The well-ordering theorem states that every set can be well-ordered. That is, there is a total order on the set in which every non-empty subset has a least element. I was wondering whether it's also true that for every set you can define a total order such that for every non-empty subset there is a greatest element and would that be equivalent to the well ordering theorem?
Thanks!
On
Yes: such an order is simply the reverse of a well-order, so every set has the one if and only if it has the other: just replace $\le$ by $\ge$.
On
Any claim about an order is immediately applicable on its reverse order by "reversing" the statement. Minimals are now maximals, and vice versa.
Any well-order can be reversed to obtain an order in which every non-empty set has a maximal element; and vice versa. So the axiom of choice is equivalent to this principle as well.
Equally the axiom of choice is equivalent to the dual Zorn's lemma:
If $(P,\leq)$ is a partially ordered set in which every chain is bounded from below, then there is a minimal element.
Yes, of course. Just reverse the order given by the well-ordering theorem.