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