Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 23 / 1 / b / Solution

Name every element $a\in A$ by a new constant $c_a$, distinct from the symbols already in $L^{\prime }$. Let $\mathrm{Diag}(A)$ be its full atomic and negated atomic diagram of a structure. A model of this diagram contains an isomorphic copy of $A$ as an $L$-substructure. We claimis satisfiable.

Otherwise compactness theorem gives a finite diagram conjunction $\delta (c_{a_1},\dots ,c_{a_m})$ inconsistent with $T^{\prime }$. Replace these new constants by variables. Since $T^{\prime }$ says nothing about the new names, it then entails the universal sentenceThe conjunction $\delta$ uses only symbols of $L$, so this is one of the universal $L$-consequences assumed true in $A$. But $A$ satisfies $\delta (a_1,\dots ,a_m)$, a contradiction. A model of the consistent enlarged theory gives an embedding of $A$ into the $L$-reduct of a model of $T^{\prime }$. Negated atomic facts and disequalities in the diagram ensure that this is a structure embedding, not merely a homomorphism.