Correct, he didn’t actually make two cubes from one here. The demonstration is just a physical analogy. Mathematically the Branagh Tarski paradox is still true.
@@pmcate2 I don't know if it is a lack of imagination from my part... but is impossible for me to try to visualize how the Banach-Tarski paradox could work. Maybe it is just one of those math theorems that can't be visualized and we have to trust the proof
In this video, I was merely trying to have some fun by pretending to illustrate the paradoxical decomposition---it was all "tongue in cheek"! In the actual Banach-Tarski partition, the pieces are nothing like these subcubes and triangular prisms, for exactly the reason you mention. In the actual partition, each piece is dense in the unit ball, although they do not overlap; those pieces are not themselves measurable---they have no coherent definite volume, and this is a requirement of the theorem, precisely because rigid motion preserves measure.
@@joeldavidhamkins5484 it's very funny to me that this is a joke where i understand neither the real nor joke explanation only that the joke explanation is wrong for the reason the first response said... good gag! XD
So, the interesting thing of the Banach-Tarski paradox is not that we can transform "one ball" into "two balls" (since we know that an "ℝ³" ball has the same cardinality as the whole "ℝ³" set), but "how" we transform "one ball" into "two balls". Isn't it? 👀
Hold on... you made two because the original 'cube' in your hands was already in manufactured into two pieces that were put together as a single; you simply separated them in an entertaining and somewhat fanciful way. How does this translate to the Barnach-Tarski Paradox? Serious question....
Hi, I am sorry if this is a stupid question, but do you have any resources or information on why this is the case specifically due to the axiom of choice?
Because if you negate the Axiom of Choice then every set is Lebesgue measurable and therefore Banach-Tarski becomes impossible (measure must be conserved!)
More precisely, if the axioms of set theory are consistent, then it is consistent with the axioms of set theory without the axiom of choice that all subsets are Lebesgue measurable.
@@VeganCookies Well, it's also consistent in absence of axiom of choice that the uncountable set of all real numbers is a union of countably many countable sets; in which case the Lebesgue measure doesn't work as you would expect it to (namely, it isn't countably additive). But good luck doing calculus in this model of set theory!
Maybe Banach tarski is a real world phenomenon in where the second cube is in existence within a parallel universe and an infinite number of universes exist.
I only know in the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach-Tarski type that uses only Euclidean congruences is impossible.
Great way of presenting it! :D I want me that cube :D I guess tue question is, can you do it again for each of the two physical qubes. I don't think so :P
Yes acording to the paradox you could however you could not do it with that cube because that cube is just a demonstration device vsauce has a much more in-depth video explaining it
@@willamdiprofio9091 If I understood correctly, it's possible only in theory. But there's no way to apply the method practically. Which means it isn't really possible. I mean, if something can't be demonstrated with a practical experiment, but only explained as a concept, we can't consider it possible.
Suponho que a "estrela" inter tenha igual perímetro que o cubo que ela forma certo? Mesmo assim não houve a conservação do volume real da forma geometriaca porque ela ficou com um espaço interno parcialmente vazio, certo? Mas na geometria de Euclides isso é possível?
Why is it a paradox? Aren't the new 'cubes' just approximations of the original cube? So, this splitting can go infinitely, just each new pair will be 'hollower' / 'thinner' ? In reality can be possible too, if we have atom-controlling beam. How do we measure the volume? What is a rigid motion here?
Banach-Tarski tells you that the 2 cubes you end up with are identical to the one you started with. Not approximations, the exact same In reality, we can never make this happen. We would need to be able to partition an object into infinitely small points, which just isn't possible in the real world
Not really. (That is, unless you mean that infinity divided by 2 is still infinity, and so the density remains the same.) A solid ball is the set of all points whose distance from the center is no greater than R. Banach-Tarski paradox is the following theorem of set theory: assuming axiom of choice, it is possible to take a solid body (for example, a solid ball), split it into finitely many subsets, move these around by translation and rotation (without overlap), and get a solid body of a different volume (for example, two copies of the original solid ball). And a solid body is the set of *all* points within the solid body; it can't contain just some of the points (that would be a subset of the solid body). In particular, it can't contain "every other point" (which isn't a meaningful notion on reals anyway - there's no such thing as adjacent points). When you move or rotate a solid body, its volume doesn't change. The trick, then, is that the pieces in question are not solid bodies and don't have a well-defined volume; the volume can't be zero, and it can't be non-zero either.
