A beginner's question:


We know: "Since order-equivalence is an equivalence relation, it partitions the class of all sets into equivalence classes." (from [Wikipedia](http://en.wikipedia.org/wiki/Order_type))


This holds since every set can be (well-)ordered by the Axiom of Choice.


But there can be many (well-)orderings of a given set. Especially, the Axiom of Choice doesn't tell us, *what* the choice function is and thus, what the well-ordering is: there can be many.


Thus, a set can belong to many order types and order-equivalence isn't an equivalence relation anymore.



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Axiom of Choice and Order Types
Order equivalence is an equivalence relations on ordered sets, not on sets. It is just the isomorphism relation on ordered structures. An ordered structure is a set, together with an order.


The Axiom of Choice says that every set has a well-order. Since the order-types of well-orders are well-ordered (given any two, one of them is uniquely isomorphic to a unique initial segment of the other), it follows under AC that for every set, we can associate to it the smallest order-type of a well-order on that set. This is called the *cardinality* of the set. 


There is another more general concept of cardinality, which does not rely on AC or on orderings at all, and it is just the equinumerosity class of the set. 


