The first paradox discovered in the modern theory of sets. Every well-ordered set has an ordinal number assigned to it. These ordinals can be compared: of any two, either they are equal, or one is smaller and one is larger. They therefore form a well-ordered set. The ordinal of this set must be greater than any ordinal within the set. Let S be the set of all ordinals. Since it is well-ordered it has an ordinal number, w, that must be greater than any element in the set. But S was the set of all ordinals, and must include w.