It seems to me that, since the set of real numbers has a total ordering, I could fairly trivially construct some choice function like “the element closest to 0” that will work no matter how many elements you remove, without needing any fancy axioms.
I don’t know what to do if the set is unordered though.
If I give you the entire real line except the point at zero, what will you pick? Whatever you decide on, there will always be a number closer to zero then that.
I guess I can pick another number x to be closest to but it has the same problem unless I can guarantee it’s in the set. And successfully picking a number in the set is the problem to begin with! Foiled again!
It seems to me that, since the set of real numbers has a total ordering, I could fairly trivially construct some choice function like “the element closest to 0” that will work no matter how many elements you remove, without needing any fancy axioms.
I don’t know what to do if the set is unordered though.
If I give you the entire real line except the point at zero, what will you pick? Whatever you decide on, there will always be a number closer to zero then that.
I guess I can pick another number x to be closest to but it has the same problem unless I can guarantee it’s in the set. And successfully picking a number in the set is the problem to begin with! Foiled again!