Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / B / 5 / a / iv

A prime model admits an elementary embedding into every model of its complete theory. In the countable setting a prime model is atomic. If a tuple in the prime model had a nonisolated complete type, the omitting types theorem would give a model omitting that type; an elementary embedding of the prime model into it would realize the omitted type, a contradiction.

Combining the four parts, all positive implications are exactly those generated byThe structure $(\mathbb{N},<,0)$ refutes (ii)$\Rightarrow$(i). The structure $(\mathbb{Q},<,0)$ is a countable atomic and hence prime model of its complete theory, while failing both (i) and (ii); it refutes every remaining reverse arrow, including (iv)$\Rightarrow$(i) and (iv)$\Rightarrow$(ii). The counterexamples therefore cover all ordered pairs of distinct properties not forced by the boxed chain.