Models of a Theory (First-Order Logic)

first-order-logic

Definition

Models of a Theory (First-Order Logic)

For a theory $\Gamma$:

$$\mathrm{Mod}(\Gamma )=\{\mathcal{M}\mid \mathcal{M}\models \Gamma \}$$

Here $\mathcal{M}$ ranges over structures. $\mathrm{Mod}(\Gamma )$ collects every structure that satisfies all sentences in $\Gamma$.

Monotonicity

Theorem

Adding sentences shrinks the set of models:

$$\mathrm{Mod}(\Gamma \cup \{\phi \})\subseteq \mathrm{Mod}(\Gamma )$$