The classification of order types In response to a previous question which I asked concerning the order type of the Rationals, a reference was made (by MDJ) to a theorem of Cantor stating that any two countable, dense, linearly ordered sets without endpoints are isomorphic as linear orderings.
My question is : If we remove the endpoints condition, say by looking at the intersection of Q and [0,1], then is this order type also unique (upto isomorphism) amongst such linear, dense, countable orderings.
More generally, have all of the (linear) order types which do not correspond to an ordinal number been classified?
 A: Up to isomorphism there are exactly four countable dense linear orders:


*

*Those without endpoints, like $\Bbb Q$ or $\Bbb Q\cap(0,1)$.  

*Those with a left endpoint but no right endpoint, like $\Bbb Q\cap[0,1)$.  

*Those with a right endpoint but no left endpoint, like $\Bbb Q\cap(0,1]$.  

*Those with both endpoints, like $\Bbb Q\cap[0,1]$.


However, there are many other countable order types that are not represented by ordinals. For starters, there are the order types $\alpha^*$ for countable ordinals $\alpha$, obtained by reversing the ordering of $\alpha$: these are the types of the countable reverse well-orderings. There is the type $\omega^*+\omega$, which is isomorphic to $\Bbb Z$ in its natural order. There are types like $\omega^*+\eta+\omega$, where $\eta$ is the order type of the rationals. And you can combine the types $\alpha$ and $\alpha^*$ for $\alpha<\omega_1$ with $\eta$ to get a huge variety. (The last three types above can be formed in this way: $\Bbb Q\cap[0,1)$ has the type $1+\eta$, and the other two have types $\eta+1$ and $1+\eta+1$, respectively.)
And when you get to uncountable order types, life gets much more complicated. For example, there is not just one dense linear order of cardinality $\omega_1$ without endpoints.
