TITLE: Is there a standard ordering of non-ordinal order types?
QUESTION [1 upvotes]: Two ordered sets have the same order type if there exists an order isomorphism between them.  The order type of a well-ordered set is called an ordinal number.  There is a standard ordering of the ordinal numbers: if $A$ and $B$ are well-ordered sets, then we say that the order type of $A$ is less than the order type of $B$ if there exists an order isomorphism between $A$ and a proper initial segment of $B$.  And this ordering is in fact a well-ordering.
My question is, is there a standard way to order the order types of all totally ordered sets, well-ordered or not?  Is there at least a standard partial order on the order types of totally ordered sets, even if there isn’t a standard total order?

REPLY [1 votes]: If $A$ and $B$ are totally ordered sets of order type $\alpha$ and $\beta$ respectively, then $\alpha\le\beta$ means that $A$ is isomorphic to a subset of $B$. (This agrees with the usual ordering of ordinal numbers if $A$ and $B$ are well-ordered sets.) This relation $\le$ is a quasi-ordering: reflexive and transitive but not antisymmetric. The notation $\alpha\lt\beta$ means that $\alpha\le\beta$ while $\beta\not\le\alpha$.
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