"Ordered sets" are coherent in the sense that you can make a data structure like a set and then define an order over its members, and call it an ordered set, bu

"Ordered sets" are coherent in the sense that you can make a data structure like a set and then define an order over its members, and call it an ordered set, but "ordered sets" are not sets. There are certain purposes sets are natural for. When you introduce ordering, you will naturally be asked to add methods for querying or perhaps modifying that ordering in various ways. Many methods later, it's worth asking why you are calling this data structure a set, and what its actual purpose is supposed to be.