I am trying to prove the statement of the question.
The only strategy that comes to my mind is to exploit the theorem of Morley saying that if a theory (in a countable language) is $\lambda$ categorical for an uncountable cardinal then it is $\lambda$ categorical for any uncountable cardinal.
Then it suffices to show, for instance that the real and the irrational numbers, seen as DLO are not isomorphic, like in The theory of dense linear orders without end-points is not $2^\omega$-categorical.
I have also seen proofs based on the notion of stable theory.
But is it there a simpler proof which avoids the theorem of Morley and the notion of stable theory (which we have not treated in the course where the excercise was proposed)?
Thanks
Best Answer
Given an uncountable cardinal $\kappa$ consider the linear order $\leq_1$ obtained by replacing every element of $\kappa$ with a copy of $\Bbb Q$, and the linear order $\leq_2$ obtained by replacing every element of $\kappa^\ast$ ($\kappa$ with the reverse order) by a copy of $\Bbb Q$.
Explicitely the first order is $\kappa\times\Bbb Q$ where $(\alpha,q)<_1(\beta,p)$ iff $\alpha<\beta$ or $\alpha=\beta$ and $p<q$, while the second is $\kappa\times\Bbb Q$ where $(\alpha,q)<_1(\beta,p)$ iff $\beta<\alpha$ or $\beta=\alpha$ and $p<q$.
Those are both dense linear orders of cardinality $\kappa$ without endpoints, do you see why they are not isomorphic? Hint: does $\omega_1$ embed into both?