rooms in other dimensions, in parallel of that list. But not inside the list.
      Thus, to say that a specific length X is amazing in itself because a combination of it coupled to another dimension has some specific properties is, I guess, incorrect. Isn't it?

    • MikeRosoftJH
      MikeRosoftJH Day ago

      No, that's not the case. (In particular there's no free room because all rooms are occupied, and no room at the end because there's no end.) But what you can do is the following: suppose that all guests would simultaneously leave their rooms (n) and each one moved to a room one higher (n+1). Then after the maneuver each guest still has a room, but the first room is now empty because there's no one to move into it.

  • Uglygodbitxh
    Uglygodbitxh 2 days ago

    bro whenever michael says, “OR CAN YOU??” i laugh

  • XDTheLaughingManXD
    XDTheLaughingManXD 2 days ago

    This is basically a video on how to play street fighter.

  • De'Vonte Rush
    De'Vonte Rush 2 days ago

    wat...

  • John Bowers
    John Bowers 2 days ago

    my head hurts

  • Paula Hillier
    Paula Hillier 2 days ago

    Owww, my brain.

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  • KittyMapletree
    KittyMapletree 2 days ago

    If you have an infinite number of $1 bills and an infinite number of $20 bills, they would be the same amount

  • dale chen
    dale chen 2 days ago

    Why is the video 360 p. Am I the only one?

  • Josh Danker
    Josh Danker 2 days ago

    If you stare at the thumbnail long enough it starts to move :)

  • The Milky Cow
    The Milky Cow 2 days ago

    I always thought of infinity, not by a forever expanding number, but a forever expanding number that is able to expand at different speeds including the speed of infinity which has different ways of working. It can expand by 1,2,3, squares, or at random, it may accelerate or decelerate but never stops.

  • ProfRaccoon
    ProfRaccoon 3 days ago

    Professor raccoon proved that mathematics is absurd, and physics is absurd as well, all laws have lawless exceptions. Grow a pair of massive Banach Tarski balls first, before you ask for the proof.

  • Idk lol
    Idk lol 3 days ago

    Bruh this is just a cover up for a glitch in the system 😂

  • Terry Bursey
    Terry Bursey 3 days ago

    It's a lot like when you divide something by zero or the fact that when considered in amounts, 2 is equal to 1 in that 1 can become 2. Also gives some insight into the big bang itself and how something can come from nothing in a state when normal time and physics beak down or are non existent. Great info!

  • Mina_ Spam
    Mina_ Spam 3 days ago

    wow its interesting that the first 10 min of the video were stuff our teacher taught us in school in the 10th grade..

  • Hey You
    Hey You 3 days ago

    My head hurts. I understood infinity, I thought? But, infinity is or isn’t infinity. If infinity can make everything, then it’s nothing. A lot of these videos I understand, this is insane.

  • Samantha Grey
    Samantha Grey 3 days ago

    What?

  • Liz Markert
    Liz Markert 3 days ago

    I cannot believe I ripped up a dollar thinking I was about to double my money smh 🤦‍♀️

  • NorDeeS NDS
    NorDeeS NDS 3 days ago

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  • Taylor Fishel
    Taylor Fishel 3 days ago

    Does this apply with bacteria, since it is asexual it must multiply itself on it's own while coexisting.

  • hi there
    hi there 3 days ago

    I just gained an infinite number of brain cells just by watching this video
    But I'm still just average cause I had a negative number of brain cells before this..

  • janska
    janska 3 days ago

    this is so unnecessary and stupid...
    Infinity is just uncountable.
    Even though we have numbers to count, they are basically uncountable like numbers between 0 and 1. It goes on and on and on and on...
    It is just how it is!
    And to the strange hotel theory: If you take the nr 1 out and replace it as shown there will br a gab going on for ever! And also... where is "1" if it's infinity?

    • Edgar Nackenson
      Edgar Nackenson 2 days ago

      You're misunderstanding uncountable. A countably infinite set is one where you can create a one to one mapping from it to the natural numbers. An uncountable set is one where this is not the case. It's as simple as that, really. There are a number of equivalent definitions. For example, a countably infinite set is one where you can create a list from its elements such that you will reach any given element in finitely many steps.