To me, the Axiom of Choice is absolutely natural – if we can pick one element of a set then we surely can pick elements from a collection of sets, who cares tha

To me, the Axiom of Choice is absolutely natural – if we can pick one element of a set then we surely can pick elements from a collection of sets, who cares that this collection can be infinite. The problem with this axiom is that it can't be formally proven from ZF axioms. But there are a lot of other natural things that can't be proven in ZF. For example, the consistency of ZF can't be proven in ZF due to Godel's incompleteness.