4.3 Elementary Equivalence
Two structures may differ as sets and operations, yet satisfy exactly the same first order sentences, and this situation is captured by the notion of elementary equivalence, which compares structures at the level of logical expressibility rather than at the level of concrete representation. Definition 4.23 (Theory of a Structure) Let $\mathcal A$ be a structure for a fixed signature. The theory of $\mathcal A$, denoted: $$ \mathrm{Th}(\mathcal A) $$...