Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2007 / B / 6 / i / Solution

A first-order theory is a set of first-order sentences in a fixed first-order language, often identified with its deductive closure. It is $\kappa$-categorical if it has a first-order model of cardinality $\kappa$ and all first-order models of that cardinality are isomorphic as first-order structures in that same first-order language. In particular, an isomorphism must preserve the named constants as well as the order.

A first-order theory is complete if it is first-order consistent and decides every first-order sentence: for each $\sigma$ it entails either $\sigma$ or $\neg \sigma$. Equivalently all its first-order models satisfy exactly the same first-order sentences. First-order consistency is included to exclude the vacuous first-order theory with no first-order models. The distinctions matter here: naming infinitely many ordered points changes countable categoricity without destroying completeness.