How the hell does the Banach-Tarski paradox work? I've read about it, but I just don't grok it.
>le i watched a youtube video but my mind cant get arround it 3hard5 me xd
popsci fag detected
>>7683959
No, I actually *read* about it. Couple different explanations, too. Perhaps I just don't have the necessary math background?
>>7683954
Exactly what part of the proof you don't understand?
Overall, it works thanks to the axiom of proof and non-measurable sets, without any of them it wouldn't be possible.
If there's a particular part of the proof that you don't get, post it and I may be able to explain it further.
>>7683954
>grok
I like you :)
>>7683954
>vsauce
>>7683971
It is just a trick involving group actions and the size of the sets.
I'm assuming you know how a group of symmetries can act on a set. The free group presents a group of symmetries (isometry of 3D space), but also provides a way to list out the points of the sets. By looking at the orbits and choice, you can pick representatives of each orbit and use them to construct the original balls.
>>7683977
>axiom of proof
axiom of choice :)
>>7683954
Consider the one-dimensional equivalent:
There are as many points between 0 and 1 then there are between 0 and 2.
Therefore, there must exist a one-to-one mapping between points on a line segment of length 1, and a line segment of length 2.
This is equivalent to saying there is some transformation that converts a length 1 segment to one of length 2 by rearranging the points.
The same holds for higher-dimensional infinite sets of points.
>>7684354
the banarch tarski paradox does not work in fewer than 3 dimensions you popsci kid
>>7684354
This is so off-point and blatantly wrong it hurts.
>>7684376
Only because of the additional complication that the mapping must be accomplished by rotations and translations, and is thereby paradoxical in that these operations ought to preserve volume; it still does the same thing.
I assumed OP's primary confusion was how transforming a small object into a large one was possible at all.
>>7684388
You have no idea what you're talking about. It's NOT the same thing, it's not even an homeomorphism, it's measure-invariant operations along with set-operations which finally change the actual measure.
Your operation is just a dumb biyection, you're just guessing it has to be the same thing, but it couldn't be more wrong.
>>7684393
Welp, it appears I have once again summited Mount Stupid.
> grok it
You dumb type 0 alien immigrants make me sick
>>7684354
The counter-intuitive thing about B-T is that it works with finitely many subsets. You can't do that in one or two dimensions because of congruence.
>>7684226
>I'm assuming you know how a group of symmetries can act on a set.
And OP was never heard from again
>>7684354
no, there's twice as many
learn basic addition, kid.
>>7684354
You're stupid and I hate you.
>>7684648
Honesty is a rare beauty. Thanks for posting.
>>7683954
Lets say you have an infinite line of CS monkeys that you want to kill but you still need an infinite line of them. Take every other CS monkey out of the line and form a new one. Now you have 2 equally infinite lines exactly the same as the original.
>>7683954
It's based on the Axiom of Choice.
>>7684354
Don't listen to the haters, bro, I like your analogy :)
>>7684684
Infinity is a beautifully strange concept.
It's logical and illogical at the same time.
>dude weed lmao
>>7684684
I've never read the Banach-Tarski paradox proof nor do I know the necessary mathematics to understand it beyond Real Analysis but I understand basic things about the real numbers (I do know that the Banach-Tarski paradox uses properties of the topology of real numbers).
What you're talking about is a bijection or rather making a concrete statement about equal countable sets (as an example, the cardinality of the integers is equal to the cardinality of the positive integers). The real numbers aren't countable so the logic you're talking about makes no sense in this context.
>>7684354
BANACH TARSKI ALLOWS NO STRETCHING YOU FUCK
ITS ALL AFFINE TRANFORMATION
>>7685833
>NO STRETCHING
>AFFINE TRANFORMATION
lel
https://en.wikipedia.org/wiki/Affine_transformation
>>7685830
>what is countably infinite
The other anon makes perfect sense, There exists a bijection between [math]\mathbb{N}[/math] and [math]\mathbb{Z}[/math].
We call the first line of CS monkeys [math]\mathbb{Z}^{+}[/math] and the second [math]\mathbb{Z}^{-} \cup {0}[/math]. And there is the bijection he was talking about.
>>7684646
>engineer detected
try gitting gud before talking shit on the internet, kid
>>7683954
Haha, Banach
https://www.youtube.com/watch?v=n12bfWTw9Hk
>>7686352
you double playing the fool. he clearly said this logic doesn't extend to uncountable sets. yes, there are (many) trivial bijections between Z and N.
How exactly do you plan on taking out "every other real number"? You cannot, they are not countable.
This really plays little role in B-T (though AC does), but in general, if you are trying to pull a trick like that (which you don't need here) on uncountable things you better try something like the back-and-forth method.
>>7685700
but it's not an analogy pertaining to this paradox. his example had to do with the size of uncountable sets, but he did it through rescaling.
It is no paradox that, allowed stretching and scaling, you could cut a sphere in two and make two smaller ones, then blow them up.
In fact, his example had to do with homeomorphism, and transformations in the B-T are NOT homeomorphisms.
The B-T pararox comes from a fact about rearranging objects in a rigid fashion, i.e. by group actions.
So, it's an ok analogy for a completely different problem, but is actually misleading here, using the very tools that would cease to make this paradoxical.
>proof: you probably didn't realize that extending his example to spheres ceases to make the situation paradoxical
>>7683954
It really isn't that amazing.
Consider that there are as many natural numbers as there are even numbers (because of the bijection N->2N: x->2x)
Or consider that there are as many reals on [0,1] as there are on [0,2] (x->2x).
Consider that there are enough points for 2 cubes in one cube. Let's say we have {(x,y,z) | x on [0,1], y on [0,1], z on [0,1]}. Observe the bijection (x,y,z) -> (2x,y,z). Now we fill 2 cubes. However, you will see that there's a problem here. If you take the original cube out, you will see that you are left with a space which is not closed. There are some tricks to get around this. (I assume you are already convinced there are the same number of points)
With balls you have the same thing, but the problems are a bit weirder. The Banach-Tarski paradox is simply that there's an actual way to fix all these problems, in other words: a bijection from one ball to two balls can be constructed.
>>7685830
But it can be done with real numbers too. Consider [0,1]. x->2x is a bijection between this and [0,2]. Throw away all the numbers which would be sent to a point in (1,2] by the bijection. You have thrown away an (uncountable) infinite number of them, yet you are left with as many numbers as there are in [0,1]
>>7683954
>How the hell does the Banach-Tarski paradox work?
It's because of those real numbers...
>>7686448
You can have a turkey too
>>7683954
yes banach banaal anaal oraal vaginaal
The key word in the Banach-Tarski problem is "finite." And it's rather disappointing to see it said only once in this 38-post thread.
A bijective mapping from [0,1] to [0,2] is NOT finite. It's the same magnitude as [0,1] (and R).
>>7687627
Why am I laughing.
>>7686438
https://www.youtube.com/watch?v=HfxfnokQuLM
>>7686441
>What you're talking about is a bijection or rather making a concrete statement about equal countable sets
>countable sets
>COUNTABLE
>>7683954
noone fucking cares, the entire point of this paradox is that it cannot be implemented in real life
