= Solution
The theory is consistent because the rational order is a model. If some sentence $\theta$ were not decided by the theory, both extensions by $\theta$ and by $\neg\theta$ would be consistent. By <Gödel's completeness theorem> and the countable-model consequence of <Löwenheim-Skolem theorem>, each would have a <countable> model. The two models would disagree on $\theta$, so could not be isomorphic. This contradicts part (iii).
Therefore the theory decides every sentence and is complete. Under the standard convention that a theory includes all its logical consequences, this makes it <maximal consistent>.
If one takes only the displayed finite list of axioms as a raw set, that list is not literally maximal consistent: one can add further theorems without inconsistency. The conclusion applies to its deductive closure, which is the complete theory of dense linear order without endpoints.
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