@@methatis3013 If I'm not wrong, it consists of dividing the sphere into 6 parts and through rotations and translations (without breaking, stretching, or reflecting them) obtaining two identical balls. Which is a stronger condition than being just equipotent. Isn't it? 👀
@@GabriTell I mean, that's the point. You partition the ball into finitely many pieces (6 to be precise), and by rearranging them, you get 2 balls in total
24 April 11;20 Nonton UA-cam Sciencephile Iceberg Of Paradox 11:42 Joel David Hamkins Banach Tarski Paradox tapi balik lagi buat komen 11:44 Lanjut Joel David Hamkins 11:48 VSauce Banach Tarski tapi balik lagi buat komen
People usually blame AC, but I think that's a result of faulty formalisation. It also has another flaw: you can only give each point of the border to one piece, so an additional construct is needed to prove that you can turn any polygon into any polygon of the same area
What you doing is creating fume. The Banach-Tarski paradox is an interesting mathematical concept, it has no practical application in the physical world. Although in video whatever magic your performing to create non-measurable sets in the physical world, the reassembly process would require an infinite number of infinitesimal transformations, which is not possible in practice due to the constraints of physical reality. Can you do again with the other two cubes??
This is a theorem of set theory: assuming axiom of choice, it is possible to take a solid body (for example, a solid ball), split it into finitely many subsets, move these around by translation and rotation (without overlap), and get a solid body of a different volume (for example, two copies of the original solid ball). When you move or rotate a solid body, its volume doesn't change. The trick, then, is that the pieces in question are not solid bodies and don't have a well-defined volume; the volume can't be zero, and it can't be non-zero either.
The BS stems from the fact that the cube has been hallowed out by 50%. Hence, NOT A CUBE ! He failed to mention that. Terminology is kind of important. A circle is not a ball. But ,I can make infinite circles out of one. But, calling a circle a ball is just wrong !!!
it was just a funny demonstration, we cannot modify matter with the precision the theorem would require. But the theorem says that you can definitely make two cubes
No. You simply assumed Infinity does not obey rules applying to the other numbers. The paradox appears only because of that assumption. When a concept such as infinity assumes many “values”, then there is no real or physical equivalence for it. Only in the thought process… MATH has mislabeled infinity as a concept applied to numbers. What MATH actually requires, is a concept of undefined number, or unknown value, which can take on only one “value”. Then there will be no paradox…
It doesn't matter if the last two solids are hollow inside? Doesn't that change their measure?
Correct, he didn’t actually make two cubes from one here. The demonstration is just a physical analogy. Mathematically the Branagh Tarski paradox is still true.
@@pmcate2 I don't know if it is a lack of imagination from my part... but is impossible for me to try to visualize how the Banach-Tarski paradox could work. Maybe it is just one of those math theorems that can't be visualized and we have to trust the proof
In this video, I was merely trying to have some fun by pretending to illustrate the paradoxical decomposition---it was all "tongue in cheek"!
In the actual Banach-Tarski partition, the pieces are nothing like these subcubes and triangular prisms, for exactly the reason you mention. In the actual partition, each piece is dense in the unit ball, although they do not overlap; those pieces are not themselves measurable---they have no coherent definite volume, and this is a requirement of the theorem, precisely because rigid motion preserves measure.
@@joeldavidhamkins5484 it's very funny to me that this is a joke where i understand neither the real nor joke explanation only that the joke explanation is wrong for the reason the first response said... good gag! XD
True, but he also wasn't applying the Axiom of Choice in this demonstration.
Obviously this is just a didactic interpretation! And a very creative one! Beautiful! 👏🏽👏🏽
Nice! Where I can buy a cube like this?
