Axiom of Choice and Order Types
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)
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.