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sav08:compactness_for_first-order_logic [2008/03/20 17:56] vkuncak |
sav08:compactness_for_first-order_logic [2008/03/20 17:58] vkuncak |
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| ====== Compactness for First-Order Logic ====== | ====== Compactness for First-Order Logic ====== | ||
| - | *Theorem (Compactness for First-Order Logic):** If every finite subset of a set $S_0$ of first-order formulas has a model, then $S_0$ has a model. | + | **Theorem (Compactness for First-Order Logic):** If every finite subset of a set $S_0$ of first-order formulas has a model, then $S_0$ has a model. |
| **Proof:** | **Proof:** | ||
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| Then $expandProp(clauses(S_0))$ has no model. | Then $expandProp(clauses(S_0))$ has no model. | ||
| - | Some finite subset $S_1 \subseteq expandProp(clauses(S_0))$ of it has no model. | + | Then by [[Compactness Theorem]] for propositional logic, some finite subset $S_1 \subseteq expandProp(clauses(S_0))$ of it has no model. |
| There is then finite subset of clauses $S_2 \subseteq clauses(S_0)$ that generate $S_1$, i.e. such that $S_1 \subseteq expandProp(S_2)$. Therefore, $S_2$ has no model. | There is then finite subset of clauses $S_2 \subseteq clauses(S_0)$ that generate $S_1$, i.e. such that $S_1 \subseteq expandProp(S_2)$. Therefore, $S_2$ has no model. | ||