I'm reading Cantor's 1883 Grundlagen, it says a set is well-ordered if the set and it's subsets have first element, the next successor (unless it's an empty set or there is no successor). Note that the first element not neccessarily a least element. "Theory of sets" by E. Kamke also give the same definition. However, "Naive Set Theory" by Paul Halmos and many other recent publications say the first element as smallest element. Why is it so?