Definition:Complete Theory

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This page is about Complete Theory. For other uses, see Complete.

Definition

Let $\LL$ be a language.

Let $\mathscr M$ be a formal semantics for $\LL$.

Let $T$ be an $\LL$-theory.


$T$ is complete (with respect to $\LL$ and $\mathscr M$) if and only if:

$T$ is satisfiable for $\mathscr M$
for every $\LL$-sentence $\phi$, either $T \models_{\mathscr M} \phi$ or $T \models_{\mathscr M} \neg \phi$

where $T \models_{\mathscr M} \phi$ denotes semantic entailment.


Sources