# The Banach–Tarski Paradox

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 4 squares by 8 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 4 by 8 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

Banach-Tarski paradox. It proves that there is, in fact, a way to

take an object and separate it into 5 different pieces. And then, with those five pieces, simply rearrange them.

No stretching required into 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-blah. And as far as we know, infinity could be real. The universe may be infinite in size and flat, extending out for ever 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 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 number, 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 0 to 1 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? 0.000000… Eventually, we would imagine a 1 going somewhere at the end, but there is no end. We could always add another 0.

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. Georg Cantor’s famous diagonal argument helps illustrate this.

Imagine listing every number between zero and one. Since they are

uncountable and can’t be listed in order, let’s imagine randomly generating them forever with no repeats. Each number regenerate 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 for ever. 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 0 and 1, 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 either

set, so this one to one correspondence shows that both sets are the same size. In other words, infinity divided by two is still infinity. Infinity plus one is also infinity. A good illustration of this is Hilbert’s

paradox up 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 1 to room number 2. And a guest in room 2 to

room 3 and 3 to 4 and 4 to 5 and so on. Because the number of rooms is never ending we cannot run out of rooms.

Infinity -1 is also infinity again. If one guest leaves the hotel, we can shift every guest the other way.

Guest 2 goes to room 1, 3 to 2, 4 to 3 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. Banach-Tarski hasn’t left

our sights 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 2Pi. 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 and

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 Point 1 moves to fill in the hole, point 2 fills in the place where point 1 used to be,

3 fills in 2 and so on. Since we have a 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 needo

consequence of infinity we should discuss before tackling Banach-Tarski.

Ian Stewart famously proposed a brilliant dictionary. One that he called the Hyperwebster.

The Hyperwebster lists every single possible word of any length formed from the 26 letters in the

English alphabet. It begins with “a,” followed by “aa,” then “aaa,” then “aaaa.” And after an infinite number of those, “ab,” then “aba,” then “abaa”, “abaaa,” and so on until “z, “za,” “zaa,” et cetera, et cetera,

until the final entry in infinite sequence of “z”s.

Such a dictionary would contain every single word.

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 happened 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 Hyperwebster? 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 lef, right, up and down as well as left 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 lenght 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 colour 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 end with up, let’s colour 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 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 point we will have indeed named and colored every single point on

the surface just once.

With the exception of poles. 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 polls 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 point’s 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 you 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 Hyperwebster. If we had the right piece and the poles of rotation and 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 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 Us. 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 poles of rotation and center are missing from this copy, but no worries.

There’s a countably infinite number of holes,

where the poles of rotations used to be, which means there is some pole around

which we can rotate this sphere such that every pole 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 centerpoint. 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 1.

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 Banach-Tarski 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. The Banach-Tarski 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 by Banach-Tarski and the way tiny tiny sub-atomic

particles can collide at high energies and turn

into more particles than we began with. 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 day, history continues to show us that the

universe isn’t strange. We are. And as always, thanks for watching. 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 kinda around Banach-Tarski. First of all, Leonard Wapner’s “The Pea and the Sun.” This book is fantastic and it’s full of lot of the preliminaries needed to understand the proof that comes later.

He also talks a lot about the ramifications of what Banach-Tarski and their

theorem might mean for mathematics. Also, if you wanna talk about math and

whether it’s discovered or invented, whether it really truly will map onto the universe,

Yanofsky’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. Brian 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’s 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. [Jake:] Hi. Thanks for filming this, by the way. Guys, I really appreciate who you all are. And as always, thanks for watching.

よーわからん

So, this is what I got out of this. I understood every individual word that Michael used in this video, right? Like every single word of it is perfectly reasonable and understandable. But, when you put all the words together, in the exact order and fashion as shown in this video, my mind breaks and I don't understand anything at all.

infinity = -1/12 ?

انا دماغي لفت يابا…

Who else is confused from this?

You could even change the length of one direction's movement in the circle

This kills meh brain

I call this the Tarski-Barski Paradox. Easier to remember.

Holy fucking shit stop talking oh fuck no please staaaahhp

Don't fuck with me Michael??

Moral of the story kids: "no hole will be left unfilled"

Another way of looking at the circle analogy is, points have no dimension and do not exist. They are a location. If a point has no dimension, (infinitely small) they can be infinitely close together and the number of them on a circle is infinite.

If we get confused by 1=0 equation, Im sure there are loads of equations which are wrong, but we can't see it.

Everything in “our” world is a paradox.

ughh!!

I am never gonna ask for extra piece of choclate.

Vesauce Michal here. Or is it?

So if I get 100$ I can get 200$?

nani!!?!?!??

Personally that video is my favorite of vsauce's video

this all comes down to one question: what is 1?