I also want to know
Me 3
rubix magic thingy
Can you further split the two cubes to an infinite number of them?😂
Yes I can
Yes acording to the paradox v sauce has a much more in-depth video of it
It is not possible to do this with the cubes shown.
In order to take on the star-like shape they must be empty.
it is possible mathematically but not physically
Sure: you can split a solid body into a countably infinite collection of pieces, move these around, and get countably many copies of the original.
Absolutely beautiful!!!
So, the interesting thing of the Banach-Tarski paradox is not that we can transform "one ball" into "two balls" (since we know that an "ℝ³" ball has the same cardinality as the whole "ℝ³" set), but "how" we transform "one ball" into "two balls". Isn't it? 👀
Hold on... you made two because the original 'cube' in your hands was already in manufactured into two pieces that were put together as a single; you simply separated them in an entertaining and somewhat fanciful way. How does this translate to the Barnach-Tarski Paradox? Serious question....
excellent. you can also demonstrate the concept of anti-time by putting the cubes back into one. bravo. 👏
Hi, I am sorry if this is a stupid question, but do you have any resources or information on why this is the case specifically due to the axiom of choice?
Because if you negate the Axiom of Choice then every set is Lebesgue measurable and therefore Banach-Tarski becomes impossible (measure must be conserved!)
More precisely, if the axioms of set theory are consistent, then it is consistent with the axioms of set theory without the axiom of choice that all subsets are Lebesgue measurable.
No question is stupid. Especially when a maxim is wrong.
@@VeganCookies Well, it's also consistent in absence of axiom of choice that the uncountable set of all real numbers is a union of countably many countable sets; in which case the Lebesgue measure doesn't work as you would expect it to (namely, it isn't countably additive). But good luck doing calculus in this model of set theory!
Maybe Banach tarski is a real world phenomenon in where the second cube is in existence within a parallel universe and an infinite number of universes exist.
thats the code
That's not how it works ....
I only know in the Euclidean plane, two figures that are equidecomposable with respect to the group of Euclidean motions are necessarily of the same area, and therefore, a paradoxical decomposition of a square or disk of Banach-Tarski type that uses only Euclidean congruences is impossible.
The only raminder question here it's. Where may I buy that cube? Please, let me know.
Amazing proof of concept. Is there somewhere one can acquire one of these cubes?
Search up “infinity cube”
WOW. As a layperson, I thought this was really interesting and educational since it is Lockdown here in Sydney, Australia - why not tune in 🤗
Great way of presenting it! :D I want me that cube :D I guess tue question is, can you do it again for each of the two physical qubes. I don't think so :P
Yes acording to the paradox you could however you could not do it with that cube because that cube is just a demonstration device vsauce has a much more in-depth video explaining it
@@willamdiprofio9091 If I understood correctly, it's possible only in theory. But there's no way to apply the method practically. Which means it isn't really possible. I mean, if something can't be demonstrated with a practical experiment, but only explained as a concept, we can't consider it possible.
@@gracefuldice1956 I agree
Thank you
Suponho que a "estrela" inter tenha igual perímetro que o cubo que ela forma certo? Mesmo assim não houve a conservação do volume real da forma geometriaca porque ela ficou com um espaço interno parcialmente vazio, certo? Mas na geometria de Euclides isso é possível?
amazing!
Love this!
OMG I don't care if it causes universe ending paradoxes --- I just want that toy. Where do I get one?
I wonder what the weight is of each cube compered to the first. Guessing that the 2 each weigh half as much as the one, if not its f*&*n magic :)
Volume within would probably be a better measure.
Why is it a paradox? Aren't the new 'cubes' just approximations of the original cube? So, this splitting can go infinitely, just each new pair will be 'hollower' / 'thinner' ?
In reality can be possible too, if we have atom-controlling beam.
How do we measure the volume?
What is a rigid motion here?
Banach-Tarski tells you that the 2 cubes you end up with are identical to the one you started with. Not approximations, the exact same
In reality, we can never make this happen. We would need to be able to partition an object into infinitely small points, which just isn't possible in the real world
Greetings, please what is the name of that rubik cube you have ? and where can i buy it?
where can I buy a model
Where can i buy such cubes ?
