Examples that show that AC is not that natural

Question

I recently took a course in set theory and I was wondering the following:

Let's "accept" that the axioms of $ZF$ are all natural and close to our intuition. As we know, $ZF$ can neither prove $AC$ nor $\neg AC$. Of course almost everyone that works outside of set theory works in $ZFC$, but I have the impression that $AC$ is somehow not as natural as $ZF$ nor that close to our intuition. What I'd like to know is, if there are any examples that point out/ defend the impression I mentioned above.

PS: I know that this question is a little too broad, so If anyone can improve it, please edit it.

For history, see G.H.Moore, Zermelo's Axiom of Choice: Its Origins, Development, and Influence. — Mauro ALLEGRANZA, Oct 11 '18 at 15:01
The problem that people forget is that the failure of choice is just as unnatural as choice. For starters, $\lnot\sf AC$ itself seems to be extremely counterintuitive. Secondly, there is almost a dichotomy between "this kind of weirdness" and "that kind of weirdness" (and sometimes a little bit of both). It's not choice which is to blame. Seriously. — Asaf Karagila, Oct 11 '18 at 15:20
@AsafKaragila Could you please expand on your last statement "It's not choice which is to blame" I didn't quite get that — Just dropped in, Oct 11 '18 at 15:26
https://math.stackexchange.com/questions/2378211/what-are-some-unintuitive-consequences-if-we-assume-that-axiom-of-choice-is-wron — Asaf Karagila, Oct 11 '18 at 15:42

score 3 · Accepted Answer · answered Oct 11 '18 at 15:47

There are a number of arguments both for and against Choice. The general attitude in the anti-Choice direction seems generally to be that it leads to existence proofs for objects which cannot be "constructed in a natural way" - e.g. ZFC proves the existence of a non-measurable set, but we have many results coming from descriptive set theory saying that such a set cannot have a simple description (especially under additional set-theoretic hypotheses).

Note that there is a similar flavor here to arguments in favor of constructive logic. However, in my opinion this similarity is actually fairly superficial. Consider, for example, the following definition: "$x=0$ if $P$ is true, and $x=1$ otherwise" (where $P$ is some statement you don't know right now is true or false - say, the Riemann hypothesis). Then this is certainly not a constructively-acceptable definition in general. However, we can separate the weirdness of a mathematical object from the weirdness of our description of it - in this case, $x$ is really not weird at all (it's either $0$ or $1$) even though we've defined it in a weird way - and so we can accept its definition while maintaining a commitment to only "concrete" mathematical objects.

Interestingly, arguments in favor of constructive logic can be invoked either for or against AC:

Against: To the extent to which we view the mathematical universe as a construction of the mind, how can we accept an axiom which yields the existence of objects we can't describe?
For: When we assert that a collection of sets is nonempty, we need to have a constructive proof of this fact. But such a proof amounts exactly to a choice function for that family!

In my view, this supports my claim that these issues - "non-constructive principles" in set-theory versus logic - are genuinely different.

All this, though, is a bit vague. It's a good idea to get acquainted with the precise mathematical issues here - both with AC and without AC - in order to appreciate the debate going on.

Herrlich's book is a good introduction to what mathematics looks like both with and without choice. Jech's book, which was recommended in the comments, is also quite good, but focuses mostly on the very advanced material - namely, proofs of consistency (e.g. showing that ZF neither proves nor disproves AC) - and may not be a good starting point.

Examples that show that AC is not that natural

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