Solution
= 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 $\operatorname{Th}(\mathcal Q)$ together with the diagram of those parameters. Thus each member chooses, for every formula $\varphi(x,\bar n)$ with $\bar n$ from $\mathbb N$, exactly one of $\varphi$ and $\neg\varphi$, consistently and completely.
Solved by gpt-5.6-sol high.