What use is the Banach–Tarski paradox?

What use is the Banach–Tarski paradox?

Why would anyone even need an infinite number of spheres?

>What use is the Banach–Tarski paradox?
demonstrates unexpected results of the axiom of choice

>Why would anyone even need an infinite number of spheres?
what does this have to do with banach-tarski?

idk it was a joke
but my head is so blown everything seems funny
set theory is a mindfuck

>What use

None, it is a manifestation of axioms. It does not mean anything for the real world. Axioms mimic our reality but the problem is that mathematics is so precise that even the universe cannot match it.

Proof:
In the real world circles have both rational diameter and circumference

In mathematics circles have either rational diameter and irrational circumference or irrational circumference and rational diameter.

Thus proving that the universe is actually a simulation. By measuring real life circles we can approximate pi, just like you could approximate pi by writing a javascript script that draws a circle and then measuring that circle,

>What use is the Banach–Tarski paradox?
it proves that you can generate an infinite number of infinitely thin wedges out of one perfect sphere. it's the weapon of the future, an inexhaustible ammunition. every gun will have a small sphere in it that generates an infinite number of small needles that are propelled out the front.

Thanks to Vsauce every brainlet with internet conection feels that he can undestand le highmath le banachtarski xdddddd without knowing shit about Real Analysis,

Amenable groups

my thoughts exactly, im no genius, scientist, or engineer but I still find this stuff interesting so I lurk here but I'm not stupid enough to make threads without googling it and doing my own research first

This particular issue is very intuitive, trivial even.
You don't need someone to tell you about it to figure it out yourself.
The only problem is the obvious lack of rigor. But that shit is for mega-autists, so who cares

In the real world circles do not exist

I never really got why banach tarski is the go to paradox and not vitali sets.
Fucking normies I swear.
It's le schrödinger's cat of popmath.

>Proof:
>In the real world circles have both rational diameter and circumference
That ain't a proof, bob.

This paradox works in a situation where the sphere has an infinite set of points across its surface area. In reality, the maximum number of points possible on a sphere is equal to the surface area/planck's distance, which means this theorem has no direct application to reality.

>In the real world circles have both rational diameter and circumference
In the real world circles dont exist

It is satire, like Schrodinger's cat. It shows the absurdity of the subject.

More importantly: What's the evolutionary benefit of the Banach-Tarsi paradox?

you do not need to know real analysis to understand banach tarski

no analysis is involved in banach tarski

basic knowledge of set theory and group theory is need for banach tarski

>This particular issue is very intuitive, trivial even
Hmm. Obviously not if the result of the theorem is so profound it is called a paradox. The rigor explains everything about why the theorem works. Without rigor you have a bunch of results and no idea of why they work and become an idiot that can only parrot results.

>which means this theorem has no direct application to reality.

I don't think you have enough imagination. It's the logic that is valuable, not this particular interpretation of the theorem. For example, what the fuck is the use of mathematics in a 50 dimensional space? Well for one, image processing where you split it up into sub blocks represented by vectors where 50 different features are described in each vector (covariance, mean, colour, etc..).

>Without rigor you have a bunch of results and no idea of why they work and become an idiot that can only parrot results.
To add to that, "without proof, it's just opinion. And then where would you be? In a humanities department."

Spheres could be used for many things, like construction, you could have spheres making up the foundations and frames of houses, buildings, schools, you could build shelters for the homeless, you could build structures on the Moon and other planets

Imagine sending these spheres to Mars, and let it self-replicate, it could build cities by itself

Circles do not exist in the real world you ultimate poof

>what the actual fuck is a measure
>no analysis required
k

But it is intuitive.
Think about circles, it must apply to them as well.

You are clearly not using your child like imagination to find wonderful applications.

Wow, you got it right.

Far too many people failed to realize Schrodinger's cat was satire, as not many bothered to read the original papers, which are surprisingly hard to find despite being so influential.

Not sure about this sphere thing, but wouldn't be surprised if it was a similar thing. After all you tend to get weird results when ever you mix finite and infinite systems.

An infinite number of spheres means an infinite amount of mass. The plan is to use the theorem atop a superstructure skyscraper, where the balls will fall down and drive turbines, creating nearly infinite energy.

All that's stopping us is that we need sharper blades to do the required amount of cuts. In the lab, scientists have managed to duplicate an entire orange using a $530 million dollar graphene blade.

The Reals are not useful or real. They are an abstraction without use. Please do not use them anymore.

Once again the universe is saved by the science of economics!

It matters not how many rules of physics are broken, so long as the economic utility of them is unfeasible to widely adopt. This is why I don't lose sleep over people flooding the planet with balls that block out the sun and/or increase the local mass to the point of creating a planet crushing black hole.

As long as it costs ~$600 million to duplicate an orange from nothingness and the majority of people are not willing to spend ~$600 million to do such, then it shall not be done. And I can continue using simplified matter conservation equations in my designs, without fear of our existence destabilizing.

You don't need to know any measure theory to understand it.

>muh infinity bullshit
yeah, nice """""""paradox"""""""

You are so much better and smarter than everybody else you special snowflake

Surely the Planck distance squared?

I am not talking about bullshit that I do not understand as the spergelords do with Banach Tarski

>the maximum number of points possible on a sphere is equal to the surface area/planck's distance
Reminder that there's no actual indication that the Planck length is exactly the minimum distance

plank length
>muh, universe pixels

Thats been disproved and there is no planck length constant

Generating interesting examples and such allows us to train our intuition and guides how we define things, and what we want to study. This in turn leads to us creating good mathematics, some of which ends up being used irl, some not.

I don't understand how you can "disprove" it since it's more of a concept than something physical.

>there is no planck length constant
Well there is, just no-one knows if it's significant for anything.

Can't just just all agree that the axiom of choice is bullshit?