Blah blah blah. Let me ask you smarty pants this….ever heard of the brady bunch paradox, I bet not

Bruh none of this make any sense

Don't try the dollar bill thing . At first your thinking you've discovered a new money making scheme …. Then you slowly realize that you are in fact an embicile …. An idiot …. Obviously it doesn't make extra money

My broke self cried at 1:32 🙁

`_,,,,_

`◑›-◑›

` .ʵ

` .●—Thx 4 the Ed'u'vacation❣

`_,,,,_

`◑›-◑›

` .ʵ

` .●—Thx 4 the Ed'u'vacation❣

4:25 … thats how i feel doing math

If a never ends how does it get to ab

Sometimes i feel like it happens in the capatalist world every decade or so

To infinity and beyond

I thought about making a huge post showing why you are so wrong to think there's "so much more" real #s than whole #s and just how much you don't understand infinity but this video is very old and I'm sure you'd never see it anyway. This video has a ton of views though so I will say please ppl do some studying and learning for yourselves. Infinity is not a number at all, there's no such thing as a greater and lesser infinity either. something is infinite or it's not. The only thing you can say is this equation or set or whatever approaches infinity faster than another but that's it! infinity is infinity or it's not. The only reason some ppl will make it seem like one "infinity" is larger than another is they don't understand what they're talking about and they merge finite concepts with infinite ones

2:14 if you keep looking at the left side of the number line you can see the word censored for a split second

Is there a point to getting this smart? I feel like once you get this smart either make a new math equation or a college and call it a day

I'm sorry but you can't make the argument that all positive whole numbers fit between 0 and 1 in the real numbers infinity and say that even numbers infinity are the same size as positive whole numbers. You can line up the even numbers with the corresponding whole numbers and have leftover whole numbers if we were somehow able to get to the end of infinity.

Also Hilbert's paradox of the Grand Hotel is wrong. If you are going to say that there is a person booked into every room than there are NO MORE ROOMS LEFT TO FILL. You are saying that whatever end infinity possesses has been reached and filled. You CAN'T than say that there is still something more to be reached. By definition saying there is a person booked in every hotel room in an infinite hotel means that there are an infinite amount of guests and that if we were to reach the "end" of the hotel there would be no more rooms left! You have to be consistent with the logic or you can come up with whatever outcome you want.

Positive whole number infinity over the same positive whole number infinity equals ONE.

A thing that is irritating about the inconsistent logic in this video is that it ignores the significance of adding or removing from infinity. If we think of infinity as something that is there already, or in other words as having an "end", than we can understand what it means to add or subtract from it. Whole numbers are there and if we had an infinite amount of time we could count them all. With that in mind imagine counting all the whole numbers and than somehow adding one to the beginning; that infinity would now be bigger. Not relatively bigger but actually bigger than an infinity without that one since the infinity we are talking about is there already. If we bring this argument to the hyperWebster example than taking out an A for the first volume would not give us the same infinity. The new dictionary would actually be 1 a short even if it is infinity. If we were to take out infinite amount of letters from the hyperWebster than we would not be left with infinity, we would be left WITH NOTHING.

Doing anything to a countable infinity actually changes it. Even uncountable infinities could be thought of as infinite infinities. The number of digits that have to go before the first 1 in a real numbers infinity is the same as all countable numbers. After reaching that first 1 (0.00000………….1) you can now count up to 1.0 with infinite whole numbers and repeat for every number after that. An infinite amount of infinites.

You CANNOT decompose pieces of an object into the whole thing. Logic does not allow it given that arguments above are right. Infinity does change because infinity is tangible if we were capable of doing so or if we were God.

I’m infinitely rich because i have a $1 in my bank account

Vsauce hurt my head >:(

don’t watch this while high

Alright yeah that dollar bill trick was cool but it was a gOOD DOLLAR

2:17 Vsauce t-shirt: "There's an infinite number of bla-bla-bla."

?i didnt get anything…i am so high

Felony

My mind is officially dead. To much info

i feel uncomfertable

All this attempt by these professors and scholars to explain infinity and relation to numbers when they are simply trying to grasp time moving forward and never moving back.

Eternity / Infinity has no beginning, just like nothing / 0 has no end. Zero does not exist neither does ten. Always (!) 1-9.

10:10 So you're saying that I, along with every comment I have uploaded to YouTube, could be in the Hyperwebster somewhere? If so, nice. I could find my future within its pages

I tried that with the money. It didn't work.☹️

Maths teacher: 2+2 = 4

Michael, out of nowhere: Or is it?

Now wait, you said there's one person in literally every room in infinity, then how can a guest get a room if every room already has someone? Are they making new rooms every guest?

The most infinity is infinite infinity

So, this is how God must have created something out of nothing…

but substract maybe 2 from infinity instead of dividing you actually have something different, still not able to count, but it changed and you can actually make a logic protokoll for it. imagine it's money but you have limited coins , or imagine you have all the numbers you can count if you would be unmortal and live forever, still maybe not enough resources to count , time might be an illusion or not, but something will break down if anything we learned is correct for this reality. what I am sayin is its not just money, its money with the attribute to it that you wanna spend it which has also attributes evtl., which you bring into the math prob in my case or me rather. I dunno, I like math only abstract or practical, and might have just chosen practical here, just for the sake of conversation, not confrontation

yep, yep, nice, didnt know the name, but wrote something about it ito get an idea straight and bit ordered, and couldnt find

infinity-infinity?

