= Solution
Use the completeness of the <first-order theory> of <dense linear orders without endpoints> and its <quantifier elimination>. After naming finitely many increasing constants, <quantifier elimination> still holds in the expanded <first-order language>: treat these constants as fixed parameters when eliminating <quantifiers>. Every <quantifier-free sentence> is a Boolean combination of <logical equalities> and order comparisons of the named constants, all decided by their stated strict increasing order. Hence $\mathrm{DLO}_n$ is complete.
An alternative <formal proof> of this finite case uses the stated general theorem that a <first-order theory> in a <countable> <first-order language> with only infinite <first-order models> and a unique <countable> <first-order model> is complete: if <first-order models> disagreed on a <first-order sentence>, downward Löwenheim-Skolem would give disagreeing <countable> <first-order models>. Part (iii) supplies the uniqueness.
Every <first-order sentence> of $L_\omega$ mentions only finitely many <constant symbols>, hence belongs to some $L_n$. Its truth is decided by $\mathrm{DLO}_n$, and every <first-order model> of $\mathrm{DLO}_\omega$ has a reduct satisfying that finite <first-order theory>. Thus the larger <first-order theory> also decides every <first-order sentence>. The rational-order examples in (ii) establish <first-order consistency>. Consequently
$$
\boxed{\mathrm{DLO}_n\text{ and }\mathrm{DLO}_\omega\text{ are complete}.}
$$
There is no <contradiction> with (ii): completeness is agreement on individual finite <first-order sentences>, while categoricity requires a global <isomorphism> respecting all constants simultaneously.
Back to article page
