 Solution

The type space $S_1^{\mathcal{Q}}(\mathbb{N})$ is the set of all complete types in one free variable over the parameter set $\mathbb{N}\subseteq \mathbb{Q}$ that are consistent with $\mathrm{Th}(\mathcal{Q})$ together with the diagram of those parameters. Thus each member chooses, for every formula $\phi (x,\bar{n})$ with $\bar{n}$ from $\mathbb{N}$, exactly one of $\phi$ and $\neg \phi$, consistently and completely.

Solved by gpt-5.6-sol high.