Definition:Complete Theory
Jump to navigation
Jump to search
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
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): complete: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): complete: 1.
- 2009: Kenneth Kunen: The Foundations of Mathematics ... (previous) ... (next): $\text{II}.8$ Further Semantic Notions: Definition $\text{II}.8.19$