Can it be proven/disproven that every set can be well-ordered to have a maximal element?

Question

Can we prove or disprove that every set can be well-ordered to have a maximal element in ZF or ZFC?

Or is it the area where different models have different answers to this?

Answer 1

Take a nonempty set $S$ and pick $x\in S$. Well order $S\backslash\{x\}$. Extend the resulting well-order to $S$ by making $x$ larger than every other element. You get a well-order with a maximum.

Answer 2

If the set is empty, then it doesn't have any elements, in particular maximal elements. But that's the trivial, and uninteresting case. So let's casually chuck it aside, and assume we're talking about non-empty sets.

Of course that without the axiom of choice we cannot prove that every set can be well-ordered to begin with. So the axiom of choice is necessary here.

Assuming it, though, we have two cases:

1. The set is finite, in which case it will definitely have a maximal element in any well-order.

2. The set is infinite, in which case we can put it in bijection with many different ordinals. Many of which have maximal elements.

So the answer is indeed positive.