Completeness Theorem

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Disambiguation

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Completeness Theorem may refer to:

Completeness Theorem for Semantic Tableaux

Let $\mathbf A$ be a WFF of propositional logic.

Let $\mathbf A$ be unsatisfiable for boolean interpretations.

Then every completed tableau for $\mathbf A$ is closed.

Completeness Theorem for Propositional Tableaux and Boolean Interpretations

Tableau proofs (in terms of propositional tableaux) are a complete proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:

$\models_{\mathrm{BI} } \mathbf A$ implies $\vdash_{\mathrm{PT} } \mathbf A$

Extended Completeness Theorem for Propositional Tableaux and Boolean Interpretations

Tableau proofs (in terms of propositional tableaux) are a strongly complete proof system for boolean interpretations.

More precisely, for every countable collection $\mathbf H$ of WFFs of propositional logic and every WFF $\mathbf A$:

$\mathbf H \models_{\mathrm{BI} } \mathbf A$ implies $\mathbf H \vdash_{\mathrm{PT} } \mathbf A$

Completeness Theorem for Hilbert Proof System Instance 2 and Boolean Interpretations

Instance 2 of the Hilbert proof systems is a complete proof system for boolean interpretations.

That is, for every WFF $\mathbf A$:

$\models_{\mathrm{BI}} \mathbf A$ implies $\vdash_{\mathscr H_2} \mathbf A$

Gödel's Completeness Theorem

Gödel's Completeness Theorem

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): completeness theorem
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): completeness theorem