Full Theory

Last updated Nov 1, 2022

Let $\mathcal{L}$ be a Language, let $\mathcal{M}$ be an $\mathcal{L}$-Language Structure, and let $\mathcal{S}$ be the Set of $\mathcal{L}$-Sentences. The Full Theory of $\mathcal{M}$, denoted $\text{Th}(\mathcal{M})$, is defined as $$\text{Th}(\mathcal{M}) := {\phi \in \mathcal{S} : \mathcal{M} \models \phi}$$