Vsauce made a great video about this https://youtu.be/s86-Z-CbaHA
Banach-Tarski and the Paradox of Infinite Cloning
61–70 of 148 posts
Re: Banach-Tarski and the Paradox of Infinite Cloning
#62Earlier quoted context omitted.
That's just because you use the word "exact", though. Exactitude doesn't exist in the universe as we understand it. There's a difference between something not being instantiated in this universe and being unscientific, though. If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it u…
> Exactitude doesn't exist in the universe as we understand it. Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge. > There's a difference between something not being instantiated in this universe and being unscientific, though. Well, science is a particular way of studying what exists. Studying something that doesn't exist is unscientific (of course, y…
> Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge.
Aren't claims like this unscientific according to your standard? You will never be able to measure that two electrons have the same charge to infinite decimal precision. You might have a theory that says they should have the same charge, but you won't be able to test that theory to infinite precision either.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#63Infinity is the axiom of paradox. Does the inclusion Infinity complete an otherwise incomplete set of axioms? It solves the halting problem for a finite Turing Machine. I don't buy the diagonalization proof as anything more than the Pythagoreom Theorom. You have infinite rows, and infinite columns. Infinity is Schrodinger's Cat. Once you check in on the state (nth row by mth column) the only thing you can say about t…
From your description, I fear you don't understand the usual diagonalisation proof that constructs an uncounted real from any attempt to count the reals. Why should "the longest" diagonal have anything to do with it?
Re: Banach-Tarski and the Paradox of Infinite Cloning
#64Earlier quoted context omitted.
What I mean is: if you imagine someone drawing up a requirements document for the team assigned to the task of axiomatising geometry, and somebody asked "Do we want our model of geometry to support cutting up a ball into five pieces, moving the pieces rigidly, and reassembling them into two copies?", I think their first idea would be to answer "no". So it isn't parallel to the intermediate value theorem, but opposite…
The whole idea of a proof system is that there are some things you can't have without also having other things. The Banach-Tarski theorem is a consequence of things we want. You don't get to pick and choose everything at once.
> The Banach-Tarski theorem is a consequence of things we want
Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski”
Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want.
You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axioms, we get nonsensical results.
In general, it seems like we’ve picked _pretty good_ axioms that mostly give us sensible and useful results. But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#65I don't understand the paradox. Obviously if you dissaemble or scamble something u can reararange it?
When's the last time you came across something that you could disassemble and could then reassemble into two things identical to the first thing you disassembled? It is natural to suspect that foundational axioms are somewhere flawed.
One thing worth pointing out is that our universe operates on the integers rather than the real numbers, and the Banach-Tarski requires operating on the reals.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#66Earlier quoted context omitted.
When's the last time you came across something that you could disassemble and could then reassemble into two things identical to the first thing you disassembled? It is natural to suspect that foundational axioms are somewhere flawed.
One thing worth pointing out is that our universe operates on the integers rather than the real numbers, and the Banach-Tarski requires operating on the reals.
Is that known? It's an appealing idea, but bearing in mind that general relativity is very resistant to quantisation, I'm not sure I'd be comfortable to declare it as fact.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#67Earlier quoted context omitted.
What about negative numbers? Or complex numbers? They are only tools, which can be quite useful to build models of the world with predictive powers but shouldn't be confused for the underlying reality. Even whole numbers are an abstraction that makes sense only when you can clearly define what is the thing you're counting.
Whole numbers can be defined and proven to be necessary to describe the world pretty easily. From there, rational numbers are trivial to define. Negative numbers are somewhat more abstract, but they have very intuitive definitions in many domains, such as accounting. It may be possible to avoid them in a theory of physics, though. The complex numbers (well, at least those with a rational imaginary part and a rational…
Infinity may be also "necessary to describe the world". But like every tool, you need to know its limits.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#68Can someone correct me if im wrong? What i see here is a splitting of the set of points in the sphere? However the set of points in the sphere is not really the sphere. A point has no volume so no matter how many you add together you don't get something with a volume. This seems more akin to splitting the natural numbers into odd and even numbers which are all equally large. The language that i see in this article an…
> The language that i see in this article and elsewhere however is suggesting that we actually duplicated the sphere (doubled the volume). > This seems incorrect. It isn't incorrect. You're right that the number of points in the sphere does not equate to the volume of the sphere. But the Banach-Tarski theorem does in fact let you double the volume. It is considered to be of interest because it does the following: 1.…
"cut the ball into 5 pieces" is not the best description.
A better one is:
2a. Split the ball into infinite pieces
2b. Divide the infinite pieces into 5 groups
Re: Banach-Tarski and the Paradox of Infinite Cloning
#69Earlier quoted context omitted.
That's just because you use the word "exact", though. Exactitude doesn't exist in the universe as we understand it. There's a difference between something not being instantiated in this universe and being unscientific, though. If we produce a model of the universe that doesn't make a single incorrect prediction given all data available, and it predicts infinities to exist in some strange but quite real cases, is it u…
> Exactitude doesn't exist in the universe as we understand it. Of course exactitude exists. For example, two electrons have exactly the same charge. A photon has exactly 0 charge. > There's a difference between something not being instantiated in this universe and being unscientific, though. Well, science is a particular way of studying what exists. Studying something that doesn't exist is unscientific (of course, y…
As the other comment implied, infinity and exactitude are two sides of the same coin. Exactitude is infinite precision. No finite amount of empirical evidence can afford infinite precision, so you’re back in math-land.
Re: Banach-Tarski and the Paradox of Infinite Cloning
#70Earlier quoted context omitted.
The whole idea of a proof system is that there are some things you can't have without also having other things. The Banach-Tarski theorem is a consequence of things we want. You don't get to pick and choose everything at once.
> The Banach-Tarski theorem is a consequence of things we want Is it? I think the parent comment is saying: “maybe we shouldn’t want things that result in Banach-Tarski” Maybe it’s a hint that the underlying axioms we’ve selected aren’t exactly what we want. You’re right that we can’t pick and choose the results of our axioms, but we do explicitly get to pick and choose the axioms we start with. If we choose bad axio…
> But maybe this result that seems somewhat… odd, is an indication that those axioms have an odd corner somewhere.
The only way you're going to avoid getting results like this is with axioms like "there is no such thing as an infinite number". At that point, the real line doesn't exist (too many points) and it becomes impossible to duplicate spheres by dividing them at a level of fineness that also doesn't exist.
But that's not a productive approach to anything.