Nevertheless it feels like the point of Banach-Tarski is that it proves math went wrong somewhere. Evidently the sets it's talking about are not objects which are interesting in reality.
Math is not reality. Physics isn't even reality, they're just theoretical frameworks that are easy to work with and somewhat align with what we've observed. "All models are wrong, but some are useful" as they say.
If you can prove something can or can't be done in math it doesn't mean shit, but it might end up being a useful guideline. Banach-Tarski assumes an infinite pointcloud (i.e. a mathematical sphere), which as you've realized, doesn't actually exist.
Yes, all models are wrong, some are useful, and the model math is using is evidently not very good. Probably there is a better one out there.
Physics isn't reality but it does use "closeness to reality" as a metric for quality of a theory, and that's by the metric by which Banach-Tarski is irritating.
1. Accepting versus rejecting the axiom of choice.
2. Classical versus intuitionistic mathematics. Intuitionism goes much further than just rejecting the Axiom of Choice, and e.g. says that you haven't proved "p or q" until you have either proved p or proved q. It denies not just the axiom of choice but the "law of the excluded middle" which says that for any proposition p, either p is true or not p is true.
3. Accepting versus rejecting "large" sets -- in your case, you say we should be suspicious of anything that assumes the existence of uncountable things. (I think this is a bit unusual; there are finitists who deny that there are any infinite sets, and ultrafinitists who go further and say there aren't even arbitrarily big finite sets, but it's not so common to accept countable but not uncountable infinities.)
I think intuitionism implies rejecting AC. I don't think any of the other possible implications between these three things holds; e.g., so far as I know most intuitionists have no particular problem with the existence of large infinite sets.
(Actually, probably some theorems of the form "if there are no large sets, then the axiom of choice is true for boring reasons" are provable.)
> and the model math is using is evidently not very good.
There isn't really one "model of math". Plenty of people study alternative foundations of math or various axioms you can attach to ZF, etc. and its great that they do - choice, and lots of other set theoretic/foundational stuff, is very weird. But the reason mathematicians have largely settled on ZFC as a default is that it (and maybe nowadays + an inaccessible cardinal) also reduces some pathology and makes math more convenient.
Let's list some things which are related to the axiom of choice:
* The cartesian product of nonempty sets is nonempty (equivalent)
* The reals can be partitioned into more parts than there are real numbers (consistent with the negation of choice)
* All vector spaces have a basis (equivalent)
* All commutative rings have a maximal ideal (equivalent)
* All fields are contained in an algebraic closure (implied by choice)
The first two, at least to me, are similar to Banach-Tarski in that they are things I would like to be true and false, respectively, and are not if we do not accept choice. The point here is that the weirdness of Banach-Tarski is as much related to the axiom of choice as it is to generally the fact that (uncountable) infinities are just very weird, and while introducing choice does introduce pathologies like Banach-Tarski, it also reduces some.
The last three illustrate a more practical perspective. Suppose a universal decree, that math is no longer allowed to be done in ZFC, you could only use ZF, was imposed on mathematicians. This doesn't really change anything - all that mathematicians will do is, if they were perfectly fine with choice before, simply replace instances of "vector space" with "vector space with a basis", or "commutative ring" with "commutative ring with a maximal ideal", because the types of mathematical objects they care about are the ones with these desirable properties - they only use the axiom of choice because it ensures the general objects they work with also have those desirable properties. And mathematicians (often even the same ones as before, just wearing a different hat) who _do_ care about choice will continue studying those weird instances of the general object just as before.
As a final point for the convenience of axiom of choice, there are lots of instances where a proof will use choice purely as a matter of convenience. Maybe with some technical set theory or a smarter argument its possible to completely eliminate the requirement of choice, or use a weaker, less-objectionable version of choice. Other times, while a general vector space having a basis requires choice, the vector spaces in your particular application have a basis regardless of your thoughts on the axiom of choice. But of course its much easier to simply assert "vector spaces have a basis" and introduce a (faux) dependence on the axiom of choice. For a concrete example of this last point, see https://mathoverflow.net/a/35772. While about inaccessible cardinals/"universes" rather than the axiom of choice, the principle I want to illustrate is the same. The proof of Fermat's last theorem, if you trace citations back, eventually depends on "universes". But the actual invocations of those theorems are applied to objects where that generality is not required.
(Why not simply only have done the non-general scenario? Because "holds in general with additional axiom, holds in specificity without" is more knowledge than just "holds in specificity")
> The last three illustrate a more practical perspective. Suppose a universal decree, that math is no longer allowed to be done in ZFC, you could only use ZF, was imposed on mathematicians. This doesn't really change anything - all that mathematicians will do is, if they were perfectly fine with choice before, simply replace instances of "vector space" with "vector space with a basis", or "commutative ring" with "commutative ring with a maximal ideal", because the types of mathematical objects they care about are the ones with these desirable properties
Lol, I would love that. That's the book I want to read. I am weird, maybe, but I find the full-generality of mathematics to be exhausting when I just want to understand how numbers and geometry work. I will never care about the details of infinite sets.
I am reminded by a quote from Jaynes' probability text: that in their opinion, infinite objects are only meaningful when explicitly provided via a limiting process from finite objects. It is not, as far as I can tell, a widespread stance, but it's the one I subscribe to.
(Thanks for the lengthy reply, though. Just, it reminds me of the stuff I already find exhausting.)
The best/worst part is that there are useful branches of mathematics which assume the Axiom of Choice is true AND there are useful branches of mathematics which assume the Axiom of Choice is false. That's one reason I take the view that mathematics does not exist: if math did exist, there would only be one set of axioms consistent with nature.
>> Universe could be itself infinite but locally everything is finite.
> You don't know that.
I think they were saying that it could be that the Universe is infinite with everything still being locally finite. I do not think they were asserting that everything is locally finite, I think it was included in the "it could be" part.
But certainly you're right that we don't know that.
No, they're just plane geometry, albeit in a cludgy notation (which conflate vectors and rotation operators on vectors because they happen to be isomorphic in R^2). But plane geometry is a real thing.
Imaginary numbers are just 2D vectors. Its such a horrible name. The imaginary part isnt any more mysterious than the real part. Theyre just orthogonal properties.
What are imaginary numbers if you remove its relation to the real numbers? Do they not become regular real numbers?
How I am seeing it, real and imaginary numbers are both equivalent constructions with the same properties/construction (you can add/substract/multiply/divide them the same way), and its only in the context of using both (in the context of the complex numbers), that they can be differentiated.
That's not quite right. Imaginary numbers are vectors with angle addition via multiplication. Real numbers all have zero angle, so their angle addition is trivial.
Is that a property of imaginary numbers, or a property of relating two orthogonal numbers? Legitimately asking because I can't find any special properties that imaginary numbers alone have in relation to themselves.
