Definition:Theory of Structure

Definition

Let $\LL$ be a logical language.

Let $\MM$ be an $\LL$-structure.

The $\LL$-theory of $\MM$ is the $\LL$-theory consisting of those $\LL$-sentences $\phi$ such that:

$\MM \models \phi$

where $\models$ denotes that $\MM$ is a model for $\phi$.

This theory can be denoted $\map {\operatorname{Th} } \MM$ when the language $\LL$ is understood.

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II}.8.21$