DATS A BIG ASS HOTEL

Totally enjoy your videos, however I must tell you I'm disappointed. I waited to the end to see how to cut the chocolate bar and you never returned to that. I also didn't understand the dollar bill. As for us being finite, our minds are infinity.

How many times can you put nothing in a bucket. 0/1

Can someone please try this? 1:28

and thats how jesus able to feed 5000 people from 5 loaves and and 2 fish

This video got my dog pregnant – and he's male

Wow a learned a lot and forgot it a few seconds later im so smart

4:23 me when im doing my homework

11:34 HOW TF DID HE CONNECT THE CIRLE SO FREAKIN FLAWLESSLY

consider the genetic system, our dna, blood-cells, hair on our bodies… the mirroring effect. quantum physics is also unpredictable at present. but if you take ordered matter, and make chaos of it, what happens about the chaos? what happens with molecular structure when say… water is exposed to air, or Carbon-Dioxide even? what happens when you add up firework ingredients to each other safely? how do diamonds come about? steel? brass? yogurt? soda? candy? plant structures? salt water? heavy water? sand? obsidian? glass? how many times can you have a full twenty-four hours? in how many places? how much time can really be spent? how many stars can be counted in the Universe? how many stars can be made? how many worlds can there be out there? how many worlds are within the Earth? what amount of diversities can be found within those worlds? voltage limit? mass limit? light limit? understanding limit? order limit? chaos limit?

many ways of saying the same thing… but if you see a bigger picture… let us know. it's all relative, but do we ever take the time to actually think and talk about it? what would you want to say if you could say what you found out, about anything related to the topic? what results would come of many responses upon many? just how many opinions are out there waiting to be voiced? i'd certainly like to know… how about ya'll? what can we bring to mind that may shed more light on the subject? and can it all be worth while?

~~-Phoenix<3-~~mind if i post the first million numbers of pi real quick?

The beginning is already fake, if you watch carefully when he spereates the one dollar into 5 pieces, you can see after he put them on the table that the dollar up top cant be like he ripped it apart. He first took the right part and than he ripped it from the left to the right in a line, but the dollar up top on the table has a ripped line through the whole dollar, which is not possible after you ripped off the complete right part.

WHAT

anyone else terrified by the way he writes his 8

The real questions is why does infinity exists or does it oh wait? edit:I think this video gave me a nose bleed tho lol

What we will get if we minus infinite with infinite

BUTT ACT lol

This is why nature doesn't do point particles. It's all waves in the quantum world.

could you assume infinity as a variable( if it is countable infinity ) then use it in an equation

what if a room is destroyed

I watched this whole video just to watch you create an infinite sphere of directions ( made up measurement to eventually calculate a point) then use energy to rotate the spheres adding another direction and say nothing happened.

I could've brought 160 Robux with the notes he teared up.

First lesson 1+1=1 written on the wall at the bottom of marston road in Oxford. Finally i join the 22million viewers who still don't know. Wtf it means

My mind is truly blown. ?

When you add an R for the first time @16:30 and then you add other "rotations" later on, isn't it there where you are adding "physical" material and not just mathematical rotations from which a double surface will pop up?

Police:Your under arrest!

Vsauce:Or am i?

Read more

I can never count the same again

13:43

I see what u did thereWhy have i not seen this till now

I simultaneously understand every word he said and understand nothing he just said. I guess I just created a paradox

I thought a circle was pointless lol

Dude what the fuck

Isn’t this just taking an infinite amount of points, putting away half of the points and boom you’ve got 2 of what you had in the beginning

proof of the uncountability of reals given here is obviously flawed

People are too smart

Juden

How smart is this GUY

What do you smoke?

what about (Infinity – Infinty)

He left and never came back

My brain hurts

Hyperwebster sounds exactly like library of Babel. What’s the difference?

You play tricks with words. your suggestions won't actually work, because they'll fail the same way you're introductory demonstration will fail. I understand that there are different types of infinity, and we use words to describe them, but I could also say almost(?) everything that grammatically makes sense, and it may be true – somewhere, in some universe.. if you get what I mean.

But to your demo about how whole numbers are "countable" and real numbers are not, you're wrong. If I assume that I have infinite amount of time, thus making the whole numbers "countable" – well, clearly you're never actually going to count them – ever… you'll just be counting for an infinite amount of time.

Same goes for real numbers – since to count them, all you'll need is an infinite amount of an infinite amount of times. It will take you an infinite amount of time to write down the zero's before the 1 you'll never get to, but that's just writing zero's for infinity, just like writing whole numbers for infinity – except for with the real numbers, you'll have to do it an infinite amount of times for all the other infinite reals, the 2nd number being

Just because you can write something down, like 1, 2, 3, .. ∞ – doesn't mean it makes any more sense than writing down 1/0 = ∞.

You're never going to finish writing down 1, 2, 3, – ∞

Boggles the mind, and calculations with ∞ may still work in some cases – but it doesn't mean they apply to anything practical… or does it..? What if there are an infinite number of universes that are literally infinitely large?

Cheers, and thanks for the vid.

We definitely can’t fly

Vsause: Or can we?

If it's an endless list, that number will eventually show up

trippy :/

Thanks Michael, now my head hurts.

Fascinating. Incredibly fascinating.