So it looks like the volume doubled but the density halved for each?
Not really. (That is, unless you mean that infinity divided by 2 is still infinity, and so the density remains the same.) A solid ball is the set of all points whose distance from the center is no greater than R. Banach-Tarski paradox is the following theorem of set theory: assuming axiom of choice, it is possible to take a solid body (for example, a solid ball), split it into finitely many subsets, move these around by translation and rotation (without overlap), and get a solid body of a different volume (for example, two copies of the original solid ball). And a solid body is the set of *all* points within the solid body; it can't contain just some of the points (that would be a subset of the solid body). In particular, it can't contain "every other point" (which isn't a meaningful notion on reals anyway - there's no such thing as adjacent points).
When you move or rotate a solid body, its volume doesn't change. The trick, then, is that the pieces in question are not solid bodies and don't have a well-defined volume; the volume can't be zero, and it can't be non-zero either.
@@MikeRosoftJH Thank you so much for answering my question. I will try to wrap my head around what you posted. Thank you again.
That was so awesome!!!
Umm, doesn't that paradox concern *infinite* division of the spheres?
No. The sphere is divided into 6 bigger parts. These part consist of infinitely many elements, but there's still only 6 of them
@@methatis3013 If I'm not wrong, it consists of dividing the sphere into 6 parts and through rotations and translations (without breaking, stretching, or reflecting them) obtaining two identical balls.
Which is a stronger condition than being just equipotent. Isn't it? 👀
@@GabriTell I mean, that's the point. You partition the ball into finitely many pieces (6 to be precise), and by rearranging them, you get 2 balls in total
Beautiful 😭
Holy Moly!
24 April
11;20 Nonton UA-cam Sciencephile Iceberg Of Paradox 11:42 Joel David Hamkins Banach Tarski Paradox tapi balik lagi buat komen 11:44 Lanjut Joel David Hamkins 11:48 VSauce Banach Tarski tapi balik lagi buat komen
Good, good.
They really shouldn't be posting videos like this on the internet and leave them unattended 😅😅 someone could get injured
People usually blame AC, but I think that's a result of faulty formalisation. It also has another flaw: you can only give each point of the border to one piece, so an additional construct is needed to prove that you can turn any polygon into any polygon of the same area
trippy!
What you doing is creating fume. The Banach-Tarski paradox is an interesting mathematical concept, it has no practical application in the physical world. Although in video whatever magic your performing to create non-measurable sets in the physical world, the reassembly process would require an infinite number of infinitesimal transformations, which is not possible in practice due to the constraints of physical reality. Can you do again with the other two cubes??
🤍
very cool
🤯
HEY PEWDSSSSSSSSSS
😄
Wait what
WHAT?!?!?!??!
This is a theorem of set theory: assuming axiom of choice, it is possible to take a solid body (for example, a solid ball), split it into finitely many subsets, move these around by translation and rotation (without overlap), and get a solid body of a different volume (for example, two copies of the original solid ball). When you move or rotate a solid body, its volume doesn't change. The trick, then, is that the pieces in question are not solid bodies and don't have a well-defined volume; the volume can't be zero, and it can't be non-zero either.
kinda like cell division
And bumm the universe was created.. .
what up chat and pewds
The BS stems from the fact that the cube has been hallowed out by 50%. Hence, NOT A CUBE !
He failed to mention that. Terminology is kind of important.
A circle is not a ball. But ,I can make infinite circles out of one. But, calling a circle a ball is just wrong !!!
Still, cool toy. 😎👍
it was just a funny demonstration, we cannot modify matter with the precision the theorem would require. But the theorem says that you can definitely make two cubes
😲
Im from pewds live stream
You're not allowed to do that
WTF
No. You simply assumed Infinity does not obey rules applying to the other numbers. The paradox appears only because of that assumption. When a concept such as infinity assumes many “values”, then there is no real or physical equivalence for it. Only in the thought process…
MATH has mislabeled infinity as a concept applied to numbers. What MATH actually requires, is a concept of undefined number, or unknown value, which can take on only one “value”. Then there will be no paradox…
This is why in calculus we approach infinity
You are rambling incoherently