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The Banach–Tarski Paradox — Transcript · YTtoTranscript

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The Banach–Tarski Paradox

Vsauce · 3,496 words · 17 min read · EN-ORIG

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Below is the complete, readable transcript of The Banach–Tarski Paradox by Vsauce on YouTube. Read the full text, copy any part you need, or generate a transcript for any video with our free tool.

hey Vsauce Michael here there's a famous way to seemingly create chocolate out of nothing maybe you've seen it before this chocolate bar is four squares by eight squares but if you cut it like this and then like this and finally like this you can rearrange the pieces like so and wind up with the same 4x8 bar but with a

leftover piece apparently created out of thin air there's a popular animation of this illusion as well I call it an illusion because it's just that fake in reality the final bar is a bit smaller it contains this much less chocolate each Square along the cut is shorter than it was in the original but the cut

makes it difficult to notice right away the animation is extra misleading because it tries to cover up its deception the Lost height of each square is surreptitiously added in while the piece moves to make it hard to notice I mean come on obviously you cannot cut up a chocolate bar and rearrange the pieces

into more than you started with or can you one of the strangest theorems in modern mathematics is the bunnock tarsky Paradox it proves that there is in fact a way to take an object and separate it into five different pieces [Music] and then with those five pieces simply rearrange them no stretching required in

two two exact copies of the original item same density same size same everything seriously to dive into the mind blow that it is and the way it fundamentally questions math and ourselves we have to start by asking a few questions first what is infinity a number I mean it's nowhere on the number line but we often say things like

there's an infinite number of blah blah and as far as we know Infinity could be real the universe may be infinite in size and flat extending out forever and ever without end Beyond even the part we can observe or ever hope to observe that's exactly what Infinity is not a number per se but rather a size the size

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of something that doesn't end Infinity is not the biggest number instead it is how many numbers there are but there are different sizes of infinity the smallest type of infinity is countable Infinity the number of hours in Forever it's also the number of whole numbers that there are natural numbers the numbers we use when counting

things like 1 2 3 4 5 6 and so on sets like these are unending but they are countable countable means that you can count them from one element to any other in a finite amount of time even if that finite amount of time is longer than you will live or the universe will exist for

it's still finite uncountable Infinity on the other hand is literally bigger too big to even count the number of real numbers that there are not just whole numbers but all numbers is uncountably infinite you literally cannot count even from zero to one in a finite amount of time by naming every real number in between

I mean where do you even start zero okay but what comes next 0.000 eventually we would imagine a one going somewhere at the end but there is no end we could always add another zero uncountability makes this set so much larger than the set of all whole numbers that even between 0 and 1 there are more

numbers than there are whole numbers on the entire endless number line getter cantor's famous diagonal argument helps illustrate this imagine listing every number between 0 and 1. since they are uncountable and can't be listed in order let's imagine randomly generating them forever with no repeats each number we generate can be paired with a whole number

if there's a one-to-one correspondence between the two that is if we can match one whole number to each real number on our list that would mean that countable and uncountable sets are the same size but we can't do that even though this list goes on forever forever isn't enough watch this if we go

diagonally down Our Endless list of real numbers and take the first decimal of the first number and the second of the second number the third of the third and so on and add one to each subtracting one if it happens to be a nine we can generate a new real number that is

obviously between zero and one but since we've defined it to be different from every number on Our Endless list and at least one place it's clearly not contained in the list in other words we've used up every single whole number the entire Infinity of them and yet we can still come up with more real numbers

here's something else that is true but counter-intuitive there are the same number of even numbers as there are even and odd numbers at first that sounds ridiculous clearly there are only half as many even numbers as all whole numbers but that intuition is wrong the set of all whole numbers is denser but

every even number can be matched with a whole number you will never run out of members of either set so this one-to-one correspondence shows that both sets are the same size in other words Infinity divided by 2 is still infinity infinity plus one is also Infinity a good illustration of this is Hilbert's Paradox of the Grand Hotel

imagine a hotel with a countably infinite number of rooms but now imagine that there is a person booked into every single room seemingly it's fully booked right no infinite sets go against common sense you see if a new guest shows up and wants a room all the hotel has to do is move the guest in room number one to

room number two and the guest in room two to room three and three to four and four to five and so on because the number of rooms is never ending we cannot run out of rooms Infinity minus one is also Infinity again if one guest leaves the hotel we can shift every guest the other way guest two goes to

