5.5 Limitations
First order logic has strong structural properties such as compactness and the Lowenheim Skolem theorems, and these properties impose intrinsic limitations on what can be expressed or uniquely characterized within the language. Definition 5.27 (Expressibility) A class of structures $\mathcal{K}$ is said to be first order definable if there exists a set of first order sentences $\Gamma$ such that: $$ \mathcal{M} \in \mathcal{K} \quad \text{if and only if} \quad \mathcal{M}...