Banach-Tarski from a constructivist/intuitionistic point of view:

"So it really seems that when we switch to constructive theory, it actually does something even better, it eliminates this pathology altogether, making balls behave more sensibly?"

https://math.stackexchange.com/questions/175675/intuitionist...

I like theories without crazy theorems. And I like theories with crazy theorems.

Edit: the link below from soVeryTired to Andrej Bauers post is better