room one three to two four to three and so on because we have an infinite amount of guests that is a never-ending supply of them no room will be left empty as it turns out you can subtract any finite number from infinity and still be left with infinity it doesn't care it's unending

banatarsky hasn't left our sites yet all of this is related we are now ready to move on to shapes Hilbert's Hotel can be applied to a circle points around the circumference can be thought of as guests if we remove one point from the circle that point is gone right Infinity tells us it doesn't matter the

circumference of a circle is irrational it's the radius times 2 pi so if we mark off points beginning from the whole every radius length along the circumference going clockwise we will never land on the same point twice ever we can count off each point we Mark with a whole number so this set is never

ending but countable just like guests in rooms in Hilbert's hotel and like those guests even though one has checked out we can just shift the rest move them counterclockwise and every room will be filled 0.1 moves to fill in the hole 0.2 fills in the place where 0.1 used to be 3 fills in two and so on

since we have an unending supply of numbered points no hole will be left unfilled the missing point is forgotten we apparently never needed it to be complete there's one last Neato consequence of infinity we should discuss before tackling bonoctarsky Stewart famously proposed a brilliant dictionary one that he called the hyper Webster the hyper Webster lists every

single possible word of any length formed from the 26 letters in the English alphabet it begins with a followed by a a then AAA then AAA and after an infinite number of those a b then a b a then a b a a a b a a and so on until z z a z a a etc etc until the

final entry an infinite sequence of Z's such a dictionary would contain every single word nay every single thought definition description truth lie name story what happened to Amelia Earhart would be in that dictionary as well as every single thing that didn't happen to Amelia Earhart everything that could be said using our alphabet

obviously it would be huge but the company publishing it might realize that they could take a shortcut if they put all the words that begin with a in a volume titled A they wouldn't have to print the initial a readers would know to just add the a because it's the a volume by removing the initial a the

publisher is left with every a word Sans the first a which has surprisingly become every possible word just one of the 26 volumes has been decomposed into the entire thing it is now that we're ready to investigate this video's titular Paradox what if we turned an object a 3D thing into a hyper-webster could we decompose

pieces of it into the whole thing yes the first thing we need to do is give every single point on the surface of the sphere one name and one name only a good way to do this is to name them after how they can be reached by a given starting point if we move this starting

point across the surface of the sphere in steps that are just the right length no matter how many times or in what direction we rotate so long as we never backtrack it will never wind up in the same place twice we only need to rotate in four directions to achieve this Paradox up down left and right around

two perpendicular axes we are going to need every single possible sequence that can be made of any finite length out of just these four rotations that means we will need left right up and down as well as left left up left down but of course not left right because well that's backtracking going left and then right

means you're the same as you were before you did anything so no left rights no right lefts and no up downs and no down UPS also notice that I'm writing the rotations in order right to left so the final rotation is the leftmost letter that will be important later on anyway a list of all possible sequences of

allowed rotations that are finite in length is well huge countably infinite in fact but if we apply each one of them to a starting point in green here and then name the point we land on after the sequence that brought us there we can name a countably infinite set of points on the surface let's look at how say

these four strings on our list would work right up left okay rotating the starting point this way takes us here let's color code the point based on the final rotation in its string in this case it's left and for that we will use purple next up down down that sequence takes us here we name the point DD and color it

blue since we ended with a down rotation RDR that will be this Point's name takes us here and for a final right rotation let's use red finally for a sequence that ends with up let's color code the point orange now if we imagine completing this process for every single sequence we will have a countably

infinite number of points named and color coded that's great but not enough there are an uncountably infinite number of points on a sphere's Surface but no worries we can just pick a point we we missed any point and color it green 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 have indeed named and colored every single point on the surface just once with the exception of polls every sequence has two poles of rotation locations on the sphere that come back to exactly where they started for any sequence of right or left rotations the

poles are the North and South Poles the problem with poles like these is that more than one sequence can lead us to them they can be named more than once and be colored in more than one color for example if you follow some other sequence to the North or South Pole any subsequent rights or lefts will be

equally valid names in order to deal with this we're going to just count them out of the normal scheme and color them all yellow every sequence has two so there are a countably infinite amount of them now with every point on the sphere given just one name and just one of six colors we are ready to take the entire

sphere apart every point on the surface corresponds to a unique line of points below it all the way to the center point and we will be dragging every points line along with it the lone center point we will set aside okay first we cut out and extract all the yellow poles the green starting

