Question

The axiom of choice is controversial. Are there solutions?

Answer 1

> Are the paradoxes actually acceptable and the AoC should not be controversial in the first place?

For the majority of working mathematicians (outside of logic and closely related areas), my impression is that the answer is *yes*. The use of AC is completely conventional now. Sure, the first time people see some of its implications they can get terribly worked up about it -- your "Undergraduate" flair has me suspecting you are in this situation. By now multiple generations of mathematicians have spent their working lives with AC available and they find it quite valuable as a tool (e.g., in algebra see the proof of existence of maximal ideals or algebraic closures).

I am not denying that some consequences of AC are very counterintuitive (e.g., a well-ordering of the real numbers), but other consequences are *extremely* valuable and that is why most mathematicians accept AC.

Answer 2

I'm not sure what you have in mind when you talk about something being "done". There's no governing body that can tell mathematicians which axioms to use. There's no way to force a mathematician to stop using the Axiom of Choice.

Mathematicians don't argue about whether axioms are true or false any more - that was an early 20th century view. Instead we just keep track of which theorems can be proved from which axioms - "This theorem can be proved without the Axiom of Choice. This one can be proved if you assume the Axiom of Choice, and we know it can't be proved without it. This one can be proved using the Axiom of Choice, and it's an open problem whether it can be proved without it."

Mathematicians talk about whether they like or do not like working with certain axioms, but it's now seen more as a matter of taste and personal preference rather than a disagreement about a matter of fact. We know from Gödel's Incompleteness Theorems that there can never be one set of axioms that is enough for all of mathematics.

As other answers have said, there are more specific versions of the AoC (the Axiom of Countable Choice, the Axiom of Dependent Choice) and alternative axioms which are inconsistent with the AoC (e.g. the Axiom of Constructability). It's interesting to see what theorems can be proved if we use those axioms instead.

If you don't like the fact that the Axiom of Choice implies the Banach-Tarski Paradox - well, there's no way to change that. You can choose to work in a set theory without the Axiom of Choice if you want. But that won't change the fact that the Banach-Tarski Paradox is derivable from the Axiom of Choice (plus the other axioms of ZF set theory).

Answer 3

I'd say that the axiom of choice isn't really controversial anymore.

Answer 4

There is nothing at all controversial about the axiom of choice. I don't think Banach-Tarski should be seen as a paradox. You're playing with non-measurable sets and then trying to apply physical intuition about measure.

Answer 5

One thing that is important and hasn't been explicitly mentioned yet. Despite its unintuitive behaviour, the axiom of choice is relatively non threatening in terms of making contradictions.

In particular, if ZFC (a common set of axioms including choice ) is inconsistent, then ZF (the same axioms without choice) is also inconsistent. That is, it's not choice that actually caused the problem. I think this makes Choice much easier to stomach.

Answer 6

I want to add something which I haven't seen in the other comments yet. If we work in specific categories then AC fails. What I mean by that is the following:

AC is equivalent to every surjective map f:A -> B has a right inverse, that is a map g:B -> A such that fog is the identity on B. This is only a map of sets though.

In other categories this statement is false. If we have a surjective continuous map f:A -> B between topological spaces there does not need to exist a CONTINUOUS right inverse. So if you want to interpret it this way: In the category of topological spaces the axiom of choice does not hold (sure it holds for the underlying sets if we assume choice but not for the topological structure).

So this is an argument why you would want to prove theorems without axiom of choice. If you need choice your argument won't work internal to a category in which choice does not hold.

People often focus too much on seemingly counter intuitive paradoxes like Banach-Tarski and so on. I think this argument here is a better one why one would want to not use it (even if one assumes it).

Answer 7

I’d say choice isn’t terribly controversial anymore. If we can prove a contradiction using it, we could have done so without it. (That is, ZFC is consistent if and only if ZF is.) I think the prevailing opinion is that this result has resolved any worry we might have over its use.

That being said, there are alternatives. For example, one can instead take the axiom of determinacy. It implies countable choice, but is inconsistent with full choice. ZF+Determinacy is an interesting theory, which differs from ZFC in a number of ways. For example, all sets of reals are measurable.   


However, I’d posit that removing choice tautologically gives us a weird paradox: the formal statement of the axiom of choice says that “the product of nonempty sets is nonempty.” If choice fails, then there exists a collection of nonempty sets whose cartesian product is empty.

Answer 8

I don't think the axiom of choice is really that controversial. Those who doubt its validity are either a very tiny minority of mathematicians... or computer scientists. (And there's no way in hell I'm letting computer scientists dictate what our foundations should be.)

Also, without the axiom of choice, we can't have some nice things, like ~~the Nullstellensatz~~, Tikhonov's theorem, or ~~“a map of sheaves over a topological space is an isomorphism if and only if it's stalkwise an isomorphism”~~. So we would have to do a lot of work to reestablish results in algebra and topology that we currently take for granted.

Answer 9

Axiom of choice is controversial because it allows you to make an infinite amount of decisions. In real life, one cannot make an infinite amount of choices so consequently, it doesn't make sense to do so in set theory. In smaller sets this isn't really a problem because you can very easily specify the objects you want to talk about, but when you get to larger cardinals this might be an issue. It allows you to do things like find maximals elements by just infinitely choosing a bigger object.

However the intuition is that you can select one object from exactly one set in an infinite collection of non-empty sets. Which makes sense because they're all non-empty, but the fact that we cannot specify exactly the objects that we are taking is bizarre and leads to counterintuitive results.

All in all the axiom of choice works and is only problematic in a philosophical sense. It is interesting to see what math one can do without the axiom tho.

Answer 10

The axiom of choice isn’t really particularly controversial anymore. I think it’s pretty well understood that there are contexts where it holds and contexts where it doesn’t, and because of the relative consistency proofs, there’s not conflict to having both kinds of contexts around. The standard approach is to take choice as an axiom and do choice-less math in restricted settings, but some people prefer to work in a setting without it (typically a constructive one) and let the things that need the axiom of choice be done as a special case where it’s added.

As far as I know, all the so-called “paradoxes” are really just taking a context where the axiom of choice isn’t appropriate, using it anyway, and being surprised that the result you get doesn’t make sense for the original context.