Banach - Tarski

Mathematical proofs are always correct by definition.
the real world, however, is not continuous and so it has limitations.

>>8222254
What are you even asking?

>>8222288
why is this paradox not rejected as stupid and not - creative

>>8222254
Why is it a problem? This is not a statement about physical reality.

>>8222296
Why is that kind of decomposition of sphere allowed since it induces non-measurable sets?

>>8222301
cuz we made an axiom that said we could
realists BTFO

I'm sorry I don't actually know I was just joking.

>>8222293
>coming up with a valid mathematical argument which uses non-measurable sets to produce a result that shatters usual intuition
>not creative

Maybe you meant "not productive"

>>8222306
Yes. Sorry. That's what i thought.
I was just trying to figure out path of thinking since paradox first appeared. Did mathematicians went

"Ok, we're gonna have to interduce non-measurable sets and...well...yeah, they can't have volume...so...No, this is not good."

or

"Huh...this axiom already produced two paradoxes...maybe we should...oh, wait, we can't just reject it, it's too logical"

>>8222301
>>8222311
Basically, the idea is that non-measurable sets are awful. This is the case beyond Banach-Tarski. The result is paradoxical because you're seemingly creating more volume, but you're using sets that have no notion of volume to accomplish this.

>>8222254
https://njwildberger.com/2015/12/03/the-banach-tarski-paradox-is-it-nonsense/

>>8222311
No, the idea of non-measurable sets already existed before this. Some people said "no, this is not good," but others asked "why not?" I mean you CAN do math without the axiom of choice, it's just really boring.

A friend of mine explained the proof of Banach-Tarski to me once but I forget because it was a while ago, though I do remember that it actually has a lot more to do with the study of amenable groups than it has to do with the notion of non-measurability.

>>8222331
>I mean you CAN do math without the axiom of choice, it's just really boring.
It's like saying you CAN live without psychedelics, but it's just really boring.

>>8222331
https://www.youtube.com/watch?v=s86-Z-CbaHA

>>8222311
analysis would be pretty shitty without existence of a basis, hahn-banach or baire, so I'd rather not do without them just to have the whole power set of the reals lebesgue measurable

>>8222358
Axiom of Choice is more like microprocessors than psychedelics: it's kind of ubiquitous without you fully realizing it

>>8222358
Algebra would kind of fall apart without the axiom of choice. It comes up directly or indirectly all the time.

>>8222426
>>8222430
Ok, I'm just a layman who overdosed on constructive logic and type theory.

>>8222552
lol, yeah, AC and constructive logic don't party together.

banach and tarski considered this a paradox about the axiom of choice, rather than measure theory
remember that tarski was a logician

>>8222254
The paradox is a nontrivial consequence of the AoC. There's no problem in Measure Theory itself.

Without AoC, it's not possible to prove the existence of nonmeasurable sets. I'm pretty sure even with DC (dependent choice), you can't construct one. So without AoC, the Banach-Tarski paradox is circumvented.

I'm no expert on this either, just some idiot who took Real Analysis back in grad school. What I want to know is why we don't just adopt countable choice or dependent choice and just call it a day.

>>8222620
because then algebra and analysis fall apart. no maximal ideal, no basis for your vector space, etc

>>8222555
It's just that treating mathematics like a programming language, a'la Martin-Löf, appeals to my plebeian soul. I wonder how the HoTT guys are going to fare.

>>8222631
What rings and vector spaces in particular need the AoC but not say DC? Isn't countable enough for the things that actual matter?

>>8222750
You absolutely need Zorn's lemma to do any serious algebra.

>>8222328

I was expecting this.

>>8222755
Alright, I realize you use Zorn's Lemma for things like guaranteeing maximal ideals and such.

What I'm asking is what particular algebra (or class of algebras, whatever) suffers from not having it? Isn't there a weaker version equivalent to DC or CC that would suffice?

Like the integers have (obviously) only countably many ideals. I guess you wouldn't have the same result for R, but do we care about maximal ideals in R?

>>8222775
It's always a pleasure to spread the word of Wildberger

>>8222795
Consider the fact that the most important rings are polynomial rings over algebraically closed fields, which have uncountably many prime ideals.

>>8222814
> caring about prime numbers
> thinking you can put infinite things all into the same set

>>8222849
I honestly don't have a clue what you're trying to say.

>>8222814
>which have uncountably many prime ideals.
Okay, this is actually not true in general, but it is true often enough.

>>8222254
Math is a discipline that is as the following:

1. Bunch of definition
2. Axioms (things I'll assume are true but can't prove)
3. Theorems (things I can prove from 1 and 2)

Just because someone can construct something counter intuitive doesn't negate your mathematics.

Banach-Tarski is equivalent to the concept of a space filling curve (which can be extended to any number of dimensions). An unexpected result based on flaws in your definitions/axioms. Your sequence of curves are countable but fill in uncountable irrationals?

Also look up the Lakes of Wada for another counter intuitive result.

>>8222898
OP here. No, it's not the same as space filling curves.