I learned that Axiom of Choice states that the cartesian product of a family of non-empty sets $X_i$ indexed by a non-empty set $I$ is non-empty.
I think I can accept this axiom.
But I don’t understand how it guarantees that cartesian product can have more than one elements! It’s because ‘non-empty’ sounds to me ‘having at least one element’ here. So I think AC just implies that there exists at least one element in the product.
How do we insure that there exist all the possible functions from $I$ to $\cup X_i$?
Thanks in advance!