I hope this is not too off-topic / soft for math.stackexchange.
My basic question is: why does the Axiom of Choice allow for some of the weirdest constructions in math? I’ll make a list of the weird constructions I know, and I’d like answers to tackle two things:
I’d like to know what exactly AoC gives us that allows to construct these sorts of strange mathematical objects, and how one should think about this.
This is a very natural question, even if a little subjective (due to the inherent subjectivity of what does it mean “weird”). But the answer is that the axiom of choice is not the cause of weirdness.
The axiom of choice is just a tool with which we can prove there is some uniform “mess” throughout the universe of sets. But it really helps us to rein down some of the stranger mess caused without the axiom of choice. So really the reason people ask this question (and they do, often) is that they are not familiar with disasters that may happen without choice.
So to paraphrase von Neumann, if people say that the axiom of choice has counterintuitive implications they really don’t know how strange can a universe without choice be.
First let’s deal with the obvious problem. The axiom of choice is a global statement. It asserts the existence of some objects for every family. The negation of the axiom of choice does not imply that these objects do not exist, and even the negation of the axiom of choice does not tell us where the axiom of choice fails and how badly.
So it is possible that the axiom of choice fails, but every countable union of countable sets is countable, and the real numbers can be well-ordered and their power set can be well-ordered, and everything anyone ever cared about can be well-ordered, and only somewhere in the very high stratosphere of the universe of sets there is some failure of choice which affects nobody “down on earth”. But of course that in this situation we can just repeat all the choice-y arguments to get all the usual counterintuitive results.
What does it mean? It means that if we want to talk about counterintuitive results without choice, we might as well assume that choice failed at some level “of interest”. What could go wrong?
You don’t want Banach-Tarski? This means that the Hahn-Banach theorem fails, and this means that there is a Banach space without any continuous functionals except $0$.
You don’t want non-measurable sets? This means that you can partition the real numbers into strictly more parts than real numbers, but no part is empty. Yes, let that sink for a moment: you can partition the reals into more parts than numbers!
You can have a situation where the real numbers is a countable union of countable sets. There measure theory goes out the window, because the Lebesgue measure cannot be $\sigma$-additive. You can salvage a little bit of measure theory by talking about codes, but it makes everything a million times more technical (as if measure theory was non-technical to begin with).
You can have the case that $\Bbb N$ is not a Lindelof space. Yes, a countable discrete space is not necessarily Lindelof. And equivalently, it turns out, that $\Bbb R$ could just as well be non-Lindelof.
It is possible that $\Bbb Q$ has two non-isomorphic algebraic closures. They are still elementary equivalent, so to some extent it’s not that bad. But this is something to consider when you want to talk about algebraic closures.
The list can go on and on and on. Every time something counterintuitive fails, you can use that failure to get something which is very counterintuitive to hold. This is some sort of dichotomy, and it is simply due to the fact that infinite sets are weird. This is not the fault of choice, it is the fault of infinity. Actually, this is nobody’s fault, this is why mathematics is so much fun! And this is why we actually bother to write proofs, and not just say “yeah, that’s reasonable, let’s continue!”