So they asked me, "Dan, what do you do for fun on a Friday night?" I answered "Well..."

I was thinking about Well-Ordered sets today and a thought struck me. I know that (assuming the Axiom of Choice) there exists as well-ordering for any set. Some sets are easy to well-order and others, like the reals, seem rather impossible.

My question is this: Has anyone ever managed to define a well-ordering on any set that satisfies the continuum hypothesis in some (other) topology? I suppose the question is roughly the same if I ask whether a set with the cardinality of the reals has ever been well-ordered (though I am aware that the two questions aren't exactly the same).

Just curious.

-Dan