points the orange up points the blue down points and the red and purple left and right points that's the entire sphere with just these pieces you could build the whole thing but take a look at the left piece it is defined by being a piece composed of every Point accessed via a sequence

ending with a left rotation if we rotate this piece right that's the same as adding an R to every Point's name but left and then right is a backtrack they cancel each other out and look what happens when we reduce them away the set becomes the same as a set of all points with names that end with L but

also u d and every Point reached with no rotation that's the full set of starting points we have turned less than a quarter of the sphere into nearly three quarters just by rotating it we added nothing it's like the hyper Webster if we add the right piece and the pulls of rotation in the center point well

we've got the entire sphere again but with stuff left over to make a second copy let's rotate the up piece down the down UPS cancel because well it's the same as going nowhere and we're left with a set of all starting points the entire up piece the right piece and the left piece but there's a problem here we

don't need this extra set of starting points we still haven't used the original ones no worries let's just start over we can just move everything from the up piece that turns into a starting point when rotated down that means every Point whose final rotation is up let's put them in the down piece

of course after rotating points named uu will just turn into points named U and that would give us a copy here and here so as it turns out we need to move all points with any name that is just a string of use we will put them in the down piece and rotate the up piece down

which makes it congruent to the up right and left pieces add in the down piece along with some up and the starting point piece and well we're almost done the pulls of rotation and Center are missing from this copy but no worries there's a countably infinite number of holes where the poles of rotation used

to be which means there is some pull around which we can rotate this sphere such that every pull hole orbits around without hitting another well this is just a bunch of circles with one point missing we fill them each like we did earlier and we do the same for the center point imagine a circle that

contains it inside the sphere and just fill in from Infinity and look what we've done we have taken one sphere and turned it into two identical spheres without adding anything one plus one equals one that took a while to go through but the implications are huge and mathematicians scientists and philosophers are still

debating them could such a process happen in the real world I mean it can happen mathematically and math allows us to abstractly predict and describe a lot of things in the real world with amazing accuracy but does the banatarsky Paradox take it too far is it a place where math and physics separate

we still don't know history is full of examples of mathematical Concepts developed in the abstract that we did not think would ever apply to the real world for years decades centuries until eventually science caught up and realized they were totally applicable and useful Paradox could actually happen in our real world the only catch of course is

that the five pieces you cut your object into aren't simple shapes they must be infinitely complex and detailed that's not possible to do in the real world where measurements can only get so small and there's only a finite amount of time to do anything but math says it's theoretically valid and some scientists think it may be

physically valid too there have been a number of papers published suggesting a link between banatarsky and the way tiny tiny subatomic particles can Collide at high energies and turn into more particles than we began with [Music] we are finite creatures our lives are small and can only scientifically consider a small part of reality

what's common for us is just a sliver of what's available we can only see so much of the electromagnetic spectrum we can only delve so deep into extensions of space Common Sense applies to that which we can access but common sense is just that common if total sense is what we want we should

be prepared to accept that we shouldn't call Infinity weird or strange the results we've arrived at by accepting it are valid true within the system we use to understand measure predict and order the universe perhaps the system still needs perfecting but at the end of the day history continues to show us that the Universe

isn't strange we are and as always thanks for watching [Music] my eyes

finally as always the description is full of links to learn more there are also a number of books linked down there that really helped me wrap my mind kind of around but notarsky first of all Leonard wapner's the p and the sun this book is fantastic and it's full of a lot of the preliminaries needed to

understand the proof that comes later he also talks a lot about the ramifications of what banatarsky and their theorem might mean for mathematics also if you want to talk about math and whether it's discovered or invented whether it really truly will map onto the universe yanovsky's the outer limits of reason is great this is the favorite book of mine

that I've read this entire year another good one is e Bryan Davies why beliefs matter this is actually Corn's favorite book as you might be able to see there it's delicious and full of lots of great information about the limits of what we can know and what science is and what mathematics is if you love infinity and

math I cannot more highly recommend Matt Parker's things to make and do in the fourth dimension he is hilarious and this book is very very great at explaining some pretty awesome things so keep reading and if you're looking for something to watch I hope you've already watched Kevin lieber's film on field day

I already did a documentary about Whittier Alaska over there Kevin's got a great short film about putting things out on the internet and having people react to them there's a rumor that Jake Roper might be doing something on field day soon so check out mine check out Kevin's And subscribe to field day for

upcoming Jake Roper action yeah he's actually in this room right now say hi Jake hi thanks for filming this by the way um guys I really appreciate who you all are and as always thanks for watching [Music]