I expect you already know some of these, but for anyone interested:
-
Asaf Karagila’s Anti-anti Banach–Tarski arguments. A short blog post whose main point is that “The axiom of choice is not at fault here. The axiom of infinity is.” As an illustration, he shows that if, instead of the axiom of choice, one assumes dependent choice and that all sets of reals are Lebesgue measurable, then there is a partition of the real line into strictly more parts than elements.
-
By the same author: Zornian Functional Analysis, or: How I Learned to Stop Worrying and Love the Axiom of Choice. A 30-page article discussing some of the (often counterintuitive) consequences of rejecting the axiom of choice.
-
ZF(C) is not the only way to axiomatize set theory as a first-order theory. Lawvere’s Elementary Theory of the Category of Sets (ETCS) is a reasonable alternative. I find it interesting that in ETCS the axiom of choice is built in without much hesitation, whereas the axiom of replacement is not part of the core theory (though it can be added to recover equivalence with ZFC). While the set theorists I have spoken to tend to regard replacement as a major axiom, I do not often see arguments that it should be rejected in order to avoid paradoxes. (For a short introduction to ETCS, I personally recommend Tom Leinster’s Rethinking set theory, an 8-page article navigating between intuitions and formalism.)
-
On the Banach–Tarski paradox, Vsauce’s Michael Stevens has made a video that gives a clear explanation and helpful visualisation: link.
Not sure what you mean by that.
It sounds a lot like the typical mind-projection fallacy Jaynes warns against.
(Do you see what I mean?)