Question. Is there a decent classification theorem for linear orders satisfying all three of:
Dense. Given a pair of elements $y,x$ with $y>x$, there exists $k$ satisfying $y>k>x$.
Complete. Given a non-empty subset $A$, if $A$ is bounded above, then $A$ has a least upper bound.
Endless. There is neither a greatest element nor a least element.
Comments:
Assuming separability, $\mathbb{R}$ is the only example..
I'm happy to assume some further conditions, like "for any two points, there is an order-automorphism mapping the first point to the second." However, I'd like models of unboundedly large cardinalities.
On the basis of Eric's comment: no.
In particular, Suslin's problem asks whether or not there exists a non-empty linearly ordered set $R$ that isn't isomorphic to $\mathbb{R}$ satisfying the 3 conditions stated in my question together with the condition: "You can't cram more than countably many disjoint non-empty open intervals into $R$." This is independent of ZFC. Therefore, it seems that no truly useful classification can exist without assuming further axioms.