Why can’t “the empty set” be selected with a choice function and placed in a set of chosen sets? Is there an alternative Axiom of Choice?

Question

Is there a variant of the Axiom of Choice that allows you to put “the empty set” in a set of chosen sets?

Summarily, my question in plain language:

Why can’t we pick an empty bag or an empty box under the Axiom of Choice?

Answer 1

Yes, you can always say that a choice function will return $\varnothing$ if you give it the empty set. There are some caveats, though.

If we think about $\prod_{i\in I}A_i$ as the collection of choice functions from $\{A_i\mid i\in I\}$, then we expect it to be empty when one of the $A_i$ is empty. Because that's the rule for products: if one of the sets is empty, the product is empty.

But now, if we modified the definition of a choice function, this is no longer the case, since every collection of sets have a choice function (under AC), but $\prod_{i\in I}A_i$ can be $\varnothing$.

So like everything else, it's a choice of "where do you want to place your inconvenience" and the answer is very much context dependent.

The analogy with the bag, by the way, is misleading, since you can choose the empty set if it is an element of some set; but you cannot choose something form inside an empty bag, because it's empty. Try it: take an empty bag and offer your friend to choose whatever candy bar they want, see if they can do it.