>> Well, the existence of a set that cannot be well-ordered contradicts >> the existence of a well-ordering of the universe of sets. Both are >> consistent, but they can't both exist. > >You mean if the universe of sets is well-ordered, then every set is >well-ordered?
Yes. To say that the universe is well-ordered implies that every set is well-ordered, since every set is a subset of the universe.
>What does "a well ordering of the universe of sets" mean, anyway?
A binary relation R(x,y) is a well-ordering of a collection C if it is a total ordering and for every subcollection C' of C, if C' is non-empty, then C' has a minimal element, under the ordering R.
A well-ordering of the universe is equivalent to the existence of a way to index elements of the universe with ordinals.
In ZFC, you can't actually talk about proper classes, so the claim "The universe is well-ordered" can't be formulated, but it can in an extension such as Morse-Kelley set theory that allows talk about proper classes.