= Solution
Use the <complete theory> of <dense linear orders without endpoints> in the <first-order language> $\{<\}$. It eliminates <quantifiers>: existential conditions on one new point reduce to finitely many <logical equality> constraints and interval bounds, and density and absence of endpoints decide whether a suitable point exists. Its countably infinite <first-order models> are all <isomorphic> to the rational order by the usual <back-and-forth method>, extending finite order-preserving maps alternately along enumerations of the two <first-order models>. Thus it is $\aleph_0$-categorical.
To distinguish <first-order models> of size $\aleph_1$, let
$$
M_1=\sum_{\alpha<\omega_1}(\mathbb Q,<),\qquad M_2=M_1+(\mathbb Q,<),
$$
with ordered sums. Both are dense without endpoints and have <cardinality> $\aleph_1$. The first has <cofinality> $\aleph_1$: countably many points lie in a bounded <set> of block indices and cannot be <cofinal>. The second has <cofinality> $\aleph_0$, supplied by its final rational block. <Cofinality of a linear order> is preserved by <isomorphism>, so they are not <isomorphic>. Therefore
$$
\boxed{\text{DLO is }\aleph_0\text{-categorical but not }\aleph_1\text{-categorical}.}
$$
For Ryll-Nardzewski, its $n$-types are precisely the finitely many equality-and-order patterns on $n$ coordinates. Each is isolated by the <conjunction> specifying that pattern, agreeing with the theorem's prediction.
Back to article page
