Is there a non-trivial countably transitive linear order?

set-theory order-theory

Yes. Every infinite structure has a strongly $\omega_1$-homogeneous elementary extension. So you can start with the rationals and find an $\omega_1$-homogeneous elementary extension $(L, <)$ which is countably transitive being $\omega_1$-homogeneous. You can find a construction here. If you also assume CH, then there is a saturated DLO without end points of size $\omega_1$ (which is clearly strongly $\omega_1$-homogeneous).

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