I tend to agree with the idea that AC is rather unfairly described as ‘to be rejected’, especially Banach-Tarski. 
We are no strangers to strange things in mathematics, especially with infinity, so I don't really understand the argumentative structure: it's strange, then I don't know what to conclude from that.
It's no stranger than many other things, it seems to me : such as the fact that there are as many even integers as there are integers, or that there are as many real numbers between 0 and 1 as there are real numbers. 
There are many strange things. 
Also, with "the argument of strange things", we come back to rejecting the axiom of infinity, rather, I would say ?

But for me, that's not the heart of the problem: 
An axiom is often seen, wrongly (in my opinion), as something that we posit, or don't posit, or posit the opposite of, and then we work within that framework. 
This is not entirely wrong, but it is the point of view that seems to me to be erroneous as a point of view: 
We do not state that ‘groups are now Abelian’; no, we are going to study Abelian groups if we want to, but we do not forget that non-Abelian groups exist nonetheless; we have not ‘stated’ Abelianity, we are simply going to look at this specific type of object. 
Similarly, we do not ‘assume’ that the ring is commutative; at best, we say that we will only consider commutative rings in the rest of the text (book, course, etc.). 
In short, an axiom is not there to be assumed or not; it is there to describe a type of object, in my opinion.

So here, it is not a question of ‘postulating’ whether or not the ZF-universe verifies AC, but rather of seeing whether in our work it is more practical to work with ZF-universes that verify AC or whether we do not need to restrict ourselves to those that verify AC in order to work, and thus produce a more general result on ZF-universes; but potentially more costly to prove because it requires more generality. 
Or even to specify results on ZF-universes that do not verify AC, even if they are potentially less frequent.

So really seeing ZF-universes as objects, and not as somewhat transcendent entities, etc.

Thank you for this article, I find the subject interesting.

In this article, I am rather surprised by the use of the word ‘value’, also in the comments, so I wondered if it was a language issue on my part. 
However, the fact that the author wonders whether human values are good is something that fits in with my initial interpretation of the word value, which is as follows: value in the deepest sense, what is most important in life. 
And my initial interpretation seems to be in line with that of the Stanford Encyclopedia of Philosophy, for example: https://plato.stanford.edu/entries/value-theory/

The term “value theory” is used in at least three different ways in philosophy. In its broadest sense, “value theory” is a catch-all label used to encompass all branches of moral philosophy, social and political philosophy, aesthetics, and sometimes feminist philosophy and the philosophy of religion — whatever areas of philosophy are deemed to encompass some “evaluative” aspect. In its narrowest sense, “value theory” is used for a relatively narrow area of normative ethical theory particularly, but not exclusively, of concern to consequentialists. In this narrow sense, “value theory” is roughly synonymous with “axiology”. Axiology can be thought of as primarily concerned with classifying what things are good, and how good they are. For instance, a traditional question of axiology concerns whether the objects of value are subjective psychological states, or objective states of the world.

So I find it difficult to understand why “value” then takes on the meaning of “what we like,” which seems to me to have nothing (or very little) to do with it.

Nevertheless, despite this potential difference in concept, I find that certain reflections remain valid even when taken in a philosophical sense.

For example, this, with which I agree, even when taking the word in the philosophical sense of “value.”

We don’t really know what human values are, or what shape they are, or even whether they’re A Thing at all. We don’t have trivial introspective access to our own values; sometimes we think we value a thing a lot, but realize in hindsight that we value it only a little. But insofar as the mental picture is pointing to a real thing at all, it does tell us how to go look for our values within our own minds.

I find this difficulty fascinating and believe it necessitates precise thought experiments on this subject in order to realize how poorly we model our own values (in the deepest sense, once again).

There is also the question of how to aggregate individuals' (moral/deep) values into human (moral/deep) values, which does not seem at all obvious to me (neither the average, nor the sum, nor any other aggregation function seems to behave well a priori?).
One idea I am currently imagining is more like creating a new global model from a collection of thought experiments (and concrete decisions to be made, to avoid problems of abstraction) that is very refined in order to distinguish subtleties, and which would be iteratively refined by proposing more and more “twisted” cases to question the foundations, on which a large number of people would express their opinions after a certain (significant) period of internal deliberation.