= Solution
The <Godel completeness theorem> says that a first-order sentence follows semantically from a theory exactly when it is formally derivable from it. The <compactness theorem> says that a theory has a model exactly when every finite subset has a model.
Fix $i$. For each $j\ne i$, the theory $T_i\cup T_j$ has no model because the model classes form a partition. By compactness, some finite $\Delta_{ij}\subseteq T_i$ is already inconsistent with $T_j$. Let
$$
\Delta_i=\bigcup_{j\ne i}\Delta_{ij},
$$
a finite subset of $T_i$. Every model of $T_i$ models $\Delta_i$. Conversely, a model of $\Delta_i$ belongs to exactly one class $T_j$; if $j\ne i$, it would model both $T_j$ and $\Delta_{ij}$, a contradiction. Hence it belongs to $T_i$, and $\Delta_i$ finitely axiomatizes $T_i$.
Back to article page
