The axiom of choice basically says given an infinite number of sets we can select one element from each of them. This assumption is used in many proofs that are the building blocks in analysis and other. What if it turns out to be unreliable?

Do you think we should just leave it to the logicians?

Or do you think we're heading to maths anarchy and at any moment we're going back to the 1900's.

Apocalypse now or millennium bug? What do you think?

It sort of depends on what you mean by unreliable: it's actually a proven statement that if ZF is consistent then so is ZFC, so from that point of view it's perfectly reliable (the Axiom of choice really is an axiom: it can't be derived from the other axioms (ZF)). What most people don't like about it is that it creates some unintuitive consequences like Banach-Tarski (this being probably the most flashy example of a large number of theorems involving unmeasurable sets that you basically can't create without AC). It's a common set theoretic "joke" that AC is obviously true, the well ordering principle (the fact that you can always find a well-ordering of a set - an order such that every set of elements will have a least element) is obviously false and that no one knows about Zorn's lemma (the fact that in any partially ordered set where an ascending chain of elements will have an upper bound has a maximal element - formally, if you have a partially ordered set for which every chain there exists an element such that for all then there is a maximal element which is greater or equal to all the elements of to which it is comparable). These three are equivalent and any of them can be taken as an axiom instead of AC to get the exact same system, but the reaction of someone who doesn't know about this would probably be that AC is "obvious" while the well ordering principle is very counterintuitive (after all, you can instantly "choose" a member out of a set but you would be hard pressed to find a well-order of - no one will arguably ever find one, although the well ordering principle stipulates its existence). Zorn's lemma may seem a bit more complex than these two but is often extremely useful in practice: for example, it's what you use to prove that every vector space has a basis or that a ring has a maximal ideal.

That said, I for one am perfectly happy to accept AC. You tend to get used to the unintuitive theorems and just consider them weird and not by any means "contradictory" whereas by not accepting AC you're making yourself unable to show things like the fact that a countable union of countable sets is countable, that any vector space has a basis, that the product of compact spaces is itself compact, etc, all of which are pretty useful. Even worse, assuming AC is not true can lead to things like vector spaces with no bases or bases of different cardinalities, violations of sequential continuity and other nasty things, which seem a bit more limiting than not being able to measure all sets, for example.

You got Zorn's lemma wrong, the maximal element is not greater or equal to all other elements as it might not be comparable to all of them, it's just never smaller than any other element.

Not sure what some of these posts have to to with the AOC, but hey.

It seems to me that the axiom of choice doesn;t necessarily assert that we can FIND a choice function, merely that one EXISTS. If we believe (as I sincerely do) that mathematical truths are not mere human constructions, but would remain true even if there did not exist a single human to say so, then things like the shoes and socks paradox do not trouble us.

Doesn't sort out Banach-Tarsky paradox though, oh well.

Assuming you mean that by find you mean find explicitly or constructively, then yes, existence is all AC affirms, but that's not very limiting as far as using AC goes (obviously it means that you won't be able to explicitly specify objects which you create with it, but that's not really a problem). If AC would allow us to construct a choice function, we arguably wouldn't need it as an axiom: we would just use it as a method of constructing choice functions (I guess you could say that the method could require additional axioms but, even so, specifying choice functions would demystify a lot of AC constructions). My point is basically, I don't really see how this resolves anything. That said, I also don't think there is anything to resolve at all: I see the shoes and socks thing as a mere analogy to explain "why" AC would be needed and "Banach-Tarski Paradox" is a misnomer as it's really a theorem and not a paradox. This goes back to what I said in my original post: this sort of consequence of AC is strange by by no means contradictory or paradoxical (in the strict sense of the word).

P.S. I also disagree with the mathematical object existence thing but I don't really want to start a debate on that given how long the previous one of this type I had lasted. :P

Truth be told, I am a little unclear as to what the precise meaning of the word "paradox" is, given that it is not, as many seem to think, synonymous with "contradiction".

Well, it depends on how strict you want your notion of "paradox" to be. Many people use it to mean counterintuitive (e.g. the "birthday paradox") in which case you can indeed call Banach-Tarski a paradox. I generally prefer to take the stricter meaning (in this context at least) which is fairly close to something like "true contradiction" I guess (e.g. Russell's paradox).

My point is that by assuming the axiom of choice can we prove something untrue or something not useful due to our own mortality. Assuming we can do something an infinite number of times might not be a good idea.

Actually it's a great idea because without that you can't really get very far in maths. There are much simpler and uncontroversial axioms that basically allow you to "do something an infinite number of times", like the axiom of infinity in ZFC or even the axiom of inducton in Peano Arithmetic. Basically, as soon as you have the natural numbers you can do something an infinite number of times (wait a second, does this question have anything to do with finite primes thing? :P).

I don't really see how "our mortality" has anything to do with that or its usefulness (it's pretty obvious that it is "useful" regardless of the finite or infinite nature of the universe/humans/toasters). As far as AC allowing us to prove something untrue, it's not really a problem: adding AC or its negation to the other axioms doesn't change whether we can derive a contradiction from them or not (as you would have known if you had read my first post instead of instantly thinking "block of text" :P).

It seems to me that a paradox is inherently the failure of a specific paradigm or perspective, mathematics has yet to produce an intractible such problem; indeed if it did, we should be all out of a job.

It was the unspoken concusion of the axiomatic movement that mathematics could and should only be the interdependence of such paradigms. A paradox represents a dead end for the compatibility of our assumptions and BTP is hardly a shocker: "if we can chop things up infinitely small then measuring them doesn't make sense". "No sh*t Sherlock" springs to mind...