Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision Last revision Both sides next revision | ||
|
sav08:compactness_for_first-order_logic [2008/03/20 17:55] vkuncak |
sav08:compactness_for_first-order_logic [2008/03/20 17:57] vkuncak |
||
|---|---|---|---|
| Line 5: | Line 5: | ||
| **Proof:** | **Proof:** | ||
| - | Let $S_0$ be a set of first-order formulas. | + | Let $S_0$ be a set of first-order formulas. We show contrapositive. |
| - | Suppose $S_0$ has no model. | + | Suppose $S_0$ has no model. |
| Then $expandProp(clauses(S_0))$ has no model. | Then $expandProp(clauses(S_0))$ has no model. | ||
| Line 17: | Line 17: | ||
| These clauses are generated by a finite subset $S_3 \subseteq S_0$, i.e. $S_2 \subseteq clauses(S_3)$. | These clauses are generated by a finite subset $S_3 \subseteq S_0$, i.e. $S_2 \subseteq clauses(S_3)$. | ||
| - | Therefore $S_3$ has no model. | + | Therefore the finite subset $S_3$ of $S_0$ has no model. |
| **End of Proof.** | **End of Proof.** | ||