Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2007 / B / 8 / iii

Adjoin to $T$ the negations of all closed-term instances of $\theta$. If the resulting set had a first-order model $\mathcal{A}$, its ground-term substructure $\mathcal{B}$ from (ii) would still first-order model $T$. The assumed existential consequence therefore supplies a tuple $\vec{b}$ in $\mathcal{B}$ satisfying $\theta$. Each $b_j$ is the value of a closed term $t_j$.

Quantifier-free agreement between a substructure and its ambient first-order structure givescontradicting the adjoined negation of this very ground instance. Thusis unsatisfiable. Compactness theorem gives a finite unsatisfiable subset. Its finitely many ground negations imply that, for some finite list of first-order term tuples,If the finite subset already shows $T$ itself inconsistent and supplies no ground negations, choose any one closed-term tuple using the available constant; then any first-order sentence is a consequence of $T$, so $m=1$ suffices. Otherwise the selected list is nonempty. This also explains why the conclusion contains a finite disjunction rather than necessarily one universal choice of a witness first-order term.