Theory of Structure is Complete

Theorem

Let $\AA$ be a structure for a signature for predicate logic $\LL$.

Let $\map {\operatorname{Th}} \AA$ be the theory of $\AA$.

Then $\map {\operatorname{Th}} \AA$ is complete.

Proof

By definition of $\map {\operatorname{Th}} \AA$ be the theory of $\AA$:

$\AA \models \map {\operatorname{Th}} \AA$

so that $\map {\operatorname{Th}} \AA$ is consistent.

Now let $\mathbf A$ be an $\LL$-sentence.

Let $\map {\operatorname{val}_\AA} {\mathbf A}$ be the value of $\mathbf A$ in $\AA$.

Then either $\map {\operatorname{val}_\AA} {\mathbf A} = T$ or $\map {\operatorname{val}_\AA} {\mathbf A} = F$.

By definition of value, the latter implies:

$\map {\operatorname{val}_\AA} {\neg \mathbf A} = T$

Therefore, by definition of the models relation $\models$:

$\AA \models \mathbf A$ or $\AA \models \mathbf A$

That is, by definition of the theory $\map {\operatorname{Th}} \AA$:

$\mathbf A \in \map {\operatorname{Th}} \AA$ or $\neg \mathbf A \in \map {\operatorname{Th}} \AA$

By Element is Semantic Consequence of Set, it follows that:

$\map {\operatorname{Th}} \AA \models \mathbf A$ or $\map {\operatorname{Th}} \AA \models \neg \mathbf A$

Since $\mathbf A$ was arbitrary, it follows that $\map {\operatorname{Th}} \AA$ is complete.

$\blacksquare$

Sources

• 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Lemma $\text{II}.8.22$