I'm sorry, but I don't think there was anything remotely "rude" or "terribly dishonest" about my previous comment. If you think I am mistaken about anything I said, just explain why. Criticizing my rhetorical style and accusing me of violating social norms is not something I find helpful.

Quite frankly, I also find criticisms of the form "you sound more confident than you should be" rather annoying. E.g:

it creates the impression that the subjective reasons you give for dismissing point 3 have weight equal to the objective reasons history has given for dismissing points 1 and 2.

That's because for me, the reasons I gave in point 3 do indeed have similar weight to the reasons I gave in points 1 and 2. If you disagree, by all means say so. But to rise up in indignation over the very listing of my reasons -- is that really necessary? Would you seriously have preferred that I just list the bullet points without explaining what I thought?

So what's wrong with reasoning like this:

Nothing at all, except for the false claim that nobody likes the consequences of the Axiom of Choice. (Some people do like them, and why shouldn't they?)

The target of my critique -- and I thought I made this clear in my response to cousin_it -- is the critique of mainstream mathematical reasoning, not the research program of exploring different axiomatic set theories. The latter could easily be done by someone fully on board with the soundness of traditional mathematics. Just as it is unnecessary to doubt the correctness of Euclid's arguments in order to be interested in non-Euclidean geometry.