Past exam of the logic and set theory course of the University of Oxford / 2010 / A / 3 / d / iv / 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.