Banach - Tarski

njwildberger.com/2015/12/03/the-banach-tarski-paradox-is-it-nonsense/

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.

>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.

youtube.com/watch?v=s86-Z-CbaHA

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

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

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

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

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

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.

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

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.

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

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

I was expecting this.

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?

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

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

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

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

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

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.

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