One of the troubling aspects of the Axiom of Choice is that both affirming and denying it results in some pretty unintuitive implications (but different ones).
This MathOverflow discussion has some interesting examples of unintuitive implications of the axiom, along with a bonus answer giving examples of unintuitive implications of not having the axiom: http://mathoverflow.net/questions/20882/most-unintuitive-app...
The history of it is full of people wrestling with both sides of that, including many participants in the debate having trouble keeping their own positions consistent—the axiom of choice is equivalent to so many other things (many only discovered fairly recently) that consistently affirming or denying it has tripped up many world-class mathematicians! I am currently about 1/3 of the way through this really interesting history that tries to trace that development, Zermelo's Axiom of Choice by G.H. Moore (2013): http://www.amazon.com/gp/product/0486488411/ref=as_li_tl?ie=... (ToC: http://www.apronus.com/math/zermelos.htm)
That's a great list! Being a grad student in mathematics, I find several of those way more upsetting than anything like Banach-Tarski or the hat problem in the linked post.
Oh, I agree. I'm totally comfortable with it, by now. I think the von-Neumann quote might apply here: "In mathematics, you don't understand things. You just get used to them."
Indeed, this is what intuition is. We often fail to notice where our childhood intuitions come from (experience), and so our adult desire to understand interferes with our ability to form new intuitions.
What's the difference? Uncountable sets does not respect out intuition about countable sets.
AC is independent from ZF, meaning it doesn't matter for any application to anything in the Universe, including physics or computing. Theory of uncountable objects is a game to play with symbols, the only thing "real" about them is that they are the firm boundary between the potentially reachable and the infinite void. The name itself conveys an important truth.
This MathOverflow discussion has some interesting examples of unintuitive implications of the axiom, along with a bonus answer giving examples of unintuitive implications of not having the axiom: http://mathoverflow.net/questions/20882/most-unintuitive-app...
The history of it is full of people wrestling with both sides of that, including many participants in the debate having trouble keeping their own positions consistent—the axiom of choice is equivalent to so many other things (many only discovered fairly recently) that consistently affirming or denying it has tripped up many world-class mathematicians! I am currently about 1/3 of the way through this really interesting history that tries to trace that development, Zermelo's Axiom of Choice by G.H. Moore (2013): http://www.amazon.com/gp/product/0486488411/ref=as_li_tl?ie=... (ToC: http://www.apronus.com/math/zermelos.htm)