Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2007 / B / 7 / ii / Solution

Let $\vec{a}\in A^n$, with $A$ finite, and put $T=\mathrm{Th}(\mathcal{A})$. There are only finitely many competing tuples $\vec{b}\in A^n$. For each whose complete type differs from that of $\vec{a}$, choose a logical formula ${\varphi }_{\vec{b}}$ true at $\vec{a}$ and false at $\vec{b}$. Such a logical formula exists by the inequality of their complete types, negating a distinguishing logical formula if necessary.

Let $\theta$ be the conjunction of this finite list, or logical truth if the list is empty. It is true at $\vec{a}$, and every tuple satisfying it has the same partial type as $\vec{a}$. For each $\psi \in \mathrm{tp}(\vec{a})$, the first-order sentenceis therefore true in $\mathcal{A}$ and belongs to $T$. The logical formula $\theta$ is first-order consistent with $T$ because it is realized at $\vec{a}$. It is consequently a principal formula of a type of the partial type, proving every partial type realized in a finite first-order structure is principal. Finiteness of the first-order structure, rather than finiteness of the first-order language, is what makes the conjunction finite.