I've been looking for an example of an empty Cartesian product whose factors are non-empty. From what I've gathered so far, this statement is equivalent to the negation of AC, ie. AC fails. So constructing an example means finding a collection of sets for which no choice function exists. But I haven't a clue how to go about such a proof.
Can you actually construct an empty cartesian product where each factor is nonempty, or is it not possible by virtue of the definition? If the latter, is there a proof of such a statement? Obviously if it is, it's probably over my head, but I'd be interested nonetheless.
Thanks in advance!
No. You cannot really construct something like that.
The negation of the axiom of choice is as non-constructive as the axiom of choice itself. All it tells us is that somewhere in the universe of sets there exists a family of non-empty sets whose product is empty.
If you want more, you will have to assume more. What does that mean? For example we know that we can assume that there is a family of sets of real numbers whose product is empty, and therefore the product of all non-empty sets of real numbers is empty. But it is also consistent that the real numbers can be well-ordered, and so the collection of all non-empty sets of real numbers does admit a choice function. So you have to assume one way or another.
But generally, you cannot point out at a non-well orderable set, or a family of non-empty sets whose product is empty.