Dedekind complete model Prove that there is no first order theory $T$ in $ L=\{<\}$ such that for every linearly order set $A$, $A$ is a model of $T$ iff $A$ is Dedekind complete. 

Here is a slight variation of Wore's argument: Suppose that $T$ would characterize Dedekind complete linear orders.
Let $T^*$ be $T$ together with $\operatorname{DLONE}$ - the theory of dense linear orders without endpoints. Now $T^*$ characterizes Dedekind complete dense linear orders without endpoints and is consistent with infinite models (as $(\mathbb R;<)$ is such a model). 
Hence, by Löwenheim-Skolem, it has a countable model $(X; \prec)$. It follows, by the $\aleph_0$-categoricity of $\operatorname{DLONE}$, that $(X; \prec) \cong (\mathbb Q; <) \models T^*$. Contradiction!
